1 線形化と方向二階微分 定義 1.1. n ≥ 1 n\geq1 n ≥ 1 0 < λ ≤ Λ 0<\lambda\leq\Lambda 0 < λ ≤ Λ U ⊆ R n U\subseteq\R^n U ⊆ R n
C ⊆ S n \mathcal C\subseteq\mathcal S^n C ⊆ S n F : C → R F:\mathcal C\to\R F : C → R C \mathcal C C λ , Λ \lambda,\Lambda λ , Λ 一様楕円型 (uniformly elliptic on a convex open set )M ∈ C M\in\mathcal C M ∈ C M + N ∈ C M+N\in\mathcal C M + N ∈ C N ≥ 0 N\geq0 N ≥ 0 M , N ∈ S n M,N\in\mathcal S^n M , N ∈ S n λ tr N ≤ F ( M + N ) − F ( M ) ≤ Λ tr N \lambda\operatorname{tr}N\leq F(M+N)-F(M)\leq\Lambda\operatorname{tr}N λ tr N ≤ F ( M + N ) − F ( M ) ≤ Λ tr N O ⊆ S n × U \mathcal O\subseteq\mathcal S^n\times U O ⊆ S n × U x ∈ U x\in U x ∈ U O x = { M ∈ S n ∣ ( M , x ) ∈ O } \mathcal O_x=\{M\in\mathcal S^n\mid(M,x)\in\mathcal O\} O x = { M ∈ S n ∣ ( M , x ) ∈ O } F : O → R F:\mathcal O\to\R F : O → R O \mathcal O O λ , Λ \lambda,\Lambda λ , Λ x ∈ U x\in U x ∈ U F ( ⋅ , x ) F(\cdot,x) F ( ⋅ , x ) O x \mathcal O_x O x λ , Λ \lambda,\Lambda λ , Λ F F F 行列変数について凹 (concave in the matrix variable )x ∈ U x\in U x ∈ U F ( ⋅ , x ) F(\cdot,x) F ( ⋅ , x ) O x \mathcal O_x O x C = S n \mathcal C=\mathcal S^n C = S n (1) の条件は§E15.42 定義 3.2 の一様楕円性と一致する。
補題 1.3. n ≥ 1 n\geq1 n ≥ 1 0 < λ ≤ Λ 0<\lambda\leq\Lambda 0 < λ ≤ Λ ξ ∈ R n \xi\in\R^n ξ ∈ R n ξ i ξ j \xi^i\xi^j ξ i ξ j ξ ⊗ ξ \xi\otimes\xi ξ ⊗ ξ M ∈ S n M\in\mathcal S^n M ∈ S n ∣ M ∣ = ( M : M ) 1 / 2 |M|=(M:M)^{1/2} ∣ M ∣ = ( M : M ) 1/2 n , λ , Λ n,\lambda,\Lambda n , λ , Λ m ≥ 1 m\geq1 m ≥ 1 ξ 1 , … , ξ m ∈ R n \xi_1,\dots,\xi_m\in\R^n ξ 1 , … , ξ m ∈ R n 0 < λ ∗ ≤ Λ ∗ 0<\lambda_*\leq\Lambda_* 0 < λ ∗ ≤ Λ ∗
λ I ≤ a ≤ Λ I \lambda I\leq a\leq\Lambda I λ I ≤ a ≤ Λ I a ∈ S n a\in\mathcal S^n a ∈ S n λ ∗ ≤ β ν ≤ Λ ∗ \lambda_*\leq\beta_\nu\leq\Lambda_* λ ∗ ≤ β ν ≤ Λ ∗ β 1 , … , β m \beta_1,\dots,\beta_m β 1 , … , β m a = ∑ ν = 1 m β ν ξ ν ⊗ ξ ν a=\sum_{\nu=1}^m\beta_\nu\,\xi_\nu\otimes\xi_\nu a = ∑ ν = 1 m β ν ξ ν ⊗ ξ ν 任意のM ∈ S n M\in\mathcal S^n M ∈ S n ∣ M ∣ ≤ ∑ ν = 1 m ∣ ξ ν ⊗ ξ ν : M ∣ |M|\leq\sum_{\nu=1}^m|\xi_\nu\otimes\xi_\nu:M| ∣ M ∣ ≤ ∑ ν = 1 m ∣ ξ ν ⊗ ξ ν : M ∣
証明. e 1 , … , e n e_1,\dots,e_n e 1 , … , e n R n \R^n R n e i e_i e i 1 ≤ i ≤ n 1\leq i\leq n 1 ≤ i ≤ n ( e i + e j ) / 2 (e_i+e_j)/\sqrt2 ( e i + e j ) / 2 ( e i − e j ) / 2 (e_i-e_j)/\sqrt2 ( e i − e j ) / 2 1 ≤ i < j ≤ n 1\leq i<j\leq n 1 ≤ i < j ≤ n n 2 n^2 n 2 η 1 , … , η n 2 \eta_1,\dots,\eta_{n^2} η 1 , … , η n 2 1 ≤ i < j ≤ n 1\leq i<j\leq n 1 ≤ i < j ≤ n
1 2 ( e i + e j ) ⊗ ( e i + e j ) + 1 2 ( e i − e j ) ⊗ ( e i − e j ) = e i ⊗ e i + e j ⊗ e j , 1 2 ( e i + e j ) ⊗ ( e i + e j ) − 1 2 ( e i − e j ) ⊗ ( e i − e j ) = e i ⊗ e j + e j ⊗ e i \tfrac12(e_i+e_j)\otimes(e_i+e_j)+\tfrac12(e_i-e_j)\otimes(e_i-e_j)=e_i\otimes e_i+e_j\otimes e_j,
\qquad
\tfrac12(e_i+e_j)\otimes(e_i+e_j)-\tfrac12(e_i-e_j)\otimes(e_i-e_j)=e_i\otimes e_j+e_j\otimes e_i 2 1 ( e i + e j ) ⊗ ( e i + e j ) + 2 1 ( e i − e j ) ⊗ ( e i − e j ) = e i ⊗ e i + e j ⊗ e j , 2 1 ( e i + e j ) ⊗ ( e i + e j ) − 2 1 ( e i − e j ) ⊗ ( e i − e j ) = e i ⊗ e j + e j ⊗ e i である。左の等式をi < j i<j i < j ∑ i e i ⊗ e i = I \sum_ie_i\otimes e_i=I ∑ i e i ⊗ e i = I ∑ k η k ⊗ η k = I + ( n − 1 ) I = n I \sum_k\eta_k\otimes\eta_k=I+(n-1)I=nI ∑ k η k ⊗ η k = I + ( n − 1 ) I = n I e i ⊗ e i e_i\otimes e_i e i ⊗ e i 1 ≤ i ≤ n 1\leq i\leq n 1 ≤ i ≤ n e i ⊗ e j + e j ⊗ e i e_i\otimes e_j+e_j\otimes e_i e i ⊗ e j + e j ⊗ e i i < j i<j i < j S n \mathcal S^n S n η k ⊗ η k \eta_k\otimes\eta_k η k ⊗ η k T : R n 2 → S n T:\R^{n^2}\to\mathcal S^n T : R n 2 → S n T ( c ) = ∑ k c k η k ⊗ η k T(c)=\sum_kc_k\,\eta_k\otimes\eta_k T ( c ) = ∑ k c k η k ⊗ η k T ∘ S = i d T\circ S=\mathrm{id} T ∘ S = id S : S n → R n 2 S:\mathcal S^n\to\R^{n^2} S : S n → R n 2 max k ∣ ( S b ) k ∣ ≤ s n ∣ b ∣ \max_k|(Sb)_k|\leq s_n|b| max k ∣ ( S b ) k ∣ ≤ s n ∣ b ∣ b ∈ S n b\in\mathcal S^n b ∈ S n s n > 0 s_n>0 s n > 0 δ n = 1 / ( 2 s n ) \delta_n=1/(2s_n) δ n = 1/ ( 2 s n ) ∣ b − n I ∣ < δ n |b-nI|<\delta_n ∣ b − n I ∣ < δ n c = ( 1 , … , 1 ) + S ( b − n I ) c=(1,\dots,1)+S(b-nI) c = ( 1 , … , 1 ) + S ( b − n I ) T ( c ) = b T(c)=b T ( c ) = b c k ∈ ( 1 / 2 , 3 / 2 ) c_k\in(1/2,3/2) c k ∈ ( 1/2 , 3/2 )
T = { a ∈ S n ∣ ( λ / 2 ) I ≤ a ≤ Λ I } \mathcal T=\{a\in\mathcal S^n\mid(\lambda/2)I\leq a\leq\Lambda I\} T = { a ∈ S n ∣ ( λ /2 ) I ≤ a ≤ Λ I } T \mathcal T T a ∈ T a\in\mathcal T a ∈ T ∣ a ∣ 2 |a|^2 ∣ a ∣ 2 n Λ 2 n\Lambda^2 n Λ 2 T \mathcal T T r = λ δ n / ( 4 n ) r=\lambda\delta_n/(4n) r = λ δ n / ( 4 n ) b ∈ T b\in\mathcal T b ∈ T q 1 ( b ) , … , q n ( b ) q_1(b),\dots,q_n(b) q 1 ( b ) , … , q n ( b ) σ 1 ( b ) , … , σ n ( b ) ∈ [ λ / 2 , Λ ] \sigma_1(b),\dots,\sigma_n(b)\in[\lambda/2,\Lambda] σ 1 ( b ) , … , σ n ( b ) ∈ [ λ /2 , Λ ] a ∈ S n a\in\mathcal S^n a ∈ S n ∣ a − b ∣ < r |a-b|<r ∣ a − b ∣ < r ∣ ( 4 n / λ ) ( a − b ) ∣ < δ n |(4n/\lambda)(a-b)|<\delta_n ∣ ( 4 n / λ ) ( a − b ) ∣ < δ n c k ∈ ( 1 / 2 , 3 / 2 ) c_k\in(1/2,3/2) c k ∈ ( 1/2 , 3/2 )
a = ∑ k = 1 n ( σ k ( b ) − λ 4 ) q k ( b ) ⊗ q k ( b ) + λ 4 n ( n I + 4 n λ ( a − b ) ) = ∑ k = 1 n ( σ k ( b ) − λ 4 ) q k ( b ) ⊗ q k ( b ) + ∑ k = 1 n 2 λ c k 4 n η k ⊗ η k a=\sum_{k=1}^n\Bigl(\sigma_k(b)-\frac\lambda4\Bigr)q_k(b)\otimes q_k(b)+\frac{\lambda}{4n}\Bigl(nI+\frac{4n}{\lambda}(a-b)\Bigr)
=\sum_{k=1}^n\Bigl(\sigma_k(b)-\frac\lambda4\Bigr)q_k(b)\otimes q_k(b)+\sum_{k=1}^{n^2}\frac{\lambda c_k}{4n}\eta_k\otimes\eta_k a = k = 1 ∑ n ( σ k ( b ) − 4 λ ) q k ( b ) ⊗ q k ( b ) + 4 n λ ( n I + λ 4 n ( a − b ) ) = k = 1 ∑ n ( σ k ( b ) − 4 λ ) q k ( b ) ⊗ q k ( b ) + k = 1 ∑ n 2 4 n λ c k η k ⊗ η k と表される。ここでσ k ( b ) − λ / 4 ∈ [ λ / 4 , Λ ] \sigma_k(b)-\lambda/4\in[\lambda/4,\Lambda] σ k ( b ) − λ /4 ∈ [ λ /4 , Λ ] λ c k / ( 4 n ) ∈ ( 0 , Λ ] \lambda c_k/(4n)\in(0,\Lambda] λ c k / ( 4 n ) ∈ ( 0 , Λ ] T \mathcal T T T ⊆ ⋃ l = 1 J { a ∣ ∣ a − b l ∣ < r } \mathcal T\subseteq\bigcup_{l=1}^J\{a\mid|a-b_l|<r\} T ⊆ ⋃ l = 1 J { a ∣ ∣ a − b l ∣ < r } b 1 , … , b J ∈ T b_1,\dots,b_J\in\mathcal T b 1 , … , b J ∈ T n , λ , Λ n,\lambda,\Lambda n , λ , Λ η 1 , … , η n 2 \eta_1,\dots,\eta_{n^2} η 1 , … , η n 2 q k ( b l ) q_k(b_l) q k ( b l ) 1 ≤ l ≤ J 1\leq l\leq J 1 ≤ l ≤ J 1 ≤ k ≤ n 1\leq k\leq n 1 ≤ k ≤ n ξ 1 , … , ξ m \xi_1,\dots,\xi_m ξ 1 , … , ξ m m = n 2 + n J m=n^2+nJ m = n 2 + n J a ′ ∈ T a'\in\mathcal T a ′ ∈ T 0 ≤ γ ν ≤ 2 Λ 0\leq\gamma_\nu\leq2\Lambda 0 ≤ γ ν ≤ 2Λ γ ν \gamma_\nu γ ν a ′ = ∑ ν γ ν ξ ν ⊗ ξ ν a'=\sum_\nu\gamma_\nu\,\xi_\nu\otimes\xi_\nu a ′ = ∑ ν γ ν ξ ν ⊗ ξ ν
主張 (1) を示す。λ ∗ = λ / ( 2 m ) \lambda_*=\lambda/(2m) λ ∗ = λ / ( 2 m ) Λ ∗ = λ / ( 2 m ) + 2 Λ \Lambda_*=\lambda/(2m)+2\Lambda Λ ∗ = λ / ( 2 m ) + 2Λ λ I ≤ a ≤ Λ I \lambda I\leq a\leq\Lambda I λ I ≤ a ≤ Λ I a ′ = a − λ ∗ ∑ ν ξ ν ⊗ ξ ν a'=a-\lambda_*\sum_\nu\xi_\nu\otimes\xi_\nu a ′ = a − λ ∗ ∑ ν ξ ν ⊗ ξ ν ν \nu ν 0 ≤ ξ ν ⊗ ξ ν ≤ I 0\leq\xi_\nu\otimes\xi_\nu\leq I 0 ≤ ξ ν ⊗ ξ ν ≤ I 0 ≤ λ ∗ ∑ ν ξ ν ⊗ ξ ν ≤ ( λ / 2 ) I 0\leq\lambda_*\sum_\nu\xi_\nu\otimes\xi_\nu\leq(\lambda/2)I 0 ≤ λ ∗ ∑ ν ξ ν ⊗ ξ ν ≤ ( λ /2 ) I a ′ ∈ T a'\in\mathcal T a ′ ∈ T a ′ = ∑ ν γ ν ξ ν ⊗ ξ ν a'=\sum_\nu\gamma_\nu\,\xi_\nu\otimes\xi_\nu a ′ = ∑ ν γ ν ξ ν ⊗ ξ ν a = ∑ ν ( λ ∗ + γ ν ) ξ ν ⊗ ξ ν a=\sum_\nu(\lambda_*+\gamma_\nu)\xi_\nu\otimes\xi_\nu a = ∑ ν ( λ ∗ + γ ν ) ξ ν ⊗ ξ ν λ ∗ ≤ λ ∗ + γ ν ≤ Λ ∗ \lambda_*\leq\lambda_*+\gamma_\nu\leq\Lambda_* λ ∗ ≤ λ ∗ + γ ν ≤ Λ ∗
主張 (2) を示す。M ∈ S n M\in\mathcal S^n M ∈ S n M i i = e i ⊗ e i : M M_{ii}=e_i\otimes e_i:M M ii = e i ⊗ e i : M i < j i<j i < j 2 M i j = 1 2 ( e i + e j ) ⊗ ( e i + e j ) : M − 1 2 ( e i − e j ) ⊗ ( e i − e j ) : M 2M_{ij}=\frac12(e_i+e_j)\otimes(e_i+e_j):M-\frac12(e_i-e_j)\otimes(e_i-e_j):M 2 M ij = 2 1 ( e i + e j ) ⊗ ( e i + e j ) : M − 2 1 ( e i − e j ) ⊗ ( e i − e j ) : M
∣ M ∣ ≤ ∑ i , j ∣ M i j ∣ = ∑ i ∣ M i i ∣ + 2 ∑ i < j ∣ M i j ∣ ≤ ∑ k = 1 n 2 ∣ η k ⊗ η k : M ∣ ≤ ∑ ν = 1 m ∣ ξ ν ⊗ ξ ν : M ∣ |M|\leq\sum_{i,j}|M_{ij}|=\sum_i|M_{ii}|+2\sum_{i<j}|M_{ij}|\leq\sum_{k=1}^{n^2}|\eta_k\otimes\eta_k:M|\leq\sum_{\nu=1}^m|\xi_\nu\otimes\xi_\nu:M| ∣ M ∣ ≤ i , j ∑ ∣ M ij ∣ = i ∑ ∣ M ii ∣ + 2 i < j ∑ ∣ M ij ∣ ≤ k = 1 ∑ n 2 ∣ η k ⊗ η k : M ∣ ≤ ν = 1 ∑ m ∣ ξ ν ⊗ ξ ν : M ∣ が成り立つ。▨
命題 1.4. n ≥ 1 n\geq1 n ≥ 1 U ⊆ R n U\subseteq\R^n U ⊆ R n O ⊆ S n × U \mathcal O\subseteq\mathcal S^n\times U O ⊆ S n × U F ∈ C 2 ( O ) F\in C^2(\mathcal O) F ∈ C 2 ( O ) ( M , x ) ∈ O (M,x)\in\mathcal O ( M , x ) ∈ O ξ , η ∈ R n \xi,\eta\in\R^n ξ , η ∈ R n N , N ′ ∈ S n N,N'\in\mathcal S^n N , N ′ ∈ S n
∂ ξ F ( M , x ) = d d t F ( M , x + t ξ ) ∣ t = 0 , D M 2 F ( M , x ) [ N , N ′ ] = ∂ 2 ∂ s ∂ t F ( M + s N + t N ′ , x ) ∣ s = t = 0 \partial_\xi F(M,x)=\frac{d}{dt}F(M,x+t\xi)\Big|_{t=0},\qquad
D_M^2F(M,x)[N,N']=\frac{\partial^2}{\partial s\,\partial t}F(M+sN+tN',x)\Big|_{s=t=0} ∂ ξ F ( M , x ) = d t d F ( M , x + t ξ ) t = 0 , D M 2 F ( M , x ) [ N , N ′ ] = ∂ s ∂ t ∂ 2 F ( M + s N + t N ′ , x ) s = t = 0 とおき、a ( M , x ) ∈ S n a(M,x)\in\mathcal S^n a ( M , x ) ∈ S n B ξ ( M , x ) ∈ S n B_\xi(M,x)\in\mathcal S^n B ξ ( M , x ) ∈ S n N ∈ S n N\in\mathcal S^n N ∈ S n d d t F ( M + t N , x ) ∣ t = 0 = a ( M , x ) : N \frac{d}{dt}F(M+tN,x)|_{t=0}=a(M,x):N d t d F ( M + tN , x ) ∣ t = 0 = a ( M , x ) : N d d t ∂ ξ F ( M + t N , x ) ∣ t = 0 = B ξ ( M , x ) : N \frac{d}{dt}\partial_\xi F(M+tN,x)|_{t=0}=B_\xi(M,x):N d t d ∂ ξ F ( M + tN , x ) ∣ t = 0 = B ξ ( M , x ) : N Ω ⊆ U \Omega\subseteq U Ω ⊆ U u ∈ C 4 ( Ω ) u\in C^4(\Omega) u ∈ C 4 ( Ω ) x ∈ Ω x\in\Omega x ∈ Ω ( D 2 u ( x ) , x ) ∈ O (D^2u(x),x)\in\mathcal O ( D 2 u ( x ) , x ) ∈ O F ( D 2 u ( x ) , x ) = 0 F(D^2u(x),x)=0 F ( D 2 u ( x ) , x ) = 0 L = a ( D 2 u ( x ) , x ) : D 2 L=a(D^2u(x),x):D^2 L = a ( D 2 u ( x ) , x ) : D 2 u ξ = ξ i D i u u_\xi=\xi^iD_iu u ξ = ξ i D i u u ξ η = ξ i η j D i j u u_{\xi\eta}=\xi^i\eta^jD_{ij}u u ξ η = ξ i η j D ij u D 2 u ξ = D 2 ( u ξ ) D^2u_\xi=D^2(u_\xi) D 2 u ξ = D 2 ( u ξ )
Ω \Omega Ω F F F ( D 2 u ( x ) , x ) (D^2u(x),x) ( D 2 u ( x ) , x ) L u ξ η = − D M 2 F [ D 2 u ξ , D 2 u η ] − B ξ : D 2 u η − B η : D 2 u ξ − ∂ ξ ∂ η F Lu_{\xi\eta}=-D_M^2F[D^2u_\xi,D^2u_\eta]-B_\xi:D^2u_\eta-B_\eta:D^2u_\xi-\partial_\xi\partial_\eta F L u ξ η = − D M 2 F [ D 2 u ξ , D 2 u η ] − B ξ : D 2 u η − B η : D 2 u ξ − ∂ ξ ∂ η F O \mathcal O O O x \mathcal O_x O x F F F Ω \Omega Ω L u ξ ξ ≥ − 2 B ξ : D 2 u ξ − ∂ ξ 2 F Lu_{\xi\xi}\geq-2B_\xi:D^2u_\xi-\partial_\xi^2F L u ξ ξ ≥ − 2 B ξ : D 2 u ξ − ∂ ξ 2 F C ⊆ S n \mathcal C\subseteq\mathcal S^n C ⊆ S n F 0 ∈ C 2 ( C ) F_0\in C^2(\mathcal C) F 0 ∈ C 2 ( C ) f ∈ C 2 ( U ) f\in C^2(U) f ∈ C 2 ( U ) O = C × U \mathcal O=\mathcal C\times U O = C × U F ( M , x ) = F 0 ( M ) − f ( x ) F(M,x)=F_0(M)-f(x) F ( M , x ) = F 0 ( M ) − f ( x ) Ω \Omega Ω L u ξ ξ = f ξ ξ − D 2 F 0 ( D 2 u ) [ D 2 u ξ , D 2 u ξ ] ≥ f ξ ξ Lu_{\xi\xi}=f_{\xi\xi}-D^2F_0(D^2u)[D^2u_\xi,D^2u_\xi]\geq f_{\xi\xi} L u ξ ξ = f ξ ξ − D 2 F 0 ( D 2 u ) [ D 2 u ξ , D 2 u ξ ] ≥ f ξ ξ 証明. a a a B ξ B_\xi B ξ ( A , N ) ↦ A : N (A,N)\mapsto A:N ( A , N ) ↦ A : N S n \mathcal S^n S n F ∈ C 2 ( O ) F\in C^2(\mathcal O) F ∈ C 2 ( O ) d d s a ( M , x + s ξ ) : N ∣ s = 0 = B ξ ( M , x ) : N \frac{d}{ds}a(M,x+s\xi):N|_{s=0}=B_\xi(M,x):N d s d a ( M , x + s ξ ) : N ∣ s = 0 = B ξ ( M , x ) : N
主張 (1) を示す。Z ( x ) = ( D 2 u ( x ) , x ) Z(x)=(D^2u(x),x) Z ( x ) = ( D 2 u ( x ) , x ) u ∈ C 4 ( Ω ) u\in C^4(\Omega) u ∈ C 4 ( Ω ) Z Z Z C 2 C^2 C 2 F ( Z ( x ) ) = 0 F(Z(x))=0 F ( Z ( x )) = 0 η \eta η
a ( Z ) : D 2 u η + ∂ η F ( Z ) = 0 a(Z):D^2u_\eta+\partial_\eta F(Z)=0 a ( Z ) : D 2 u η + ∂ η F ( Z ) = 0 を得る。左辺の第一項のξ \xi ξ D M 2 F ( Z ) [ D 2 u ξ , D 2 u η ] + B ξ ( Z ) : D 2 u η + a ( Z ) : D 2 u ξ η D_M^2F(Z)[D^2u_\xi,D^2u_\eta]+B_\xi(Z):D^2u_\eta+a(Z):D^2u_{\xi\eta} D M 2 F ( Z ) [ D 2 u ξ , D 2 u η ] + B ξ ( Z ) : D 2 u η + a ( Z ) : D 2 u ξ η ξ \xi ξ B η ( Z ) : D 2 u ξ + ∂ ξ ∂ η F ( Z ) B_\eta(Z):D^2u_\xi+\partial_\xi\partial_\eta F(Z) B η ( Z ) : D 2 u ξ + ∂ ξ ∂ η F ( Z ) 0 0 0 a ( Z ) : D 2 u ξ η = L u ξ η a(Z):D^2u_{\xi\eta}=Lu_{\xi\eta} a ( Z ) : D 2 u ξ η = L u ξ η 主張 (1) の等式を得る。
主張 (2) を示す。x ∈ Ω x\in\Omega x ∈ Ω M = D 2 u ( x ) M=D^2u(x) M = D 2 u ( x ) N = D 2 u ξ ( x ) N=D^2u_\xi(x) N = D 2 u ξ ( x ) O x \mathcal O_x O x M M M t ↦ F ( M + t N , x ) t\mapsto F(M+tN,x) t ↦ F ( M + tN , x ) 0 0 0 C 2 C^2 C 2 D M 2 F ( M , x ) [ N , N ] D_M^2F(M,x)[N,N] D M 2 F ( M , x ) [ N , N ] 0 0 0 主張 (1) でη = ξ \eta=\xi η = ξ
主張 (3) を示す。F ( M , x ) = F 0 ( M ) − f ( x ) F(M,x)=F_0(M)-f(x) F ( M , x ) = F 0 ( M ) − f ( x ) ∂ ξ F ( M , x ) = − f ξ ( x ) \partial_\xi F(M,x)=-f_\xi(x) ∂ ξ F ( M , x ) = − f ξ ( x ) M M M B ξ = 0 B_\xi=0 B ξ = 0 ∂ ξ 2 F = − f ξ ξ \partial_\xi^2F=-f_{\xi\xi} ∂ ξ 2 F = − f ξ ξ D M 2 F = D 2 F 0 D_M^2F=D^2F_0 D M 2 F = D 2 F 0 主張 (1) でη = ξ \eta=\xi η = ξ F 0 F_0 F 0 C \mathcal C C C 2 C^2 C 2 D 2 F 0 ( D 2 u ) [ D 2 u ξ , D 2 u ξ ] ≤ 0 D^2F_0(D^2u)[D^2u_\xi,D^2u_\xi]\leq0 D 2 F 0 ( D 2 u ) [ D 2 u ξ , D 2 u ξ ] ≤ 0 ▨
2 振動の減衰と反復 補題 2.1. n ≥ 1 n\geq1 n ≥ 1 m ≥ 1 m\geq1 m ≥ 1 0 < λ ≤ Λ 0<\lambda\leq\Lambda 0 < λ ≤ Λ 0 < λ ∗ ≤ Λ ∗ 0<\lambda_*\leq\Lambda_* 0 < λ ∗ ≤ Λ ∗ n , λ , Λ , m , λ ∗ , Λ ∗ n,\lambda,\Lambda,m,\lambda_*,\Lambda_* n , λ , Λ , m , λ ∗ , Λ ∗ C ∗ ≥ 1 C_*\geq1 C ∗ ≥ 1
x 0 ∈ R n x_0\in\R^n x 0 ∈ R n r > 0 r>0 r > 0 B ρ = B ρ ( x 0 ) B_\rho=B_\rho(x_0) B ρ = B ρ ( x 0 ) a = ( a i j ) a=(a^{ij}) a = ( a ij ) B 2 r B_{2r} B 2 r λ I ≤ a ≤ Λ I \lambda I\leq a\leq\Lambda I λ I ≤ a ≤ Λ I L = a i j D i j L=a^{ij}D_{ij} L = a ij D ij c 0 ≥ 0 c_0\geq0 c 0 ≥ 0 E ≥ 0 E\geq0 E ≥ 0 w 1 , … , w m ∈ C 2 ( B 2 r ) w_1,\dots,w_m\in C^2(B_{2r}) w 1 , … , w m ∈ C 2 ( B 2 r ) ν \nu ν B 2 r B_{2r} B 2 r L w ν ≥ − c 0 Lw_\nu\geq-c_0 L w ν ≥ − c 0 x ∈ B r x\in B_r x ∈ B r y ∈ B 2 r y\in B_{2r} y ∈ B 2 r λ ∗ ≤ β ν ≤ Λ ∗ \lambda_*\leq\beta_\nu\leq\Lambda_* λ ∗ ≤ β ν ≤ Λ ∗ β 1 , … , β m \beta_1,\dots,\beta_m β 1 , … , β m
∑ ν = 1 m β ν ( w ν ( y ) − w ν ( x ) ) ≥ − E \sum_{\nu=1}^m\beta_\nu\bigl(w_\nu(y)-w_\nu(x)\bigr)\geq-E ν = 1 ∑ m β ν ( w ν ( y ) − w ν ( x ) ) ≥ − E が成り立つとする。0 < s ≤ 2 r 0<s\leq2r 0 < s ≤ 2 r ω ( s ) = ∑ ν = 1 m ( sup B s w ν − inf B s w ν ) \omega(s)=\sum_{\nu=1}^m\bigl(\sup_{B_s}w_\nu-\inf_{B_s}w_\nu\bigr) ω ( s ) = ∑ ν = 1 m ( sup B s w ν − inf B s w ν )
ω ( 2 r ) ≤ C ∗ ( ω ( 2 r ) − ω ( r ) ) + C ∗ ( E + r 2 c 0 ) \omega(2r)\leq C_*\bigl(\omega(2r)-\omega(r)\bigr)+C_*\bigl(E+r^2c_0\bigr) ω ( 2 r ) ≤ C ∗ ( ω ( 2 r ) − ω ( r ) ) + C ∗ ( E + r 2 c 0 ) である。
証明. §E15.42 定理 2.1 のn , λ , Λ n,\lambda,\Lambda n , λ , Λ p 0 p_0 p 0 β = γ = 0 \beta=\gamma=0 β = γ = 0 C 1 C_1 C 1 p = min { p 0 , 1 } p=\min\{p_0,1\} p = min { p 0 , 1 } ν \nu ν M ν = sup B 2 r w ν M_\nu=\sup_{B_{2r}}w_\nu M ν = sup B 2 r w ν m ν = inf B 2 r w ν m_\nu=\inf_{B_{2r}}w_\nu m ν = inf B 2 r w ν M ν ′ = sup B r w ν M'_\nu=\sup_{B_r}w_\nu M ν ′ = sup B r w ν m ν ′ = inf B r w ν m'_\nu=\inf_{B_r}w_\nu m ν ′ = inf B r w ν P ν = M ν − w ν P_\nu=M_\nu-w_\nu P ν = M ν − w ν P ν ∈ C 2 ( B 2 r ) P_\nu\in C^2(B_{2r}) P ν ∈ C 2 ( B 2 r ) B 2 r B_{2r} B 2 r P ν ≥ 0 P_\nu\geq0 P ν ≥ 0 L P ν = − L w ν ≤ c 0 LP_\nu=-Lw_\nu\leq c_0 L P ν = − L w ν ≤ c 0 c 0 c_0 c 0 L n ( B 2 r ) L^n(B_{2r}) L n ( B 2 r ) 2 ∣ B 1 ∣ 1 / n c 0 r 2|B_1|^{1/n}c_0r 2∣ B 1 ∣ 1/ n c 0 r §E15.42 定理 2.1 をR = r R=r R = r b = 0 b=0 b = 0 c = 0 c=0 c = 0 c 0 c_0 c 0 p p p
( 1 ∣ B r ∣ ∫ B r P ν p ) 1 / p ≤ ( 1 ∣ B r ∣ ∫ B r P ν p 0 ) 1 / p 0 ≤ C 1 ( M ν − M ν ′ + 2 ∣ B 1 ∣ 1 / n r 2 c 0 ) \Bigl(\frac1{|B_r|}\int_{B_r}P_\nu^{p}\Bigr)^{1/p}\leq\Bigl(\frac1{|B_r|}\int_{B_r}P_\nu^{p_0}\Bigr)^{1/p_0}\leq C_1\bigl(M_\nu-M'_\nu+2|B_1|^{1/n}r^2c_0\bigr) ( ∣ B r ∣ 1 ∫ B r P ν p ) 1/ p ≤ ( ∣ B r ∣ 1 ∫ B r P ν p 0 ) 1/ p 0 ≤ C 1 ( M ν − M ν ′ + 2∣ B 1 ∣ 1/ n r 2 c 0 ) を得る。
1 ≤ ℓ ≤ m 1\leq\ell\leq m 1 ≤ ℓ ≤ m x ∈ B r x\in B_r x ∈ B r y ∈ B 2 r y\in B_{2r} y ∈ B 2 r β ν \beta_\nu β ν ν ≠ ℓ \nu\neq\ell ν = ℓ w ν ( y ) ≤ M ν w_\nu(y)\leq M_\nu w ν ( y ) ≤ M ν
β ℓ ( w ℓ ( x ) − w ℓ ( y ) ) ≤ E + ∑ ν ≠ ℓ β ν ( w ν ( y ) − w ν ( x ) ) ≤ E + Λ ∗ ∑ ν ≠ ℓ P ν ( x ) \beta_\ell\bigl(w_\ell(x)-w_\ell(y)\bigr)\leq E+\sum_{\nu\neq\ell}\beta_\nu\bigl(w_\nu(y)-w_\nu(x)\bigr)\leq E+\Lambda_*\sum_{\nu\neq\ell}P_\nu(x) β ℓ ( w ℓ ( x ) − w ℓ ( y ) ) ≤ E + ν = ℓ ∑ β ν ( w ν ( y ) − w ν ( x ) ) ≤ E + Λ ∗ ν = ℓ ∑ P ν ( x ) である。右辺は0 0 0 β ℓ ≥ λ ∗ \beta_\ell\geq\lambda_* β ℓ ≥ λ ∗ w ℓ ( x ) − w ℓ ( y ) ≤ λ ∗ − 1 ( E + Λ ∗ ∑ ν ≠ ℓ P ν ( x ) ) w_\ell(x)-w_\ell(y)\leq\lambda_*^{-1}\bigl(E+\Lambda_*\sum_{\nu\neq\ell}P_\nu(x)\bigr) w ℓ ( x ) − w ℓ ( y ) ≤ λ ∗ − 1 ( E + Λ ∗ ∑ ν = ℓ P ν ( x ) ) y ∈ B 2 r y\in B_{2r} y ∈ B 2 r M ℓ − w ℓ ( x ) = P ℓ ( x ) M_\ell-w_\ell(x)=P_\ell(x) M ℓ − w ℓ ( x ) = P ℓ ( x )
M ℓ − m ℓ ≤ P ℓ ( x ) + Λ ∗ λ ∗ ∑ ν ≠ ℓ P ν ( x ) + E λ ∗ M_\ell-m_\ell\leq P_\ell(x)+\frac{\Lambda_*}{\lambda_*}\sum_{\nu\neq\ell}P_\nu(x)+\frac{E}{\lambda_*} M ℓ − m ℓ ≤ P ℓ ( x ) + λ ∗ Λ ∗ ν = ℓ ∑ P ν ( x ) + λ ∗ E を得る。ℓ \ell ℓ C 2 = 1 + m ( 1 + Λ ∗ ) / λ ∗ C_2=1+m(1+\Lambda_*)/\lambda_* C 2 = 1 + m ( 1 + Λ ∗ ) / λ ∗ x ∈ B r x\in B_r x ∈ B r ω ( 2 r ) ≤ C 2 ( ∑ ν P ν ( x ) + E ) \omega(2r)\leq C_2\bigl(\sum_\nu P_\nu(x)+E\bigr) ω ( 2 r ) ≤ C 2 ( ∑ ν P ν ( x ) + E ) 0 < p ≤ 1 0<p\leq1 0 < p ≤ 1 s , t ≥ 0 s,t\geq0 s , t ≥ 0 ( s + t ) p ≤ s p + t p (s+t)^p\leq s^p+t^p ( s + t ) p ≤ s p + t p p p p B r B_r B r
ω ( 2 r ) p ≤ C 2 p ( ∑ ν 1 ∣ B r ∣ ∫ B r P ν p + E p ) \omega(2r)^p\leq C_2^p\Bigl(\sum_\nu\frac1{|B_r|}\int_{B_r}P_\nu^p+E^p\Bigr) ω ( 2 r ) p ≤ C 2 p ( ν ∑ ∣ B r ∣ 1 ∫ B r P ν p + E p ) であり、右辺の括弧内のm + 1 m+1 m + 1
ω ( 2 r ) ≤ C 2 ( m + 1 ) 1 / p ( ∑ ν ( 1 ∣ B r ∣ ∫ B r P ν p ) 1 / p + E ) \omega(2r)\leq C_2(m+1)^{1/p}\Bigl(\sum_\nu\Bigl(\frac1{|B_r|}\int_{B_r}P_\nu^p\Bigr)^{1/p}+E\Bigr) ω ( 2 r ) ≤ C 2 ( m + 1 ) 1/ p ( ν ∑ ( ∣ B r ∣ 1 ∫ B r P ν p ) 1/ p + E ) を得る。m ν ′ ≥ m ν m'_\nu\geq m_\nu m ν ′ ≥ m ν M ν − M ν ′ ≤ ( M ν − m ν ) − ( M ν ′ − m ν ′ ) M_\nu-M'_\nu\leq(M_\nu-m_\nu)-(M'_\nu-m'_\nu) M ν − M ν ′ ≤ ( M ν − m ν ) − ( M ν ′ − m ν ′ ) ∑ ν ( M ν − M ν ′ ) ≤ ω ( 2 r ) − ω ( r ) \sum_\nu(M_\nu-M'_\nu)\leq\omega(2r)-\omega(r) ∑ ν ( M ν − M ν ′ ) ≤ ω ( 2 r ) − ω ( r ) P ν P_\nu P ν ω ( 2 r ) \omega(2r) ω ( 2 r ) ∑ ν ( M ν − M ν ′ ) \sum_\nu(M_\nu-M'_\nu) ∑ ν ( M ν − M ν ′ ) C ∗ = C 2 ( m + 1 ) 1 / p max { C 1 , 2 ∣ B 1 ∣ 1 / n m C 1 , 1 } C_*=C_2(m+1)^{1/p}\max\{C_1,2|B_1|^{1/n}mC_1,1\} C ∗ = C 2 ( m + 1 ) 1/ p max { C 1 , 2∣ B 1 ∣ 1/ n m C 1 , 1 } ▨
補題 2.2. 0 < θ < 1 0<\theta<1 0 < θ < 1 B ≥ 0 B\geq0 B ≥ 0 R 0 > 0 R_0>0 R 0 > 0 α = 1 2 min { log 2 ( 1 / θ ) , 1 } \alpha=\frac12\min\{\log_2(1/\theta),1\} α = 2 1 min { log 2 ( 1/ θ ) , 1 } θ \theta θ C C C ω : ( 0 , R 0 ] → [ 0 , ∞ ) \omega:(0,R_0]\to[0,\infty) ω : ( 0 , R 0 ] → [ 0 , ∞ ) 0 < r ≤ R 0 / 2 0<r\leq R_0/2 0 < r ≤ R 0 /2 r r r ω ( r ) ≤ θ ω ( 2 r ) + B r \omega(r)\leq\theta\omega(2r)+Br ω ( r ) ≤ θ ω ( 2 r ) + B r 0 < ρ ≤ R 0 0<\rho\leq R_0 0 < ρ ≤ R 0
ω ( ρ ) ≤ C ( ρ R 0 ) α ( ω ( R 0 ) + B R 0 ) \omega(\rho)\leq C\Bigl(\frac\rho{R_0}\Bigr)^\alpha\bigl(\omega(R_0)+BR_0\bigr) ω ( ρ ) ≤ C ( R 0 ρ ) α ( ω ( R 0 ) + B R 0 ) である。
証明. a 0 = log 2 ( 1 / θ ) a_0=\log_2(1/\theta) a 0 = log 2 ( 1/ θ ) 0 < α ≤ a 0 / 2 < a 0 0<\alpha\leq a_0/2<a_0 0 < α ≤ a 0 /2 < a 0 α ≤ 1 / 2 \alpha\leq1/2 α ≤ 1/2 α ≤ 1 \alpha\leq1 α ≤ 1 2 α θ = 2 α − a 0 < 1 2^\alpha\theta=2^{\alpha-a_0}<1 2 α θ = 2 α − a 0 < 1 §E15.40 補題 3.3 をε = ω \varepsilon=\omega ε = ω A = B A=B A = B γ = 1 \gamma=1 γ = 1 ρ 0 = R 0 \rho_0=R_0 ρ 0 = R 0 θ \theta θ C = 2 α max { 1 , ( 1 − 2 α θ ) − 1 } C=2^\alpha\max\{1,(1-2^\alpha\theta)^{-1}\} C = 2 α max { 1 , ( 1 − 2 α θ ) − 1 } ▨
3 加法的な方程式の振動評価 定理 3.1. n ≥ 1 n\geq1 n ≥ 1 0 < λ ≤ Λ 0<\lambda\leq\Lambda 0 < λ ≤ Λ n , λ , Λ n,\lambda,\Lambda n , λ , Λ α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) C C C
B 1 = B 1 ( 0 ) ⊆ R n B_1=B_1(0)\subseteq\R^n B 1 = B 1 ( 0 ) ⊆ R n M ∈ S n M\in\mathcal S^n M ∈ S n ∣ M ∣ = ( M : M ) 1 / 2 |M|=(M:M)^{1/2} ∣ M ∣ = ( M : M ) 1/2
C ⊆ S n \mathcal C\subseteq\mathcal S^n C ⊆ S n F 0 ∈ C 2 ( C ) F_0\in C^2(\mathcal C) F 0 ∈ C 2 ( C ) C \mathcal C C λ , Λ \lambda,\Lambda λ , Λ f ∈ C 2 ( B 1 ) f\in C^2(B_1) f ∈ C 2 ( B 1 ) u ∈ C 4 ( B 1 ) u\in C^4(B_1) u ∈ C 4 ( B 1 ) x ∈ B 1 x\in B_1 x ∈ B 1 D 2 u ( x ) ∈ C D^2u(x)\in\mathcal C D 2 u ( x ) ∈ C F 0 ( D 2 u ( x ) ) = f ( x ) F_0(D^2u(x))=f(x) F 0 ( D 2 u ( x )) = f ( x ) このとき
sup B 1 / 2 ∣ D 2 u ∣ + sup x , y ∈ B 1 / 2 x ≠ y ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ∣ x − y ∣ α ≤ C ( sup B 1 ∣ D 2 u ∣ + sup B 1 ∣ D f ∣ + sup B 1 ∣ D 2 f ∣ ) \sup_{B_{1/2}}|D^2u|+\sup_{\substack{x,y\in B_{1/2}\\x\neq y}}\frac{|D^2u(x)-D^2u(y)|}{|x-y|^\alpha}
\leq C\Bigl(\sup_{B_1}|D^2u|+\sup_{B_1}|Df|+\sup_{B_1}|D^2f|\Bigr) B 1/2 sup ∣ D 2 u ∣ + x , y ∈ B 1/2 x = y sup ∣ x − y ∣ α ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ≤ C ( B 1 sup ∣ D 2 u ∣ + B 1 sup ∣ D f ∣ + B 1 sup ∣ D 2 f ∣ ) である。右辺は+ ∞ +\infty + ∞
証明. K 0 = sup B 1 ∣ D 2 u ∣ K_0=\sup_{B_1}|D^2u| K 0 = sup B 1 ∣ D 2 u ∣ φ 1 = sup B 1 ∣ D f ∣ \varphi_1=\sup_{B_1}|Df| φ 1 = sup B 1 ∣ D f ∣ φ 2 = sup B 1 ∣ D 2 f ∣ \varphi_2=\sup_{B_1}|D^2f| φ 2 = sup B 1 ∣ D 2 f ∣ ξ 1 , … , ξ m \xi_1,\dots,\xi_m ξ 1 , … , ξ m λ ∗ , Λ ∗ \lambda_*,\Lambda_* λ ∗ , Λ ∗ 補題 1.3 のn , λ , Λ n,\lambda,\Lambda n , λ , Λ w ν = u ξ ν ξ ν = ξ ν ⊗ ξ ν : D 2 u w_\nu=u_{\xi_\nu\xi_\nu}=\xi_\nu\otimes\xi_\nu:D^2u w ν = u ξ ν ξ ν = ξ ν ⊗ ξ ν : D 2 u ∣ ξ ν ⊗ ξ ν ∣ = 1 |\xi_\nu\otimes\xi_\nu|=1 ∣ ξ ν ⊗ ξ ν ∣ = 1 ∣ w ν ∣ ≤ K 0 |w_\nu|\leq K_0 ∣ w ν ∣ ≤ K 0 F 0 F_0 F 0 C × B 1 \mathcal C\times B_1 C × B 1 §E15.42 命題 3.4 (1) を適用し、その行列をa ( M ) a(M) a ( M ) A ( x ) = a ( D 2 u ( x ) ) A(x)=a(D^2u(x)) A ( x ) = a ( D 2 u ( x )) A A A λ I ≤ A ( x ) ≤ Λ I \lambda I\leq A(x)\leq\Lambda I λ I ≤ A ( x ) ≤ Λ I L = A i j ( x ) D i j L=A^{ij}(x)D_{ij} L = A ij ( x ) D ij 命題 1.4 (3) によりB 1 B_1 B 1 L w ν ≥ f ξ ν ξ ν ≥ − φ 2 Lw_\nu\geq f_{\xi_\nu\xi_\nu}\geq-\varphi_2 L w ν ≥ f ξ ν ξ ν ≥ − φ 2
x 0 ∈ B 1 / 2 x_0\in B_{1/2} x 0 ∈ B 1/2 B ρ = B ρ ( x 0 ) B_\rho=B_\rho(x_0) B ρ = B ρ ( x 0 ) 0 < s ≤ 1 / 2 0<s\leq1/2 0 < s ≤ 1/2 ω x 0 ( s ) = ∑ ν ( sup B s w ν − inf B s w ν ) \omega_{x_0}(s)=\sum_\nu(\sup_{B_s}w_\nu-\inf_{B_s}w_\nu) ω x 0 ( s ) = ∑ ν ( sup B s w ν − inf B s w ν ) 0 < r ≤ 1 / 4 0<r\leq1/4 0 < r ≤ 1/4 B 2 r ⊆ B 1 B_{2r}\subseteq B_1 B 2 r ⊆ B 1 x ∈ B r x\in B_r x ∈ B r y ∈ B 2 r y\in B_{2r} y ∈ B 2 r C \mathcal C C F 0 F_0 F 0 F 0 ( D 2 u ( y ) ) ≤ F 0 ( D 2 u ( x ) ) + A ( x ) : ( D 2 u ( y ) − D 2 u ( x ) ) F_0(D^2u(y))\leq F_0(D^2u(x))+A(x):(D^2u(y)-D^2u(x)) F 0 ( D 2 u ( y )) ≤ F 0 ( D 2 u ( x )) + A ( x ) : ( D 2 u ( y ) − D 2 u ( x )) ∣ x − y ∣ < 3 r |x-y|<3r ∣ x − y ∣ < 3 r
A ( x ) : ( D 2 u ( y ) − D 2 u ( x ) ) ≥ f ( y ) − f ( x ) ≥ − 3 r φ 1 A(x):\bigl(D^2u(y)-D^2u(x)\bigr)\geq f(y)-f(x)\geq-3r\varphi_1 A ( x ) : ( D 2 u ( y ) − D 2 u ( x ) ) ≥ f ( y ) − f ( x ) ≥ − 3 r φ 1 である。補題 1.3 (1) によりA ( x ) = ∑ ν β ν ξ ν ⊗ ξ ν A(x)=\sum_\nu\beta_\nu\,\xi_\nu\otimes\xi_\nu A ( x ) = ∑ ν β ν ξ ν ⊗ ξ ν λ ∗ ≤ β ν ≤ Λ ∗ \lambda_*\leq\beta_\nu\leq\Lambda_* λ ∗ ≤ β ν ≤ Λ ∗ ∑ ν β ν ( w ν ( y ) − w ν ( x ) ) ≥ − 3 r φ 1 \sum_\nu\beta_\nu(w_\nu(y)-w_\nu(x))\geq-3r\varphi_1 ∑ ν β ν ( w ν ( y ) − w ν ( x )) ≥ − 3 r φ 1 補題 2.1 をc 0 = φ 2 c_0=\varphi_2 c 0 = φ 2 E = 3 r φ 1 E=3r\varphi_1 E = 3 r φ 1 C ∗ C_* C ∗ C ∗ C_* C ∗ n , λ , Λ n,\lambda,\Lambda n , λ , Λ r ≤ 1 r\leq1 r ≤ 1
ω x 0 ( r ) ≤ ( 1 − 1 C ∗ ) ω x 0 ( 2 r ) + 3 r φ 1 + r 2 φ 2 ≤ θ ω x 0 ( 2 r ) + ( 3 φ 1 + φ 2 ) r , θ = 1 − 1 2 C ∗ \omega_{x_0}(r)\leq\Bigl(1-\frac1{C_*}\Bigr)\omega_{x_0}(2r)+3r\varphi_1+r^2\varphi_2\leq\theta\,\omega_{x_0}(2r)+(3\varphi_1+\varphi_2)r,
\qquad\theta=1-\frac1{2C_*} ω x 0 ( r ) ≤ ( 1 − C ∗ 1 ) ω x 0 ( 2 r ) + 3 r φ 1 + r 2 φ 2 ≤ θ ω x 0 ( 2 r ) + ( 3 φ 1 + φ 2 ) r , θ = 1 − 2 C ∗ 1 が成り立つ。ω x 0 \omega_{x_0} ω x 0 ( 0 , 1 / 2 ] (0,1/2] ( 0 , 1/2 ] ω x 0 ( 1 / 2 ) ≤ 2 m K 0 \omega_{x_0}(1/2)\leq2mK_0 ω x 0 ( 1/2 ) ≤ 2 m K 0 補題 2.2 をR 0 = 1 / 2 R_0=1/2 R 0 = 1/2 θ \theta θ α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) C 3 C_3 C 3 0 < ρ ≤ 1 / 2 0<\rho\leq1/2 0 < ρ ≤ 1/2
ω x 0 ( ρ ) ≤ C 3 ( 2 ρ ) α ( 2 m K 0 + ( 3 φ 1 + φ 2 ) / 2 ) \omega_{x_0}(\rho)\leq C_3(2\rho)^\alpha\bigl(2mK_0+(3\varphi_1+\varphi_2)/2\bigr) ω x 0 ( ρ ) ≤ C 3 ( 2 ρ ) α ( 2 m K 0 + ( 3 φ 1 + φ 2 ) /2 ) である。
x , y ∈ B 1 / 2 x,y\in B_{1/2} x , y ∈ B 1/2 x ≠ y x\neq y x = y ∣ x − y ∣ < 1 / 2 |x-y|<1/2 ∣ x − y ∣ < 1/2 ∣ x − y ∣ < ρ ≤ 1 / 2 |x-y|<\rho\leq1/2 ∣ x − y ∣ < ρ ≤ 1/2 ρ \rho ρ y ∈ B ρ ( x ) y\in B_\rho(x) y ∈ B ρ ( x ) 補題 1.3 (2) により
∣ D 2 u ( x ) − D 2 u ( y ) ∣ ≤ ∑ ν ∣ w ν ( x ) − w ν ( y ) ∣ ≤ ω x ( ρ ) |D^2u(x)-D^2u(y)|\leq\sum_\nu|w_\nu(x)-w_\nu(y)|\leq\omega_x(\rho) ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ≤ ν ∑ ∣ w ν ( x ) − w ν ( y ) ∣ ≤ ω x ( ρ ) であり、ρ → ∣ x − y ∣ \rho\to|x-y| ρ → ∣ x − y ∣ ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ≤ 2 α C 3 ( 2 m K 0 + 3 φ 1 + φ 2 ) ∣ x − y ∣ α |D^2u(x)-D^2u(y)|\leq2^\alpha C_3(2mK_0+3\varphi_1+\varphi_2)|x-y|^\alpha ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ≤ 2 α C 3 ( 2 m K 0 + 3 φ 1 + φ 2 ) ∣ x − y ∣ α ∣ x − y ∣ ≥ 1 / 2 |x-y|\geq1/2 ∣ x − y ∣ ≥ 1/2 ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ≤ 2 K 0 ≤ 2 1 + α K 0 ∣ x − y ∣ α |D^2u(x)-D^2u(y)|\leq2K_0\leq2^{1+\alpha}K_0|x-y|^\alpha ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ≤ 2 K 0 ≤ 2 1 + α K 0 ∣ x − y ∣ α sup B 1 / 2 ∣ D 2 u ∣ ≤ K 0 \sup_{B_{1/2}}|D^2u|\leq K_0 sup B 1/2 ∣ D 2 u ∣ ≤ K 0 n , λ , Λ n,\lambda,\Lambda n , λ , Λ C C C ▨
4 空間依存の誤差 補題 4.1. n ≥ 1 n\geq1 n ≥ 1 0 < λ ≤ Λ 0<\lambda\leq\Lambda 0 < λ ≤ Λ U ⊆ R n U\subseteq\R^n U ⊆ R n O ⊆ S n × U \mathcal O\subseteq\mathcal S^n\times U O ⊆ S n × U O x \mathcal O_x O x F ∈ C 2 ( O ) F\in C^2(\mathcal O) F ∈ C 2 ( O ) O \mathcal O O λ , Λ \lambda,\Lambda λ , Λ a a a B ξ B_\xi B ξ ∂ ξ F \partial_\xi F ∂ ξ F 命題 1.4 のとおりとし、D x 2 F = ( ∂ e i ∂ e j F ) i , j D_x^2F=(\partial_{e_i}\partial_{e_j}F)_{i,j} D x 2 F = ( ∂ e i ∂ e j F ) i , j ∣ M ∣ = ( M : M ) 1 / 2 |M|=(M:M)^{1/2} ∣ M ∣ = ( M : M ) 1/2 Ω ⊆ U \Omega\subseteq U Ω ⊆ U u ∈ C 4 ( Ω ) u\in C^4(\Omega) u ∈ C 4 ( Ω ) x ∈ Ω x\in\Omega x ∈ Ω ( D 2 u ( x ) , x ) ∈ O (D^2u(x),x)\in\mathcal O ( D 2 u ( x ) , x ) ∈ O F ( D 2 u ( x ) , x ) = 0 F(D^2u(x),x)=0 F ( D 2 u ( x ) , x ) = 0 L = a ( D 2 u ( x ) , x ) : D 2 L=a(D^2u(x),x):D^2 L = a ( D 2 u ( x ) , x ) : D 2 K 0 , μ 2 ≥ 0 K_0,\mu_2\geq0 K 0 , μ 2 ≥ 0 x ∈ Ω x\in\Omega x ∈ Ω ∣ D 2 u ( x ) ∣ ≤ K 0 |D^2u(x)|\leq K_0 ∣ D 2 u ( x ) ∣ ≤ K 0 ( D 2 u ( x ) , x ) (D^2u(x),x) ( D 2 u ( x ) , x ) ∣ D x 2 F ∣ ≤ μ 2 |D_x^2F|\leq\mu_2 ∣ D x 2 F ∣ ≤ μ 2 ∣ ξ ∣ = 1 |\xi|=1 ∣ ξ ∣ = 1 ξ ∈ R n \xi\in\R^n ξ ∈ R n ∣ B ξ ∣ ≤ μ 2 |B_\xi|\leq\mu_2 ∣ B ξ ∣ ≤ μ 2 K ≥ K 0 K\geq K_0 K ≥ K 0 ε > 0 \varepsilon>0 ε > 0 K 1 = K 0 + n K K_1=K_0+\sqrt nK K 1 = K 0 + n K
C ε = μ 2 2 ( 1 + 2 ε n K 1 ) 2 2 ε λ + μ 2 ( 1 + 2 ε K 1 ) C_\varepsilon=\frac{\mu_2^2(1+2\varepsilon\sqrt nK_1)^2}{2\varepsilon\lambda}+\mu_2(1+2\varepsilon K_1) C ε = 2 ε λ μ 2 2 ( 1 + 2 ε n K 1 ) 2 + μ 2 ( 1 + 2 ε K 1 ) とおく。このとき、∣ ξ ∣ = 1 |\xi|=1 ∣ ξ ∣ = 1 ξ ∈ R n \xi\in\R^n ξ ∈ R n w = u ξ ξ + ε ∣ D 2 u + K I ∣ 2 w=u_{\xi\xi}+\varepsilon|D^2u+KI|^2 w = u ξ ξ + ε ∣ D 2 u + K I ∣ 2 Ω \Omega Ω L w ≥ − C ε Lw\geq-C_\varepsilon L w ≥ − C ε
証明. u i j k = D i D j D k u u_{ijk}=D_iD_jD_ku u ij k = D i D j D k u ∣ D 3 u ∣ 2 = ∑ i , j , k u i j k 2 |D^3u|^2=\sum_{i,j,k}u_{ijk}^2 ∣ D 3 u ∣ 2 = ∑ i , j , k u ij k 2 u i = D i u u_i=D_iu u i = D i u F F F ( D 2 u ( x ) , x ) (D^2u(x),x) ( D 2 u ( x ) , x )
∣ D 2 u ξ ∣ 2 = ∑ i , j ( ∑ k ξ k u i j k ) 2 ≤ ∣ D 3 u ∣ 2 |D^2u_\xi|^2=\sum_{i,j}(\sum_k\xi^ku_{ijk})^2\leq|D^3u|^2 ∣ D 2 u ξ ∣ 2 = ∑ i , j ( ∑ k ξ k u ij k ) 2 ≤ ∣ D 3 u ∣ 2 ∣ ∂ ξ 2 F ∣ = ∣ ξ ⊗ ξ : D x 2 F ∣ ≤ μ 2 |\partial_\xi^2F|=|\xi\otimes\xi:D_x^2F|\leq\mu_2 ∣ ∂ ξ 2 F ∣ = ∣ ξ ⊗ ξ : D x 2 F ∣ ≤ μ 2 命題 1.4 (2) により
L u ξ ξ ≥ − 2 ∣ B ξ ∣ ∣ D 2 u ξ ∣ − ∣ ∂ ξ 2 F ∣ ≥ − 2 μ 2 ∣ D 3 u ∣ − μ 2 Lu_{\xi\xi}\geq-2|B_\xi||D^2u_\xi|-|\partial_\xi^2F|\geq-2\mu_2|D^3u|-\mu_2 L u ξ ξ ≥ − 2∣ B ξ ∣∣ D 2 u ξ ∣ − ∣ ∂ ξ 2 F ∣ ≥ − 2 μ 2 ∣ D 3 u ∣ − μ 2 である。
A = D 2 u + K I A=D^2u+KI A = D 2 u + K I h = ∣ A ∣ 2 h=|A|^2 h = ∣ A ∣ 2 D k h = 2 A : D 2 u k D_kh=2A:D^2u_k D k h = 2 A : D 2 u k D k D l h = 2 D 2 u k : D 2 u l + 2 A : D 2 ( D k D l u ) D_kD_lh=2D^2u_k:D^2u_l+2A:D^2(D_kD_lu) D k D l h = 2 D 2 u k : D 2 u l + 2 A : D 2 ( D k D l u ) u ∈ C 4 ( Ω ) u\in C^4(\Omega) u ∈ C 4 ( Ω )
L h = 2 ∑ k , l a k l D 2 u k : D 2 u l + 2 ∑ i , j A i j L u e i e j Lh=2\sum_{k,l}a^{kl}\,D^2u_k:D^2u_l+2\sum_{i,j}A_{ij}\,Lu_{e_ie_j} L h = 2 k , l ∑ a k l D 2 u k : D 2 u l + 2 i , j ∑ A ij L u e i e j を得る。各i , j i,j i , j ∑ k , l a k l u i j k u i j l ≥ λ ∑ k u i j k 2 \sum_{k,l}a^{kl}u_{ijk}u_{ijl}\geq\lambda\sum_ku_{ijk}^2 ∑ k , l a k l u ij k u ij l ≥ λ ∑ k u ij k 2 2 λ ∣ D 3 u ∣ 2 2\lambda|D^3u|^2 2 λ ∣ D 3 u ∣ 2 A i j = A j i A_{ij}=A_{ji} A ij = A j i 命題 1.4 (1) により
∑ i , j A i j L u e i e j = − ∑ i , j A i j D M 2 F [ D 2 u i , D 2 u j ] − 2 ∑ i , j A i j B e i : D 2 u j − A : D x 2 F \sum_{i,j}A_{ij}Lu_{e_ie_j}=-\sum_{i,j}A_{ij}D_M^2F[D^2u_i,D^2u_j]-2\sum_{i,j}A_{ij}B_{e_i}:D^2u_j-A:D_x^2F i , j ∑ A ij L u e i e j = − i , j ∑ A ij D M 2 F [ D 2 u i , D 2 u j ] − 2 i , j ∑ A ij B e i : D 2 u j − A : D x 2 F である。K ≥ K 0 K\geq K_0 K ≥ K 0 D 2 u D^2u D 2 u − K 0 -K_0 − K 0 A ≥ 0 A\geq0 A ≥ 0 A A A q 1 , … , q n q_1,\dots,q_n q 1 , … , q n σ 1 , … , σ n ≥ 0 \sigma_1,\dots,\sigma_n\geq0 σ 1 , … , σ n ≥ 0 N k = ∑ i q k i D 2 u i N_k=\sum_iq_k^iD^2u_i N k = ∑ i q k i D 2 u i D M 2 F D_M^2F D M 2 F ∑ i , j A i j D M 2 F [ D 2 u i , D 2 u j ] = ∑ k σ k D M 2 F [ N k , N k ] \sum_{i,j}A_{ij}D_M^2F[D^2u_i,D^2u_j]=\sum_k\sigma_kD_M^2F[N_k,N_k] ∑ i , j A ij D M 2 F [ D 2 u i , D 2 u j ] = ∑ k σ k D M 2 F [ N k , N k ] F ( ⋅ , x ) F(\cdot,x) F ( ⋅ , x ) O x \mathcal O_x O x C 2 C^2 C 2 D M 2 F [ N k , N k ] D_M^2F[N_k,N_k] D M 2 F [ N k , N k ] 0 0 0 0 0 0 ∣ A ∣ ≤ ∣ D 2 u ∣ + K ∣ I ∣ ≤ K 1 |A|\leq|D^2u|+K|I|\leq K_1 ∣ A ∣ ≤ ∣ D 2 u ∣ + K ∣ I ∣ ≤ K 1
∣ ∑ i , j A i j B e i : D 2 u j ∣ ≤ μ 2 ∑ i , j ∣ A i j ∣ ∣ D 2 u j ∣ ≤ μ 2 ∣ A ∣ ( n ∑ j ∣ D 2 u j ∣ 2 ) 1 / 2 ≤ n μ 2 K 1 ∣ D 3 u ∣ \Bigl|\sum_{i,j}A_{ij}B_{e_i}:D^2u_j\Bigr|\leq\mu_2\sum_{i,j}|A_{ij}||D^2u_j|\leq\mu_2|A|\Bigl(n\sum_j|D^2u_j|^2\Bigr)^{1/2}\leq\sqrt n\mu_2K_1|D^3u| i , j ∑ A ij B e i : D 2 u j ≤ μ 2 i , j ∑ ∣ A ij ∣∣ D 2 u j ∣ ≤ μ 2 ∣ A ∣ ( n j ∑ ∣ D 2 u j ∣ 2 ) 1/2 ≤ n μ 2 K 1 ∣ D 3 u ∣ であり、∣ A : D x 2 F ∣ ≤ μ 2 K 1 |A:D_x^2F|\leq\mu_2K_1 ∣ A : D x 2 F ∣ ≤ μ 2 K 1
L h ≥ 2 λ ∣ D 3 u ∣ 2 − 4 n μ 2 K 1 ∣ D 3 u ∣ − 2 μ 2 K 1 Lh\geq2\lambda|D^3u|^2-4\sqrt n\mu_2K_1|D^3u|-2\mu_2K_1 L h ≥ 2 λ ∣ D 3 u ∣ 2 − 4 n μ 2 K 1 ∣ D 3 u ∣ − 2 μ 2 K 1 を得る。
t = ∣ D 3 u ∣ t=|D^3u| t = ∣ D 3 u ∣ β = 2 μ 2 ( 1 + 2 ε n K 1 ) \beta=2\mu_2(1+2\varepsilon\sqrt nK_1) β = 2 μ 2 ( 1 + 2 ε n K 1 )
L w = L u ξ ξ + ε L h ≥ 2 ε λ t 2 − β t − μ 2 ( 1 + 2 ε K 1 ) Lw=Lu_{\xi\xi}+\varepsilon Lh\geq2\varepsilon\lambda t^2-\beta t-\mu_2(1+2\varepsilon K_1) L w = L u ξ ξ + ε L h ≥ 2 ε λ t 2 − β t − μ 2 ( 1 + 2 ε K 1 ) である。2 ε λ t 2 − β t ≥ − β 2 / ( 8 ε λ ) 2\varepsilon\lambda t^2-\beta t\geq-\beta^2/(8\varepsilon\lambda) 2 ε λ t 2 − β t ≥ − β 2 / ( 8 ε λ ) β 2 / ( 8 ε λ ) = μ 2 2 ( 1 + 2 ε n K 1 ) 2 / ( 2 ε λ ) \beta^2/(8\varepsilon\lambda)=\mu_2^2(1+2\varepsilon\sqrt nK_1)^2/(2\varepsilon\lambda) β 2 / ( 8 ε λ ) = μ 2 2 ( 1 + 2 ε n K 1 ) 2 / ( 2 ε λ ) L w ≥ − C ε Lw\geq-C_\varepsilon L w ≥ − C ε ▨
5 一般形の評価 定理 5.1 (Evans–Krylov の定理). n ≥ 1 n\geq1 n ≥ 1 0 < λ ≤ Λ 0<\lambda\leq\Lambda 0 < λ ≤ Λ K 0 ≥ 0 K_0\geq0 K 0 ≥ 0 δ > 0 \delta>0 δ > 0 μ 1 , μ 2 ≥ 0 \mu_1,\mu_2\geq0 μ 1 , μ 2 ≥ 0 n , λ , Λ , K 0 , δ , μ 1 , μ 2 n,\lambda,\Lambda,K_0,\delta,\mu_1,\mu_2 n , λ , Λ , K 0 , δ , μ 1 , μ 2 α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) C C C
B 1 = B 1 ( 0 ) ⊆ R n B_1=B_1(0)\subseteq\R^n B 1 = B 1 ( 0 ) ⊆ R n M ∈ S n M\in\mathcal S^n M ∈ S n ∣ M ∣ = ( M : M ) 1 / 2 |M|=(M:M)^{1/2} ∣ M ∣ = ( M : M ) 1/2 U ⊆ R n U\subseteq\R^n U ⊆ R n B 1 ⊆ U B_1\subseteq U B 1 ⊆ U O ⊆ S n × U \mathcal O\subseteq\mathcal S^n\times U O ⊆ S n × U O x \mathcal O_x O x F ∈ C 2 ( O ) F\in C^2(\mathcal O) F ∈ C 2 ( O ) a a a B ξ B_\xi B ξ ∂ ξ F \partial_\xi F ∂ ξ F 命題 1.4 のとおりとし、D x F = ( ∂ e 1 F , … , ∂ e n F ) D_xF=(\partial_{e_1}F,\dots,\partial_{e_n}F) D x F = ( ∂ e 1 F , … , ∂ e n F ) D x 2 F = ( ∂ e i ∂ e j F ) i , j D_x^2F=(\partial_{e_i}\partial_{e_j}F)_{i,j} D x 2 F = ( ∂ e i ∂ e j F ) i , j
F F F O \mathcal O O λ , Λ \lambda,\Lambda λ , Λ u ∈ C 4 ( B 1 ) u\in C^4(B_1) u ∈ C 4 ( B 1 ) x ∈ B 1 x\in B_1 x ∈ B 1 ∣ D 2 u ( x ) ∣ ≤ K 0 |D^2u(x)|\leq K_0 ∣ D 2 u ( x ) ∣ ≤ K 0 F ( D 2 u ( x ) , x ) = 0 F(D^2u(x),x)=0 F ( D 2 u ( x ) , x ) = 0 ∣ x − y ∣ < δ |x-y|<\delta ∣ x − y ∣ < δ x , y ∈ B 1 x,y\in B_1 x , y ∈ B 1 ( D 2 u ( x ) , y ) ∈ O (D^2u(x),y)\in\mathcal O ( D 2 u ( x ) , y ) ∈ O 集合K = { ( D 2 u ( x ) , y ) ∣ x , y ∈ B 1 , ∣ x − y ∣ < δ } \mathcal K=\{(D^2u(x),y)\mid x,y\in B_1,\ |x-y|<\delta\} K = {( D 2 u ( x ) , y ) ∣ x , y ∈ B 1 , ∣ x − y ∣ < δ } ∣ D x F ∣ ≤ μ 1 |D_xF|\leq\mu_1 ∣ D x F ∣ ≤ μ 1 ∣ D x 2 F ∣ ≤ μ 2 |D_x^2F|\leq\mu_2 ∣ D x 2 F ∣ ≤ μ 2 ∣ ξ ∣ = 1 |\xi|=1 ∣ ξ ∣ = 1 ξ ∈ R n \xi\in\R^n ξ ∈ R n ∣ B ξ ∣ ≤ μ 2 |B_\xi|\leq\mu_2 ∣ B ξ ∣ ≤ μ 2
このとき
sup B 1 / 2 ∣ D 2 u ∣ + sup x , y ∈ B 1 / 2 x ≠ y ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ∣ x − y ∣ α ≤ C \sup_{B_{1/2}}|D^2u|+\sup_{\substack{x,y\in B_{1/2}\\x\neq y}}\frac{|D^2u(x)-D^2u(y)|}{|x-y|^\alpha}\leq C B 1/2 sup ∣ D 2 u ∣ + x , y ∈ B 1/2 x = y sup ∣ x − y ∣ α ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ≤ C である。
証明. ξ 1 , … , ξ m \xi_1,\dots,\xi_m ξ 1 , … , ξ m λ ∗ , Λ ∗ \lambda_*,\Lambda_* λ ∗ , Λ ∗ 補題 1.3 のn , λ , Λ n,\lambda,\Lambda n , λ , Λ C ∗ C_* C ∗ 補題 2.1 のn , λ , Λ , m , λ ∗ , Λ ∗ n,\lambda,\Lambda,m,\lambda_*,\Lambda_* n , λ , Λ , m , λ ∗ , Λ ∗ K = K 0 K=K_0 K = K 0 K 1 = ( 1 + n ) K 0 K_1=(1+\sqrt n)K_0 K 1 = ( 1 + n ) K 0 ε = 1 / ( 8 m ( 1 + C ∗ Λ ∗ ) ( 1 + K 1 ) ) \varepsilon=1/\bigl(8m(1+C_*\Lambda_*)(1+K_1)\bigr) ε = 1/ ( 8 m ( 1 + C ∗ Λ ∗ ) ( 1 + K 1 ) ) C ε C_\varepsilon C ε 補題 4.1 の定数とする。h = ∣ D 2 u + K I ∣ 2 h=|D^2u+KI|^2 h = ∣ D 2 u + K I ∣ 2 w ν = u ξ ν ξ ν + ε h w_\nu=u_{\xi_\nu\xi_\nu}+\varepsilon h w ν = u ξ ν ξ ν + ε h K \mathcal K K ( D 2 u ( x ) , x ) (D^2u(x),x) ( D 2 u ( x ) , x ) x ∈ B 1 x\in B_1 x ∈ B 1 補題 4.1 をΩ = B 1 \Omega=B_1 Ω = B 1 L = a ( D 2 u ( x ) , x ) : D 2 L=a(D^2u(x),x):D^2 L = a ( D 2 u ( x ) , x ) : D 2 B 1 B_1 B 1 L w ν ≥ − C ε Lw_\nu\geq-C_\varepsilon L w ν ≥ − C ε L L L §E15.42 命題 3.4 (1) によりλ I ≤ a ( D 2 u ( x ) , x ) ≤ Λ I \lambda I\leq a(D^2u(x),x)\leq\Lambda I λ I ≤ a ( D 2 u ( x ) , x ) ≤ Λ I
x , y ∈ B 1 x,y\in B_1 x , y ∈ B 1 ∣ A ( y ) ∣ , ∣ A ( x ) ∣ ≤ K 1 |A(y)|,|A(x)|\leq K_1 ∣ A ( y ) ∣ , ∣ A ( x ) ∣ ≤ K 1 A = D 2 u + K I A=D^2u+KI A = D 2 u + K I ∣ h ( y ) − h ( x ) ∣ ≤ 2 K 1 ∣ D 2 u ( y ) − D 2 u ( x ) ∣ |h(y)-h(x)|\leq2K_1|D^2u(y)-D^2u(x)| ∣ h ( y ) − h ( x ) ∣ ≤ 2 K 1 ∣ D 2 u ( y ) − D 2 u ( x ) ∣ 補題 1.3 (2) により
∣ D 2 u ( y ) − D 2 u ( x ) ∣ ≤ ∑ ν ∣ u ξ ν ξ ν ( y ) − u ξ ν ξ ν ( x ) ∣ ≤ ∑ ν ∣ w ν ( y ) − w ν ( x ) ∣ + 2 m ε K 1 ∣ D 2 u ( y ) − D 2 u ( x ) ∣ |D^2u(y)-D^2u(x)|\leq\sum_\nu|u_{\xi_\nu\xi_\nu}(y)-u_{\xi_\nu\xi_\nu}(x)|\leq\sum_\nu|w_\nu(y)-w_\nu(x)|+2m\varepsilon K_1|D^2u(y)-D^2u(x)| ∣ D 2 u ( y ) − D 2 u ( x ) ∣ ≤ ν ∑ ∣ u ξ ν ξ ν ( y ) − u ξ ν ξ ν ( x ) ∣ ≤ ν ∑ ∣ w ν ( y ) − w ν ( x ) ∣ + 2 m ε K 1 ∣ D 2 u ( y ) − D 2 u ( x ) ∣ であり、2 m ε K 1 ≤ 1 / 4 2m\varepsilon K_1\leq1/4 2 m ε K 1 ≤ 1/4
∣ D 2 u ( y ) − D 2 u ( x ) ∣ ≤ 4 3 ∑ ν ∣ w ν ( y ) − w ν ( x ) ∣ , ∣ h ( y ) − h ( x ) ∣ ≤ 3 K 1 ∑ ν ∣ w ν ( y ) − w ν ( x ) ∣ |D^2u(y)-D^2u(x)|\leq\frac43\sum_\nu|w_\nu(y)-w_\nu(x)|,\qquad|h(y)-h(x)|\leq3K_1\sum_\nu|w_\nu(y)-w_\nu(x)| ∣ D 2 u ( y ) − D 2 u ( x ) ∣ ≤ 3 4 ν ∑ ∣ w ν ( y ) − w ν ( x ) ∣ , ∣ h ( y ) − h ( x ) ∣ ≤ 3 K 1 ν ∑ ∣ w ν ( y ) − w ν ( x ) ∣ を得る。
r 0 = min { 1 / 4 , δ / 4 } r_0=\min\{1/4,\delta/4\} r 0 = min { 1/4 , δ /4 } x 0 ∈ B 1 / 2 x_0\in B_{1/2} x 0 ∈ B 1/2 B ρ = B ρ ( x 0 ) B_\rho=B_\rho(x_0) B ρ = B ρ ( x 0 ) 0 < s ≤ 2 r 0 0<s\leq2r_0 0 < s ≤ 2 r 0 ω x 0 ( s ) = ∑ ν ( sup B s w ν − inf B s w ν ) \omega_{x_0}(s)=\sum_\nu(\sup_{B_s}w_\nu-\inf_{B_s}w_\nu) ω x 0 ( s ) = ∑ ν ( sup B s w ν − inf B s w ν ) 0 < r ≤ r 0 0<r\leq r_0 0 < r ≤ r 0 B 2 r ⊆ B 1 B_{2r}\subseteq B_1 B 2 r ⊆ B 1 x ∈ B r x\in B_r x ∈ B r y ∈ B 2 r y\in B_{2r} y ∈ B 2 r ∣ x − y ∣ < 3 r < δ |x-y|<3r<\delta ∣ x − y ∣ < 3 r < δ 条件 (c) により( D 2 u ( x ) , y ) (D^2u(x),y) ( D 2 u ( x ) , y ) ( D 2 u ( y ) , y ) (D^2u(y),y) ( D 2 u ( y ) , y ) O \mathcal O O O y \mathcal O_y O y F ( ⋅ , y ) F(\cdot,y) F ( ⋅ , y )
F ( D 2 u ( y ) , y ) ≤ F ( D 2 u ( x ) , y ) + a ( D 2 u ( x ) , y ) : ( D 2 u ( y ) − D 2 u ( x ) ) F(D^2u(y),y)\leq F(D^2u(x),y)+a(D^2u(x),y):\bigl(D^2u(y)-D^2u(x)\bigr) F ( D 2 u ( y ) , y ) ≤ F ( D 2 u ( x ) , y ) + a ( D 2 u ( x ) , y ) : ( D 2 u ( y ) − D 2 u ( x ) ) である。線分[ x , y ] [x,y] [ x , y ] z z z z ∈ B 1 z\in B_1 z ∈ B 1 ∣ z − x ∣ < δ |z-x|<\delta ∣ z − x ∣ < δ ( D 2 u ( x ) , z ) ∈ K (D^2u(x),z)\in\mathcal K ( D 2 u ( x ) , z ) ∈ K F ( D 2 u ( x ) , y ) − F ( D 2 u ( x ) , x ) ≤ μ 1 ∣ x − y ∣ F(D^2u(x),y)-F(D^2u(x),x)\leq\mu_1|x-y| F ( D 2 u ( x ) , y ) − F ( D 2 u ( x ) , x ) ≤ μ 1 ∣ x − y ∣ F ( D 2 u ( y ) , y ) = F ( D 2 u ( x ) , x ) = 0 F(D^2u(y),y)=F(D^2u(x),x)=0 F ( D 2 u ( y ) , y ) = F ( D 2 u ( x ) , x ) = 0
a ( D 2 u ( x ) , y ) : ( D 2 u ( y ) − D 2 u ( x ) ) ≥ − 3 r μ 1 a(D^2u(x),y):\bigl(D^2u(y)-D^2u(x)\bigr)\geq-3r\mu_1 a ( D 2 u ( x ) , y ) : ( D 2 u ( y ) − D 2 u ( x ) ) ≥ − 3 r μ 1 である。λ I ≤ a ( D 2 u ( x ) , y ) ≤ Λ I \lambda I\leq a(D^2u(x),y)\leq\Lambda I λ I ≤ a ( D 2 u ( x ) , y ) ≤ Λ I 補題 1.3 (1) によりa ( D 2 u ( x ) , y ) = ∑ ν β ν ξ ν ⊗ ξ ν a(D^2u(x),y)=\sum_\nu\beta_\nu\,\xi_\nu\otimes\xi_\nu a ( D 2 u ( x ) , y ) = ∑ ν β ν ξ ν ⊗ ξ ν λ ∗ ≤ β ν ≤ Λ ∗ \lambda_*\leq\beta_\nu\leq\Lambda_* λ ∗ ≤ β ν ≤ Λ ∗ ∑ ν ∣ w ν ( y ) − w ν ( x ) ∣ ≤ ω x 0 ( 2 r ) \sum_\nu|w_\nu(y)-w_\nu(x)|\leq\omega_{x_0}(2r) ∑ ν ∣ w ν ( y ) − w ν ( x ) ∣ ≤ ω x 0 ( 2 r )
∑ ν β ν ( w ν ( y ) − w ν ( x ) ) = ∑ ν β ν ( u ξ ν ξ ν ( y ) − u ξ ν ξ ν ( x ) ) + ε ∑ ν β ν ( h ( y ) − h ( x ) ) ≥ − 3 r μ 1 − 3 ε m Λ ∗ K 1 ω x 0 ( 2 r ) \sum_\nu\beta_\nu\bigl(w_\nu(y)-w_\nu(x)\bigr)
=\sum_\nu\beta_\nu\bigl(u_{\xi_\nu\xi_\nu}(y)-u_{\xi_\nu\xi_\nu}(x)\bigr)+\varepsilon\sum_\nu\beta_\nu\bigl(h(y)-h(x)\bigr)
\geq-3r\mu_1-3\varepsilon m\Lambda_*K_1\,\omega_{x_0}(2r) ν ∑ β ν ( w ν ( y ) − w ν ( x ) ) = ν ∑ β ν ( u ξ ν ξ ν ( y ) − u ξ ν ξ ν ( x ) ) + ε ν ∑ β ν ( h ( y ) − h ( x ) ) ≥ − 3 r μ 1 − 3 ε m Λ ∗ K 1 ω x 0 ( 2 r ) を得る。補題 2.1 をc 0 = C ε c_0=C_\varepsilon c 0 = C ε E = 3 r μ 1 + 3 ε m Λ ∗ K 1 ω x 0 ( 2 r ) E=3r\mu_1+3\varepsilon m\Lambda_*K_1\omega_{x_0}(2r) E = 3 r μ 1 + 3 ε m Λ ∗ K 1 ω x 0 ( 2 r )
ω x 0 ( 2 r ) ≤ C ∗ ( ω x 0 ( 2 r ) − ω x 0 ( r ) ) + C ∗ ( 3 r μ 1 + r 2 C ε ) + 3 C ∗ ε m Λ ∗ K 1 ω x 0 ( 2 r ) \omega_{x_0}(2r)\leq C_*\bigl(\omega_{x_0}(2r)-\omega_{x_0}(r)\bigr)+C_*\bigl(3r\mu_1+r^2C_\varepsilon\bigr)+3C_*\varepsilon m\Lambda_*K_1\,\omega_{x_0}(2r) ω x 0 ( 2 r ) ≤ C ∗ ( ω x 0 ( 2 r ) − ω x 0 ( r ) ) + C ∗ ( 3 r μ 1 + r 2 C ε ) + 3 C ∗ ε m Λ ∗ K 1 ω x 0 ( 2 r ) である。3 C ∗ ε m Λ ∗ K 1 ≤ 1 / 2 3C_*\varepsilon m\Lambda_*K_1\leq1/2 3 C ∗ ε m Λ ∗ K 1 ≤ 1/2 r ≤ 1 r\leq1 r ≤ 1 θ = 1 − 1 / ( 2 C ∗ ) \theta=1-1/(2C_*) θ = 1 − 1/ ( 2 C ∗ )
ω x 0 ( r ) ≤ θ ω x 0 ( 2 r ) + ( 3 μ 1 + C ε ) r \omega_{x_0}(r)\leq\theta\,\omega_{x_0}(2r)+(3\mu_1+C_\varepsilon)r ω x 0 ( r ) ≤ θ ω x 0 ( 2 r ) + ( 3 μ 1 + C ε ) r が成り立つ。ω x 0 \omega_{x_0} ω x 0 ( 0 , 2 r 0 ] (0,2r_0] ( 0 , 2 r 0 ] ∣ w ν ∣ ≤ K 0 + ε K 1 2 ≤ K 0 + K 1 |w_\nu|\leq K_0+\varepsilon K_1^2\leq K_0+K_1 ∣ w ν ∣ ≤ K 0 + ε K 1 2 ≤ K 0 + K 1 ω x 0 ( 2 r 0 ) ≤ 2 m ( K 0 + K 1 ) \omega_{x_0}(2r_0)\leq2m(K_0+K_1) ω x 0 ( 2 r 0 ) ≤ 2 m ( K 0 + K 1 ) 補題 2.2 をR 0 = 2 r 0 R_0=2r_0 R 0 = 2 r 0 B = 3 μ 1 + C ε B=3\mu_1+C_\varepsilon B = 3 μ 1 + C ε θ \theta θ α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) C 3 C_3 C 3 0 < ρ ≤ 2 r 0 0<\rho\leq2r_0 0 < ρ ≤ 2 r 0
ω x 0 ( ρ ) ≤ C 3 ( ρ 2 r 0 ) α ( 2 m ( K 0 + K 1 ) + 2 r 0 ( 3 μ 1 + C ε ) ) \omega_{x_0}(\rho)\leq C_3\Bigl(\frac\rho{2r_0}\Bigr)^\alpha\bigl(2m(K_0+K_1)+2r_0(3\mu_1+C_\varepsilon)\bigr) ω x 0 ( ρ ) ≤ C 3 ( 2 r 0 ρ ) α ( 2 m ( K 0 + K 1 ) + 2 r 0 ( 3 μ 1 + C ε ) ) である。
x , y ∈ B 1 / 2 x,y\in B_{1/2} x , y ∈ B 1/2 x ≠ y x\neq y x = y ∣ x − y ∣ < 2 r 0 |x-y|<2r_0 ∣ x − y ∣ < 2 r 0 ∣ x − y ∣ < ρ ≤ 2 r 0 |x-y|<\rho\leq2r_0 ∣ x − y ∣ < ρ ≤ 2 r 0 ρ \rho ρ y ∈ B ρ ( x ) y\in B_\rho(x) y ∈ B ρ ( x ) ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ≤ 4 3 ω x ( ρ ) |D^2u(x)-D^2u(y)|\leq\frac43\omega_x(\rho) ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ≤ 3 4 ω x ( ρ ) ρ → ∣ x − y ∣ \rho\to|x-y| ρ → ∣ x − y ∣ ∣ D 2 u ( x ) − D 2 u ( y ) ∣ |D^2u(x)-D^2u(y)| ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ∣ x − y ∣ α |x-y|^\alpha ∣ x − y ∣ α ∣ x − y ∣ ≥ 2 r 0 |x-y|\geq2r_0 ∣ x − y ∣ ≥ 2 r 0 ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ≤ 2 K 0 ≤ 2 K 0 ( 2 r 0 ) − α ∣ x − y ∣ α |D^2u(x)-D^2u(y)|\leq2K_0\leq2K_0(2r_0)^{-\alpha}|x-y|^\alpha ∣ D 2 u ( x ) − D 2 u ( y ) ∣ ≤ 2 K 0 ≤ 2 K 0 ( 2 r 0 ) − α ∣ x − y ∣ α sup B 1 / 2 ∣ D 2 u ∣ ≤ K 0 \sup_{B_{1/2}}|D^2u|\leq K_0 sup B 1/2 ∣ D 2 u ∣ ≤ K 0 n , λ , Λ , K 0 , δ , μ 1 , μ 2 n,\lambda,\Lambda,K_0,\delta,\mu_1,\mu_2 n , λ , Λ , K 0 , δ , μ 1 , μ 2 ▨
注意 5.2. U ⊆ R n U\subseteq\R^n U ⊆ R n B 1 ‾ ⊆ U \overline{B_1}\subseteq U B 1 ⊆ U F ∈ C 2 ( S n × U ) F\in C^2(\mathcal S^n\times U) F ∈ C 2 ( S n × U ) x ∈ U x\in U x ∈ U F ( ⋅ , x ) F(\cdot,x) F ( ⋅ , x ) λ , Λ \lambda,\Lambda λ , Λ O = S n × U \mathcal O=\mathcal S^n\times U O = S n × U δ = 2 \delta=2 δ = 2 定理 5.1 条件 (c) は自動的に成り立ち、K \mathcal K K { M ∣ ∣ M ∣ ≤ K 0 } × B 1 ‾ \{M\mid|M|\leq K_0\}\times\overline{B_1} { M ∣ ∣ M ∣ ≤ K 0 } × B 1 F ( D 2 u , x ) = 0 F(D^2u,x)=0 F ( D 2 u , x ) = 0 ∣ D 2 u ∣ ≤ K 0 |D^2u|\leq K_0 ∣ D 2 u ∣ ≤ K 0 u ∈ C 4 ( B 1 ) u\in C^4(B_1) u ∈ C 4 ( B 1 ) 定理 5.1 の評価が成り立ち、α \alpha α C C C n , λ , Λ , K 0 n,\lambda,\Lambda,K_0 n , λ , Λ , K 0 F F F x x x M , x M,x M , x D M 2 F D_M^2F D M 2 F α \alpha α C C C
6 高階正則性 定理 6.1. n ≥ 1 n\geq1 n ≥ 1 0 < α < 1 0<\alpha<1 0 < α < 1 0 < λ ≤ Λ 0<\lambda\leq\Lambda 0 < λ ≤ Λ Ω ⊆ R n \Omega\subseteq\R^n Ω ⊆ R n O ⊆ S n × Ω \mathcal O\subseteq\mathcal S^n\times\Omega O ⊆ S n × Ω O x \mathcal O_x O x F ∈ C ∞ ( O ) F\in C^\infty(\mathcal O) F ∈ C ∞ ( O ) O \mathcal O O λ , Λ \lambda,\Lambda λ , Λ u ∈ C 2 ( Ω ) u\in C^2(\Omega) u ∈ C 2 ( Ω ) x ∈ Ω x\in\Omega x ∈ Ω ( D 2 u ( x ) , x ) ∈ O (D^2u(x),x)\in\mathcal O ( D 2 u ( x ) , x ) ∈ O F ( D 2 u ( x ) , x ) = 0 F(D^2u(x),x)=0 F ( D 2 u ( x ) , x ) = 0 B ‾ ⊆ Ω \overline B\subseteq\Omega B ⊆ Ω B B B u ∈ C 2 , α ( B ‾ ) u\in C^{2,\alpha}(\overline B) u ∈ C 2 , α ( B ) u ∈ C ∞ ( Ω ) u\in C^\infty(\Omega) u ∈ C ∞ ( Ω )
証明. S n × R n \mathcal S^n\times\R^n S n × R n ∣ ( M , x ) ∣ = ( ∣ M ∣ 2 + ∣ x ∣ 2 ) 1 / 2 |(M,x)|=(|M|^2+|x|^2)^{1/2} ∣ ( M , x ) ∣ = ( ∣ M ∣ 2 + ∣ x ∣ 2 ) 1/2 ∣ M ∣ = ( M : M ) 1 / 2 |M|=(M:M)^{1/2} ∣ M ∣ = ( M : M ) 1/2 a ( M , x ) a(M,x) a ( M , x ) §E15.42 命題 3.4 (1) の行列とすると、a i i = d d t F ( M + t e i ⊗ e i , x ) ∣ t = 0 a^{ii}=\frac{d}{dt}F(M+t\,e_i\otimes e_i,x)|_{t=0} a ii = d t d F ( M + t e i ⊗ e i , x ) ∣ t = 0 i ≠ j i\neq j i = j 2 a i j = d d t F ( M + t ( e i ⊗ e j + e j ⊗ e i ) , x ) ∣ t = 0 2a^{ij}=\frac{d}{dt}F(M+t(e_i\otimes e_j+e_j\otimes e_i),x)|_{t=0} 2 a ij = d t d F ( M + t ( e i ⊗ e j + e j ⊗ e i ) , x ) ∣ t = 0 a a a C ∞ ( O ) C^\infty(\mathcal O) C ∞ ( O ) §E15.42 命題 3.4 (1) によりO \mathcal O O λ I ≤ a ≤ Λ I \lambda I\leq a\leq\Lambda I λ I ≤ a ≤ Λ I O \mathcal O O ( M , x ) (M,x) ( M , x ) F F F x l x_l x l ∂ l F ( M , x ) \partial_lF(M,x) ∂ l F ( M , x )
主張 6.1.1. W ⊆ S n × R n W\subseteq\mathcal S^n\times\R^n W ⊆ S n × R n Γ ⊆ W \Gamma\subseteq W Γ ⊆ W ρ ≥ 0 \rho\geq0 ρ ≥ 0 Γ ρ = { P ∣ dist ( P , Γ ) ≤ ρ } \Gamma_\rho=\{P\mid\operatorname{dist}(P,\Gamma)\leq\rho\} Γ ρ = { P ∣ dist ( P , Γ ) ≤ ρ } Γ 2 ρ ⊆ W \Gamma_{2\rho}\subseteq W Γ 2 ρ ⊆ W ρ > 0 \rho>0 ρ > 0 G ∈ C 1 ( W ) G\in C^1(W) G ∈ C 1 ( W ) Γ ρ \Gamma_\rho Γ ρ
証明. Γ \Gamma Γ W W W Γ \Gamma Γ W W W Γ 2 ρ ⊆ W \Gamma_{2\rho}\subseteq W Γ 2 ρ ⊆ W ρ > 0 \rho>0 ρ > 0 Γ ρ \Gamma_\rho Γ ρ Γ 2 ρ \Gamma_{2\rho} Γ 2 ρ S = max Γ ρ ∣ G ∣ S=\max_{\Gamma_\rho}|G| S = max Γ ρ ∣ G ∣ L = max Γ 2 ρ ∣ D G ∣ L=\max_{\Gamma_{2\rho}}|DG| L = max Γ 2 ρ ∣ D G ∣ P , Q ∈ Γ ρ P,Q\in\Gamma_\rho P , Q ∈ Γ ρ ∣ P − Q ∣ ≤ ρ |P-Q|\leq\rho ∣ P − Q ∣ ≤ ρ [ P , Q ] [P,Q] [ P , Q ] P P P ρ \rho ρ Γ 2 ρ \Gamma_{2\rho} Γ 2 ρ ∣ G ( P ) − G ( Q ) ∣ ≤ L ∣ P − Q ∣ |G(P)-G(Q)|\leq L|P-Q| ∣ G ( P ) − G ( Q ) ∣ ≤ L ∣ P − Q ∣ ∣ P − Q ∣ > ρ |P-Q|>\rho ∣ P − Q ∣ > ρ ∣ G ( P ) − G ( Q ) ∣ ≤ 2 S < ( 2 S / ρ ) ∣ P − Q ∣ |G(P)-G(Q)|\leq2S<(2S/\rho)|P-Q| ∣ G ( P ) − G ( Q ) ∣ ≤ 2 S < ( 2 S / ρ ) ∣ P − Q ∣ ▨
主張 6.1.2. B B B k ≥ 0 k\geq0 k ≥ 0 V : B ‾ → O V:\overline B\to\mathcal O V : B → O S n × R n \mathcal S^n\times\R^n S n × R n C k , α ( B ‾ ) C^{k,\alpha}(\overline B) C k , α ( B ) G ∈ C ∞ ( O ) G\in C^\infty(\mathcal O) G ∈ C ∞ ( O ) G ∘ V ∈ C k , α ( B ‾ ) G\circ V\in C^{k,\alpha}(\overline B) G ∘ V ∈ C k , α ( B )
証明. k = 0 k=0 k = 0 V V V B ‾ \overline B B Γ = V ( B ‾ ) \Gamma=V(\overline B) Γ = V ( B ) 主張 6.1.1 によりG G G Γ \Gamma Γ V V V α \alpha α G ∘ V G\circ V G ∘ V B ‾ \overline B B α \alpha α k ≥ 1 k\geq1 k ≥ 1 k − 1 k-1 k − 1 §E15.35 定理 2.2 によりC k , α ( B ‾ ) ⊆ C k − 1 , α ( B ‾ ) C^{k,\alpha}(\overline B)\subseteq C^{k-1,\alpha}(\overline B) C k , α ( B ) ⊆ C k − 1 , α ( B ) G ∘ V ∈ C 1 ( B ) G\circ V\in C^1(B) G ∘ V ∈ C 1 ( B ) D i ( G ∘ V ) D_i(G\circ V) D i ( G ∘ V ) ( ∂ p G ) ∘ V (\partial_pG)\circ V ( ∂ p G ) ∘ V V V V D i D_i D i ( ∂ p G ) ∘ V ∈ C k − 1 , α ( B ‾ ) (\partial_pG)\circ V\in C^{k-1,\alpha}(\overline B) ( ∂ p G ) ∘ V ∈ C k − 1 , α ( B ) V V V C k − 1 , α ( B ‾ ) C^{k-1,\alpha}(\overline B) C k − 1 , α ( B ) §E15.35 命題 1.6 (1) によりD i ( G ∘ V ) ∈ C k − 1 , α ( B ‾ ) D_i(G\circ V)\in C^{k-1,\alpha}(\overline B) D i ( G ∘ V ) ∈ C k − 1 , α ( B ) G ∘ V G\circ V G ∘ V B ‾ \overline B B G ∘ V ∈ C k , α ( B ‾ ) G\circ V\in C^{k,\alpha}(\overline B) G ∘ V ∈ C k , α ( B ) ▨
整数m ≥ 0 m\geq0 m ≥ 0 ( S m ) (\mathrm S_m) ( S m ) z ∈ Ω z\in\Omega z ∈ Ω B r ( z ) ‾ ⊆ Ω \overline{B_r(z)}\subseteq\Omega B r ( z ) ⊆ Ω u ∈ C m + 2 , α ( B r ( z ) ‾ ) u\in C^{m+2,\alpha}(\overline{B_r(z)}) u ∈ C m + 2 , α ( B r ( z ) ) r > 0 r>0 r > 0
主張 6.1.3. ( S 0 ) (\mathrm S_0) ( S 0 ) ( S 1 ) (\mathrm S_1) ( S 1 )
証明. z ∈ Ω z\in\Omega z ∈ Ω ( S 0 ) (\mathrm S_0) ( S 0 ) r > 0 r>0 r > 0 B r ( z ) ‾ ⊆ Ω \overline{B_r(z)}\subseteq\Omega B r ( z ) ⊆ Ω u ∈ C 2 , α ( B r ( z ) ‾ ) u\in C^{2,\alpha}(\overline{B_r(z)}) u ∈ C 2 , α ( B r ( z ) ) B ′ = B r ( z ) B'=B_r(z) B ′ = B r ( z ) B = B r / 2 ( z ) B=B_{r/2}(z) B = B r /2 ( z ) B ′ ′ = B r / 4 ( z ) B''=B_{r/4}(z) B ′′ = B r /4 ( z ) H = sup { ∣ D 2 u ( x ) − D 2 u ( y ) ∣ / ∣ x − y ∣ α ∣ x , y ∈ B ′ ‾ , x ≠ y } H=\sup\{|D^2u(x)-D^2u(y)|/|x-y|^\alpha\mid x,y\in\overline{B'},\ x\neq y\} H = sup { ∣ D 2 u ( x ) − D 2 u ( y ) ∣/∣ x − y ∣ α ∣ x , y ∈ B ′ , x = y } Γ = { ( D 2 u ( x ) , x ) ∣ x ∈ B ′ ‾ } \Gamma=\{(D^2u(x),x)\mid x\in\overline{B'}\} Γ = {( D 2 u ( x ) , x ) ∣ x ∈ B ′ } O \mathcal O O 主張 6.1.1 をW = O W=\mathcal O W = O ρ > 0 \rho>0 ρ > 0 a a a ∂ l F \partial_lF ∂ l F 1 ≤ l ≤ n 1\leq l\leq n 1 ≤ l ≤ n Γ ρ \Gamma_\rho Γ ρ S ′ S' S ′ L ′ L' L ′ D 2 u D^2u D 2 u B ′ ‾ \overline{B'} B ′ x , x ′ ∈ B ′ ‾ x,x'\in\overline{B'} x , x ′ ∈ B ′ ∣ x − x ′ ∣ ≤ h 0 |x-x'|\leq h_0 ∣ x − x ′ ∣ ≤ h 0 ∣ D 2 u ( x ′ ) − D 2 u ( x ) ∣ + ∣ x ′ − x ∣ ≤ ρ |D^2u(x')-D^2u(x)|+|x'-x|\leq\rho ∣ D 2 u ( x ′ ) − D 2 u ( x ) ∣ + ∣ x ′ − x ∣ ≤ ρ h 0 ∈ ( 0 , r / 2 ] h_0\in(0,r/2] h 0 ∈ ( 0 , r /2 ]
1 ≤ l ≤ n 1\leq l\leq n 1 ≤ l ≤ n 0 < ∣ h ∣ ≤ h 0 0<|h|\leq h_0 0 < ∣ h ∣ ≤ h 0 x ∈ B ‾ x\in\overline B x ∈ B x + h e l ∈ B ′ ‾ x+he_l\in\overline{B'} x + h e l ∈ B ′ t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ]
Z t ( x ) = ( ( 1 − t ) D 2 u ( x ) + t D 2 u ( x + h e l ) , x + t h e l ) Z_t(x)=\bigl((1-t)D^2u(x)+tD^2u(x+he_l),\ x+the_l\bigr) Z t ( x ) = ( ( 1 − t ) D 2 u ( x ) + t D 2 u ( x + h e l ) , x + t h e l ) は∣ Z t ( x ) − ( D 2 u ( x ) , x ) ∣ ≤ ∣ D 2 u ( x + h e l ) − D 2 u ( x ) ∣ + ∣ h ∣ ≤ ρ |Z_t(x)-(D^2u(x),x)|\leq|D^2u(x+he_l)-D^2u(x)|+|h|\leq\rho ∣ Z t ( x ) − ( D 2 u ( x ) , x ) ∣ ≤ ∣ D 2 u ( x + h e l ) − D 2 u ( x ) ∣ + ∣ h ∣ ≤ ρ Γ ρ ⊆ O \Gamma_\rho\subseteq\mathcal O Γ ρ ⊆ O F ( Z 1 ( x ) ) = F ( Z 0 ( x ) ) = 0 F(Z_1(x))=F(Z_0(x))=0 F ( Z 1 ( x )) = F ( Z 0 ( x )) = 0 t ↦ F ( Z t ( x ) ) t\mapsto F(Z_t(x)) t ↦ F ( Z t ( x )) a ( Z t ( x ) ) : ( D 2 u ( x + h e l ) − D 2 u ( x ) ) + h ∂ l F ( Z t ( x ) ) a(Z_t(x)):(D^2u(x+he_l)-D^2u(x))+h\,\partial_lF(Z_t(x)) a ( Z t ( x )) : ( D 2 u ( x + h e l ) − D 2 u ( x )) + h ∂ l F ( Z t ( x )) [ 0 , 1 ] [0,1] [ 0 , 1 ] h h h v h ( x ) = ( u ( x + h e l ) − u ( x ) ) / h v_h(x)=(u(x+he_l)-u(x))/h v h ( x ) = ( u ( x + h e l ) − u ( x )) / h
a h ( x ) = ∫ 0 1 a ( Z t ( x ) ) d t , f h ( x ) = ∫ 0 1 ∂ l F ( Z t ( x ) ) d t a_h(x)=\int_0^1a(Z_t(x))\,dt,\qquad f_h(x)=\int_0^1\partial_lF(Z_t(x))\,dt a h ( x ) = ∫ 0 1 a ( Z t ( x )) d t , f h ( x ) = ∫ 0 1 ∂ l F ( Z t ( x )) d t について、B ‾ \overline B B a h : D 2 v h + f h = 0 a_h:D^2v_h+f_h=0 a h : D 2 v h + f h = 0 a h a_h a h λ I ≤ a h ≤ Λ I \lambda I\leq a_h\leq\Lambda I λ I ≤ a h ≤ Λ I x , y ∈ B ‾ x,y\in\overline B x , y ∈ B ∣ x − y ∣ ≤ r |x-y|\leq r ∣ x − y ∣ ≤ r
∣ Z t ( x ) − Z t ( y ) ∣ ≤ ∣ D 2 u ( x ) − D 2 u ( y ) ∣ + ∣ D 2 u ( x + h e l ) − D 2 u ( y + h e l ) ∣ + ∣ x − y ∣ ≤ ( 2 H + r 1 − α ) ∣ x − y ∣ α |Z_t(x)-Z_t(y)|\leq|D^2u(x)-D^2u(y)|+|D^2u(x+he_l)-D^2u(y+he_l)|+|x-y|\leq(2H+r^{1-\alpha})|x-y|^\alpha ∣ Z t ( x ) − Z t ( y ) ∣ ≤ ∣ D 2 u ( x ) − D 2 u ( y ) ∣ + ∣ D 2 u ( x + h e l ) − D 2 u ( y + h e l ) ∣ + ∣ x − y ∣ ≤ ( 2 H + r 1 − α ) ∣ x − y ∣ α である。したがってa h a_h a h f h f_h f h B ‾ \overline B B S ′ S' S ′ α \alpha α L ′ ( 2 H + r 1 − α ) L'(2H+r^{1-\alpha}) L ′ ( 2 H + r 1 − α ) K ∗ = S ′ + L ′ ( 2 H + r 1 − α ) K_*=S'+L'(2H+r^{1-\alpha}) K ∗ = S ′ + L ′ ( 2 H + r 1 − α ) a h a_h a h f h f_h f h C 0 , α ( B ) C^{0,\alpha}(B) C 0 , α ( B ) C 0 , α ( B ) C^{0,\alpha}(B) C 0 , α ( B ) K ∗ K_* K ∗ K ∗ K_* K ∗ h h h v h ∈ C 2 ( B ) v_h\in C^2(B) v h ∈ C 2 ( B ) §E15.36 定理 3.1 を整数k = 0 k=0 k = 0 B B B B ′ ′ ⋐ B B''\Subset B B ′′ ⋐ B A = a h A=a_h A = a h b = 0 b=0 b = 0 c = 0 c=0 c = 0 K ∗ K_* K ∗ − f h -f_h − f h n , α , λ , Λ , B ′ ′ , B , K ∗ n,\alpha,\lambda,\Lambda,B'',B,K_* n , α , λ , Λ , B ′′ , B , K ∗ h h h C C C
∥ v h ∥ C 2 , α ( B ′ ′ ) ≤ C ( ∥ v h ∥ C 0 ( B ) + ∥ f h ∥ C 0 , α ( B ) ) \|v_h\|_{C^{2,\alpha}(B'')}\leq C\bigl(\|v_h\|_{C^0(B)}+\|f_h\|_{C^{0,\alpha}(B)}\bigr) ∥ v h ∥ C 2 , α ( B ′′ ) ≤ C ( ∥ v h ∥ C 0 ( B ) + ∥ f h ∥ C 0 , α ( B ) ) が成り立つ。平均値の定理により∥ v h ∥ C 0 ( B ) ≤ sup B ′ ‾ ∣ D l u ∣ \|v_h\|_{C^0(B)}\leq\sup_{\overline{B'}}|D_lu| ∥ v h ∥ C 0 ( B ) ≤ sup B ′ ∣ D l u ∣ { v h } 0 < ∣ h ∣ ≤ h 0 \{v_h\}_{0<|h|\leq h_0} { v h } 0 < ∣ h ∣ ≤ h 0 C 2 , α ( B ′ ′ ‾ ) C^{2,\alpha}(\overline{B''}) C 2 , α ( B ′′ ) h j → 0 h_j\to0 h j → 0 §E15.35 定理 2.2 により、ある部分列v h j ′ v_{h_{j'}} v h j ′ C 2 ( B ′ ′ ‾ ) C^2(\overline{B''}) C 2 ( B ′′ ) v v v u ∈ C 1 ( Ω ) u\in C^1(\Omega) u ∈ C 1 ( Ω ) v h → D l u v_h\to D_lu v h → D l u v = D l u v=D_lu v = D l u D l u ∈ C 2 ( B ′ ′ ) D_lu\in C^2(B'') D l u ∈ C 2 ( B ′′ ) D 2 v h j ′ → D 2 D l u D^2v_{h_{j'}}\to D^2D_lu D 2 v h j ′ → D 2 D l u B ′ ′ B'' B ′′ x , y ∈ B ′ ′ x,y\in B'' x , y ∈ B ′′ ∣ D 2 v h j ′ ( x ) − D 2 v h j ′ ( y ) ∣ |D^2v_{h_{j'}}(x)-D^2v_{h_{j'}}(y)| ∣ D 2 v h j ′ ( x ) − D 2 v h j ′ ( y ) ∣ sup h ∥ v h ∥ C 2 , α ( B ′ ′ ) ∣ x − y ∣ α \sup_h\|v_h\|_{C^{2,\alpha}(B'')}|x-y|^\alpha sup h ∥ v h ∥ C 2 , α ( B ′′ ) ∣ x − y ∣ α D l u ∈ C 2 , α ( B ′ ′ ‾ ) D_lu\in C^{2,\alpha}(\overline{B''}) D l u ∈ C 2 , α ( B ′′ ) l l l u ∈ C 3 , α ( B r / 4 ( z ) ‾ ) u\in C^{3,\alpha}(\overline{B_{r/4}(z)}) u ∈ C 3 , α ( B r /4 ( z ) ) ( S 1 ) (\mathrm S_1) ( S 1 ) ▨
主張 6.1.4. m ≥ 1 m\geq1 m ≥ 1 ( S m ) (\mathrm S_m) ( S m ) ( S m + 1 ) (\mathrm S_{m+1}) ( S m + 1 )
証明. z ∈ Ω z\in\Omega z ∈ Ω ( S m ) (\mathrm S_m) ( S m ) r > 0 r>0 r > 0 B r ( z ) ‾ ⊆ Ω \overline{B_r(z)}\subseteq\Omega B r ( z ) ⊆ Ω u ∈ C m + 2 , α ( B r ( z ) ‾ ) u\in C^{m+2,\alpha}(\overline{B_r(z)}) u ∈ C m + 2 , α ( B r ( z ) ) B = B r ( z ) B=B_r(z) B = B r ( z ) m + 2 ≥ 3 m+2\geq3 m + 2 ≥ 3 x ↦ F ( D 2 u ( x ) , x ) x\mapsto F(D^2u(x),x) x ↦ F ( D 2 u ( x ) , x ) B B B C 1 C^1 C 1 F ( D 2 u ( x ) , x ) = 0 F(D^2u(x),x)=0 F ( D 2 u ( x ) , x ) = 0 x l x_l x l B B B
a ( D 2 u ( x ) , x ) : D 2 ( D l u ) ( x ) + ∂ l F ( D 2 u ( x ) , x ) = 0 a(D^2u(x),x):D^2(D_lu)(x)+\partial_lF(D^2u(x),x)=0 a ( D 2 u ( x ) , x ) : D 2 ( D l u ) ( x ) + ∂ l F ( D 2 u ( x ) , x ) = 0 が成り立つ。V ( x ) = ( D 2 u ( x ) , x ) V(x)=(D^2u(x),x) V ( x ) = ( D 2 u ( x ) , x ) C m , α ( B ‾ ) C^{m,\alpha}(\overline B) C m , α ( B ) V ( B ‾ ) ⊆ O V(\overline B)\subseteq\mathcal O V ( B ) ⊆ O 主張 6.1.2 によりA ( x ) = a ( D 2 u ( x ) , x ) A(x)=a(D^2u(x),x) A ( x ) = a ( D 2 u ( x ) , x ) ∂ l F ( D 2 u ( x ) , x ) \partial_lF(D^2u(x),x) ∂ l F ( D 2 u ( x ) , x ) C m , α ( B ‾ ) C^{m,\alpha}(\overline B) C m , α ( B ) D l u ∈ C 2 ( B ) D_lu\in C^2(B) D l u ∈ C 2 ( B ) A A A λ I ≤ A ≤ Λ I \lambda I\leq A\leq\Lambda I λ I ≤ A ≤ Λ I §E15.36 定理 3.1 を整数k = m k=m k = m B B B b = 0 b=0 b = 0 c = 0 c=0 c = 0 ∥ D l u ∥ C m + 2 , α ( B r / 2 ( z ) ) \|D_lu\|_{C^{m+2,\alpha}(B_{r/2}(z))} ∥ D l u ∥ C m + 2 , α ( B r /2 ( z )) l l l u ∈ C m + 3 , α ( B r / 2 ( z ) ‾ ) u\in C^{m+3,\alpha}(\overline{B_{r/2}(z)}) u ∈ C m + 3 , α ( B r /2 ( z ) ) ( S m + 1 ) (\mathrm S_{m+1}) ( S m + 1 ) ▨
仮定により( S 0 ) (\mathrm S_0) ( S 0 ) 主張 6.1.3 と主張 6.1.4 により、すべての整数m ≥ 0 m\geq0 m ≥ 0 ( S m ) (\mathrm S_m) ( S m ) j ≥ 0 j\geq0 j ≥ 0 u ∈ C j ( Ω ) u\in C^j(\Omega) u ∈ C j ( Ω ) u ∈ C ∞ ( Ω ) u\in C^\infty(\Omega) u ∈ C ∞ ( Ω ) ▨
7 凹な作用素の例 補題 7.1. n ≥ 1 n\geq1 n ≥ 1 G G G n n n H H H n n n G + t H G+tH G + t H t t t 0 0 0 J J J φ ( t ) = log det ( G + t H ) \varphi(t)=\log\det(G+tH) φ ( t ) = log det ( G + t H ) J J J C ∞ C^\infty C ∞
φ ′ ( 0 ) = tr ( G − 1 H ) , φ ′ ′ ( 0 ) = − tr ( G − 1 H G − 1 H ) ≤ 0 \varphi'(0)=\operatorname{tr}(G^{-1}H),\qquad\varphi''(0)=-\operatorname{tr}(G^{-1}HG^{-1}H)\leq0 φ ′ ( 0 ) = tr ( G − 1 H ) , φ ′′ ( 0 ) = − tr ( G − 1 H G − 1 H ) ≤ 0 を満たす。さらに、0 < κ ≤ K 0<\kappa\leq K 0 < κ ≤ K κ I ≤ G ≤ K I \kappa I\leq G\leq KI κ I ≤ G ≤ K I H ≥ 0 H\geq0 H ≥ 0 K − 1 tr H ≤ tr ( G − 1 H ) ≤ κ − 1 tr H K^{-1}\operatorname{tr}H\leq\operatorname{tr}(G^{-1}H)\leq\kappa^{-1}\operatorname{tr}H K − 1 tr H ≤ tr ( G − 1 H ) ≤ κ − 1 tr H
証明. G 1 / 2 G^{1/2} G 1/2 G G G X = G − 1 / 2 H G − 1 / 2 X=G^{-1/2}HG^{-1/2} X = G − 1/2 H G − 1/2 X X X μ 1 , … , μ n \mu_1,\dots,\mu_n μ 1 , … , μ n G + t H = G 1 / 2 ( I + t X ) G 1 / 2 G+tH=G^{1/2}(I+tX)G^{1/2} G + t H = G 1/2 ( I + tX ) G 1/2 G + t H G+tH G + t H i i i 1 + t μ i > 0 1+t\mu_i>0 1 + t μ i > 0 J J J 0 0 0 t ∈ J t\in J t ∈ J det ( G + t H ) = det G ∏ i ( 1 + t μ i ) \det(G+tH)=\det G\prod_i(1+t\mu_i) det ( G + t H ) = det G ∏ i ( 1 + t μ i ) φ ( t ) = log det G + ∑ i log ( 1 + t μ i ) \varphi(t)=\log\det G+\sum_i\log(1+t\mu_i) φ ( t ) = log det G + ∑ i log ( 1 + t μ i ) φ \varphi φ C ∞ C^\infty C ∞ φ ′ ( 0 ) = ∑ i μ i = tr X = tr ( G − 1 H ) \varphi'(0)=\sum_i\mu_i=\operatorname{tr}X=\operatorname{tr}(G^{-1}H) φ ′ ( 0 ) = ∑ i μ i = tr X = tr ( G − 1 H ) φ ′ ′ ( 0 ) = − ∑ i μ i 2 = − tr ( X 2 ) = − tr ( G − 1 H G − 1 H ) \varphi''(0)=-\sum_i\mu_i^2=-\operatorname{tr}(X^2)=-\operatorname{tr}(G^{-1}HG^{-1}H) φ ′′ ( 0 ) = − ∑ i μ i 2 = − tr ( X 2 ) = − tr ( G − 1 H G − 1 H )
κ I ≤ G ≤ K I \kappa I\leq G\leq KI κ I ≤ G ≤ K I K − 1 I ≤ G − 1 ≤ κ − 1 I K^{-1}I\leq G^{-1}\leq\kappa^{-1}I K − 1 I ≤ G − 1 ≤ κ − 1 I H ≥ 0 H\geq0 H ≥ 0 tr ( G − 1 H ) = tr ( H 1 / 2 G − 1 H 1 / 2 ) \operatorname{tr}(G^{-1}H)=\operatorname{tr}(H^{1/2}G^{-1}H^{1/2}) tr ( G − 1 H ) = tr ( H 1/2 G − 1 H 1/2 ) K − 1 H ≤ H 1 / 2 G − 1 H 1 / 2 ≤ κ − 1 H K^{-1}H\leq H^{1/2}G^{-1}H^{1/2}\leq\kappa^{-1}H K − 1 H ≤ H 1/2 G − 1 H 1/2 ≤ κ − 1 H ▨
例 7.2. U ⊆ R n U\subseteq\R^n U ⊆ R n B 1 ‾ ⊆ U \overline{B_1}\subseteq U B 1 ⊆ U A ∈ C 2 ( U ; S n ) A\in C^2(U;\mathcal S^n) A ∈ C 2 ( U ; S n ) U U U λ I ≤ A ≤ Λ I \lambda I\leq A\leq\Lambda I λ I ≤ A ≤ Λ I f ∈ C 2 ( U ) f\in C^2(U) f ∈ C 2 ( U ) F ( M , x ) = A ( x ) : M − f ( x ) F(M,x)=A(x):M-f(x) F ( M , x ) = A ( x ) : M − f ( x ) M M M a ( M , x ) = A ( x ) a(M,x)=A(x) a ( M , x ) = A ( x ) §E15.42 命題 3.4 (1) によりF F F S n × U \mathcal S^n\times U S n × U λ , Λ \lambda,\Lambda λ , Λ ∂ ξ F ( M , x ) = ∂ ξ A ( x ) : M − f ξ ( x ) \partial_\xi F(M,x)=\partial_\xi A(x):M-f_\xi(x) ∂ ξ F ( M , x ) = ∂ ξ A ( x ) : M − f ξ ( x ) B ξ = ∂ ξ A B_\xi=\partial_\xi A B ξ = ∂ ξ A ∂ ξ ∂ η F = ∂ ξ ∂ η A : M − f ξ η \partial_\xi\partial_\eta F=\partial_\xi\partial_\eta A:M-f_{\xi\eta} ∂ ξ ∂ η F = ∂ ξ ∂ η A : M − f ξ η 注意 5.2 の上界はK 0 K_0 K 0 A , f A,f A , f B 1 ‾ \overline{B_1} B 1 A : D 2 u = f A:D^2u=f A : D 2 u = f C 4 ( B 1 ) C^4(B_1) C 4 ( B 1 ) ∣ D 2 u ∣ ≤ K 0 |D^2u|\leq K_0 ∣ D 2 u ∣ ≤ K 0 定理 5.1 の評価が成り立つ。
この線形の場合には、A A A f f f B 1 ‾ \overline{B_1} B 1 C 1 C^1 C 1 α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) C 0 , α ( B 1 ) C^{0,\alpha}(B_1) C 0 , α ( B 1 ) §E15.36 定理 3.1 をk = 0 k=0 k = 0 A : D 2 u = f A:D^2u=f A : D 2 u = f C 2 ( B 1 ) C^2(B_1) C 2 ( B 1 ) ∥ u ∥ C 2 , α ( B 1 / 2 ) \|u\|_{C^{2,\alpha}(B_{1/2})} ∥ u ∥ C 2 , α ( B 1/2 ) ∥ u ∥ C 0 ( B 1 ) + ∥ f ∥ C 0 , α ( B 1 ) \|u\|_{C^0(B_1)}+\|f\|_{C^{0,\alpha}(B_1)} ∥ u ∥ C 0 ( B 1 ) + ∥ f ∥ C 0 , α ( B 1 ) A A A a ( D 2 u ( x ) , x ) a(D^2u(x),x) a ( D 2 u ( x ) , x ) D 2 u D^2u D 2 u 定理 5.1 の証明では、線形化した係数a ( D 2 u ( x ) , x ) a(D^2u(x),x) a ( D 2 u ( x ) , x ) §E15.42 定理 2.1 から二階 Hölder 評価を導く。
例 7.3. J ≥ 1 J\geq1 J ≥ 1 a 1 , … , a J ∈ S n a_1,\dots,a_J\in\mathcal S^n a 1 , … , a J ∈ S n λ I ≤ a ν ≤ Λ I \lambda I\leq a_\nu\leq\Lambda I λ I ≤ a ν ≤ Λ I b 1 , … , b J ∈ R b_1,\dots,b_J\in\R b 1 , … , b J ∈ R ℓ ν ( M ) = a ν : M + b ν \ell_\nu(M)=a_\nu:M+b_\nu ℓ ν ( M ) = a ν : M + b ν F ( M ) = min ν ℓ ν ( M ) F(M)=\min_\nu\ell_\nu(M) F ( M ) = min ν ℓ ν ( M ) F F F M ∈ S n M\in\mathcal S^n M ∈ S n N ≥ 0 N\geq0 N ≥ 0 F ( M ) = ℓ ν ( M ) F(M)=\ell_\nu(M) F ( M ) = ℓ ν ( M ) F ( M + N ) = ℓ μ ( M + N ) F(M+N)=\ell_\mu(M+N) F ( M + N ) = ℓ μ ( M + N ) ν , μ \nu,\mu ν , μ F ( M + N ) ≤ ℓ ν ( M + N ) = F ( M ) + a ν : N ≤ F ( M ) + Λ tr N F(M+N)\leq\ell_\nu(M+N)=F(M)+a_\nu:N\leq F(M)+\Lambda\operatorname{tr}N F ( M + N ) ≤ ℓ ν ( M + N ) = F ( M ) + a ν : N ≤ F ( M ) + Λ tr N F ( M ) ≤ ℓ μ ( M ) = F ( M + N ) − a μ : N ≤ F ( M + N ) − λ tr N F(M)\leq\ell_\mu(M)=F(M+N)-a_\mu:N\leq F(M+N)-\lambda\operatorname{tr}N F ( M ) ≤ ℓ μ ( M ) = F ( M + N ) − a μ : N ≤ F ( M + N ) − λ tr N F F F λ , Λ \lambda,\Lambda λ , Λ n = 1 n=1 n = 1 J = 2 J=2 J = 2 λ < Λ \lambda<\Lambda λ < Λ a 1 = λ a_1=\lambda a 1 = λ a 2 = Λ a_2=\Lambda a 2 = Λ b 1 = b 2 = 0 b_1=b_2=0 b 1 = b 2 = 0 F ( M ) F(M) F ( M ) M ≥ 0 M\geq0 M ≥ 0 λ M \lambda M λ M M ≤ 0 M\leq0 M ≤ 0 Λ M \Lambda M Λ M M = 0 M=0 M = 0
ε > 0 \varepsilon>0 ε > 0 F ε ( M ) = − ε log ∑ ν exp ( − ℓ ν ( M ) / ε ) F_\varepsilon(M)=-\varepsilon\log\sum_\nu\exp(-\ell_\nu(M)/\varepsilon) F ε ( M ) = − ε log ∑ ν exp ( − ℓ ν ( M ) / ε ) exp ( − F ( M ) / ε ) ≤ ∑ ν exp ( − ℓ ν ( M ) / ε ) ≤ J exp ( − F ( M ) / ε ) \exp(-F(M)/\varepsilon)\leq\sum_\nu\exp(-\ell_\nu(M)/\varepsilon)\leq J\exp(-F(M)/\varepsilon) exp ( − F ( M ) / ε ) ≤ ∑ ν exp ( − ℓ ν ( M ) / ε ) ≤ J exp ( − F ( M ) / ε ) F − ε log J ≤ F ε ≤ F F-\varepsilon\log J\leq F_\varepsilon\leq F F − ε log J ≤ F ε ≤ F p ν ( M ) = exp ( − ℓ ν ( M ) / ε ) / ∑ μ exp ( − ℓ μ ( M ) / ε ) p_\nu(M)=\exp(-\ell_\nu(M)/\varepsilon)/\sum_\mu\exp(-\ell_\mu(M)/\varepsilon) p ν ( M ) = exp ( − ℓ ν ( M ) / ε ) / ∑ μ exp ( − ℓ μ ( M ) / ε ) p ν > 0 p_\nu>0 p ν > 0 ∑ ν p ν = 1 \sum_\nu p_\nu=1 ∑ ν p ν = 1
d d t F ε ( M + t N ) ∣ t = 0 = ∑ ν p ν ( M ) a ν : N , d 2 d t 2 F ε ( M + t N ) ∣ t = 0 = − 1 ε ( ∑ ν p ν ( a ν : N ) 2 − ( ∑ ν p ν a ν : N ) 2 ) \frac{d}{dt}F_\varepsilon(M+tN)\Big|_{t=0}=\sum_\nu p_\nu(M)\,a_\nu:N,\qquad
\frac{d^2}{dt^2}F_\varepsilon(M+tN)\Big|_{t=0}=-\frac1\varepsilon\Bigl(\sum_\nu p_\nu(a_\nu:N)^2-\Bigl(\sum_\nu p_\nu\,a_\nu:N\Bigr)^2\Bigr) d t d F ε ( M + tN ) t = 0 = ν ∑ p ν ( M ) a ν : N , d t 2 d 2 F ε ( M + tN ) t = 0 = − ε 1 ( ν ∑ p ν ( a ν : N ) 2 − ( ν ∑ p ν a ν : N ) 2 ) である。第二式の括弧内は確率p ν p_\nu p ν a ν : N a_\nu:N a ν : N 0 0 0 F ε F_\varepsilon F ε D F ε ( M ) [ N ] = a ( M ) : N DF_\varepsilon(M)[N]=a(M):N D F ε ( M ) [ N ] = a ( M ) : N a ( M ) = ∑ ν p ν ( M ) a ν a(M)=\sum_\nu p_\nu(M)a_\nu a ( M ) = ∑ ν p ν ( M ) a ν λ I ≤ a ( M ) ≤ Λ I \lambda I\leq a(M)\leq\Lambda I λ I ≤ a ( M ) ≤ Λ I §E15.42 命題 3.4 (2) によりF ε F_\varepsilon F ε λ , Λ \lambda,\Lambda λ , Λ 定理 3.1 のα \alpha α C C C n , λ , Λ n,\lambda,\Lambda n , λ , Λ F ε ( D 2 u ) = f F_\varepsilon(D^2u)=f F ε ( D 2 u ) = f C 4 ( B 1 ) C^4(B_1) C 4 ( B 1 ) ε \varepsilon ε
例 7.4. 0 < κ < K 0<\kappa<K 0 < κ < K C = { M ∈ S n ∣ κ I < M < K I } \mathcal C=\{M\in\mathcal S^n\mid\kappa I<M<KI\} C = { M ∈ S n ∣ κ I < M < K I } C \mathcal C C 補題 7.1 により、F ( M ) = log det M F(M)=\log\det M F ( M ) = log det M C \mathcal C C C ∞ C^\infty C ∞ C \mathcal C C 0 0 0 D F ( M ) [ N ] = tr ( M − 1 N ) DF(M)[N]=\operatorname{tr}(M^{-1}N) D F ( M ) [ N ] = tr ( M − 1 N ) a ( M ) = M − 1 a(M)=M^{-1} a ( M ) = M − 1 K − 1 I ≤ a ( M ) ≤ κ − 1 I K^{-1}I\leq a(M)\leq\kappa^{-1}I K − 1 I ≤ a ( M ) ≤ κ − 1 I §E15.42 命題 3.4 (1) によりF F F C \mathcal C C K − 1 , κ − 1 K^{-1},\kappa^{-1} K − 1 , κ − 1 f ∈ C 2 ( B 1 ) f\in C^2(B_1) f ∈ C 2 ( B 1 ) x ∈ B 1 x\in B_1 x ∈ B 1 κ I < D 2 u ( x ) < K I \kappa I<D^2u(x)<KI κ I < D 2 u ( x ) < K I det D 2 u = e f \det D^2u=e^f det D 2 u = e f u ∈ C 4 ( B 1 ) u\in C^4(B_1) u ∈ C 4 ( B 1 ) 定理 3.1 の評価が成り立つ。
実対称な正定値行列全体P \mathcal P P F F F 0 < λ ≤ Λ 0<\lambda\leq\Lambda 0 < λ ≤ Λ P \mathcal P P t > 0 t>0 t > 0 a ( t I ) = t − 1 I a(tI)=t^{-1}I a ( t I ) = t − 1 I t > 1 / λ t>1/\lambda t > 1/ λ a ( t I ) ≥ λ I a(tI)\geq\lambda I a ( t I ) ≥ λ I §E15.42 命題 3.4 (1) によりF F F P \mathcal P P λ , Λ \lambda,\Lambda λ , Λ
8 複素 Monge–Ampère 方程式への局所適用 命題 8.1. n ≥ 1 n\geq1 n ≥ 1 R 2 n \R^{2n} R 2 n ( x 1 , … , x n , y 1 , … , y n ) (x_1,\dots,x_n,y_1,\dots,y_n) ( x 1 , … , x n , y 1 , … , y n ) z j = x j + i y j z_j=x_j+iy_j z j = x j + i y j M ∈ S 2 n M\in\mathcal S^{2n} M ∈ S 2 n n n n P ( M ) P(M) P ( M )
P ( M ) i j ˉ = 1 4 ( M x i x j + M y i y j + i ( M x i y j − M y i x j ) ) P(M)_{i\bar j}=\tfrac14\bigl(M_{x_ix_j}+M_{y_iy_j}+i(M_{x_iy_j}-M_{y_ix_j})\bigr) P ( M ) i j ˉ = 4 1 ( M x i x j + M y i y j + i ( M x i y j − M y i x j ) ) で定める。n n n G G G G > 0 G>0 G > 0 tr R \operatorname{tr}_{\R} tr R tr C \operatorname{tr}_{\C} tr C S 2 n \mathcal S^{2n} S 2 n n n n
P P P P ( M ) P(M) P ( M ) ζ = α + i β ∈ C n \zeta=\alpha+i\beta\in\C^n ζ = α + i β ∈ C n α , β ∈ R n \alpha,\beta\in\R^n α , β ∈ R n v = ( α , β ) v=(\alpha,\beta) v = ( α , β ) w = ( β , − α ) ∈ R 2 n w=(\beta,-\alpha)\in\R^{2n} w = ( β , − α ) ∈ R 2 n ∑ i , j P ( M ) i j ˉ ζ i ζ ˉ j = 1 4 ( v ⋅ M v + w ⋅ M w ) \sum_{i,j}P(M)_{i\bar j}\zeta_i\bar\zeta_j=\frac14(v\cdot Mv+w\cdot Mw) ∑ i , j P ( M ) i j ˉ ζ i ζ ˉ j = 4 1 ( v ⋅ M v + w ⋅ M w ) N ≥ 0 N\geq0 N ≥ 0 P ( N ) ≥ 0 P(N)\geq0 P ( N ) ≥ 0 tr C P ( N ) = 1 4 tr R N \operatorname{tr}_{\C}P(N)=\frac14\operatorname{tr}_{\R}N tr C P ( N ) = 4 1 tr R N g g g n n n G = { M ∈ S 2 n ∣ g + P ( M ) > 0 } \mathcal G=\{M\in\mathcal S^{2n}\mid g+P(M)>0\} G = { M ∈ S 2 n ∣ g + P ( M ) > 0 } G \mathcal G G Φ ( M ) = log det ( g + P ( M ) ) \Phi(M)=\log\det(g+P(M)) Φ ( M ) = log det ( g + P ( M )) G \mathcal G G C ∞ C^\infty C ∞ D Φ ( M ) [ N ] = tr C ( ( g + P ( M ) ) − 1 P ( N ) ) D\Phi(M)[N]=\operatorname{tr}_{\C}\bigl((g+P(M))^{-1}P(N)\bigr) D Φ ( M ) [ N ] = tr C ( ( g + P ( M ) ) − 1 P ( N ) ) 0 < κ ≤ K 0<\kappa\leq K 0 < κ ≤ K M ∈ G M\in\mathcal G M ∈ G κ I ≤ g + P ( M ) ≤ K I \kappa I\leq g+P(M)\leq KI κ I ≤ g + P ( M ) ≤ K I N ≥ 0 N\geq0 N ≥ 0 N ∈ S 2 n N\in\mathcal S^{2n} N ∈ S 2 n 1 4 K tr R N ≤ D Φ ( M ) [ N ] ≤ 1 4 κ tr R N \frac1{4K}\operatorname{tr}_{\R}N\leq D\Phi(M)[N]\leq\frac1{4\kappa}\operatorname{tr}_{\R}N 4 K 1 tr R N ≤ D Φ ( M ) [ N ] ≤ 4 κ 1 tr R N 証明. 主張 (1) を示す。P P P M M M M y j x i = M x i y j M_{y_jx_i}=M_{x_iy_j} M y j x i = M x i y j M x j y i = M y i x j M_{x_jy_i}=M_{y_ix_j} M x j y i = M y i x j P ( M ) j i ˉ ‾ = 1 4 ( M x i x j + M y i y j − i ( M x j y i − M y j x i ) ) = P ( M ) i j ˉ \overline{P(M)_{j\bar i}}=\frac14(M_{x_ix_j}+M_{y_iy_j}-i(M_{x_jy_i}-M_{y_jx_i}))=P(M)_{i\bar j} P ( M ) j i ˉ = 4 1 ( M x i x j + M y i y j − i ( M x j y i − M y j x i )) = P ( M ) i j ˉ P ( M ) P(M) P ( M ) X i j = M x i x j X_{ij}=M_{x_ix_j} X ij = M x i x j Y i j = M y i y j Y_{ij}=M_{y_iy_j} Y ij = M y i y j Z i j = M x i y j Z_{ij}=M_{x_iy_j} Z ij = M x i y j M y i x j = Z j i M_{y_ix_j}=Z_{ji} M y i x j = Z j i ζ i ζ ˉ j = ( α i α j + β i β j ) + i ( β i α j − α i β j ) \zeta_i\bar\zeta_j=(\alpha_i\alpha_j+\beta_i\beta_j)+i(\beta_i\alpha_j-\alpha_i\beta_j) ζ i ζ ˉ j = ( α i α j + β i β j ) + i ( β i α j − α i β j ) P ( M ) P(M) P ( M ) ∑ i , j P ( M ) i j ˉ ζ i ζ ˉ j \sum_{i,j}P(M)_{i\bar j}\zeta_i\bar\zeta_j ∑ i , j P ( M ) i j ˉ ζ i ζ ˉ j
1 4 ∑ i , j ( ( X i j + Y i j ) ( α i α j + β i β j ) − ( Z i j − Z j i ) ( β i α j − α i β j ) ) = 1 4 ( α ⋅ X α + β ⋅ X β + α ⋅ Y α + β ⋅ Y β + 2 α ⋅ Z β − 2 β ⋅ Z α ) \frac14\sum_{i,j}\Bigl((X_{ij}+Y_{ij})(\alpha_i\alpha_j+\beta_i\beta_j)-(Z_{ij}-Z_{ji})(\beta_i\alpha_j-\alpha_i\beta_j)\Bigr)
=\frac14\bigl(\alpha\cdot X\alpha+\beta\cdot X\beta+\alpha\cdot Y\alpha+\beta\cdot Y\beta+2\alpha\cdot Z\beta-2\beta\cdot Z\alpha\bigr) 4 1 i , j ∑ ( ( X ij + Y ij ) ( α i α j + β i β j ) − ( Z ij − Z j i ) ( β i α j − α i β j ) ) = 4 1 ( α ⋅ X α + β ⋅ X β + α ⋅ Y α + β ⋅ Y β + 2 α ⋅ Z β − 2 β ⋅ Z α ) である。一方v ⋅ M v = α ⋅ X α + 2 α ⋅ Z β + β ⋅ Y β v\cdot Mv=\alpha\cdot X\alpha+2\alpha\cdot Z\beta+\beta\cdot Y\beta v ⋅ M v = α ⋅ X α + 2 α ⋅ Z β + β ⋅ Y β w ⋅ M w = β ⋅ X β − 2 β ⋅ Z α + α ⋅ Y α w\cdot Mw=\beta\cdot X\beta-2\beta\cdot Z\alpha+\alpha\cdot Y\alpha w ⋅ M w = β ⋅ X β − 2 β ⋅ Z α + α ⋅ Y α η = ζ ˉ \eta=\bar\zeta η = ζ ˉ η ∗ P ( M ) η \eta^*P(M)\eta η ∗ P ( M ) η N ≥ 0 N\geq0 N ≥ 0 M = N M=N M = N 0 0 0 ζ \zeta ζ P ( N ) ≥ 0 P(N)\geq0 P ( N ) ≥ 0 tr C P ( N ) = 1 4 ∑ i ( N x i x i + N y i y i ) = 1 4 tr R N \operatorname{tr}_{\C}P(N)=\frac14\sum_i(N_{x_ix_i}+N_{y_iy_i})=\frac14\operatorname{tr}_{\R}N tr C P ( N ) = 4 1 ∑ i ( N x i x i + N y i y i ) = 4 1 tr R N
主張 (2) を示す。正定値 Hermite 行列の全体は凸な開集合であり、P P P G \mathcal G G det \det det Φ \Phi Φ C ∞ C^\infty C ∞ M ∈ G M\in\mathcal G M ∈ G N ∈ S 2 n N\in\mathcal S^{2n} N ∈ S 2 n G = g + P ( M ) G=g+P(M) G = g + P ( M ) H = P ( N ) H=P(N) H = P ( N ) Φ ( M + t N ) = log det ( G + t H ) \Phi(M+tN)=\log\det(G+tH) Φ ( M + tN ) = log det ( G + t H ) 補題 7.1 により、t = 0 t=0 t = 0 tr C ( G − 1 H ) \operatorname{tr}_{\C}(G^{-1}H) tr C ( G − 1 H ) 0 0 0 M ∈ G M\in\mathcal G M ∈ G G \mathcal G G Φ \Phi Φ 0 0 0 Φ \Phi Φ κ I ≤ G ≤ K I \kappa I\leq G\leq KI κ I ≤ G ≤ K I N ≥ 0 N\geq0 N ≥ 0 主張 (1) によりH ≥ 0 H\geq0 H ≥ 0 tr C H = 1 4 tr R N \operatorname{tr}_{\C}H=\frac14\operatorname{tr}_{\R}N tr C H = 4 1 tr R N 補題 7.1 の最後の不等式から主張の評価を得る。▨
系 8.2. n ≥ 1 n\geq1 n ≥ 1 0 < κ ≤ K 0<\kappa\leq K 0 < κ ≤ K K 0 ≥ 0 K_0\geq0 K 0 ≥ 0 Θ ≥ 0 \Theta\geq0 Θ ≥ 0 P P P 命題 8.1 の写像とする。B 1 = B 1 ( 0 ) ⊆ R 2 n B_1=B_1(0)\subseteq\R^{2n} B 1 = B 1 ( 0 ) ⊆ R 2 n M ∈ S 2 n M\in\mathcal S^{2n} M ∈ S 2 n ∣ M ∣ = ( M : M ) 1 / 2 |M|=(M:M)^{1/2} ∣ M ∣ = ( M : M ) 1/2 U ⊆ R 2 n U\subseteq\R^{2n} U ⊆ R 2 n B 1 ‾ ⊆ U \overline{B_1}\subseteq U B 1 ⊆ U g ∈ C 2 ( U ) g\in C^2(U) g ∈ C 2 ( U ) n n n h ∈ C 2 ( U ) h\in C^2(U) h ∈ C 2 ( U ) g g g h h h B 1 B_1 B 1 Θ \Theta Θ u ∈ C 4 ( B 1 ) u\in C^4(B_1) u ∈ C 4 ( B 1 ) p ∈ B 1 p\in B_1 p ∈ B 1 ∣ D 2 u ( p ) ∣ ≤ K 0 |D^2u(p)|\leq K_0 ∣ D 2 u ( p ) ∣ ≤ K 0 κ I ≤ G ( p ) ≤ K I \kappa I\leq G(p)\leq KI κ I ≤ G ( p ) ≤ K I log det G ( p ) = h ( p ) \log\det G(p)=h(p) log det G ( p ) = h ( p ) G ( p ) = g ( p ) + P ( D 2 u ( p ) ) G(p)=g(p)+P(D^2u(p)) G ( p ) = g ( p ) + P ( D 2 u ( p )) n , κ , K , K 0 , Θ n,\kappa,K,K_0,\Theta n , κ , K , K 0 , Θ α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) C C C
sup B 1 / 2 ∣ D 2 u ∣ + sup p , q ∈ B 1 / 2 p ≠ q ∣ D 2 u ( p ) − D 2 u ( q ) ∣ ∣ p − q ∣ α ≤ C \sup_{B_{1/2}}|D^2u|+\sup_{\substack{p,q\in B_{1/2}\\p\neq q}}\frac{|D^2u(p)-D^2u(q)|}{|p-q|^\alpha}\leq C B 1/2 sup ∣ D 2 u ∣ + p , q ∈ B 1/2 p = q sup ∣ p − q ∣ α ∣ D 2 u ( p ) − D 2 u ( q ) ∣ ≤ C である。
証明. O = { ( M , q ) ∈ S 2 n × U ∣ ( κ / 2 ) I < g ( q ) + P ( M ) < 2 K I } \mathcal O=\{(M,q)\in\mathcal S^{2n}\times U\mid(\kappa/2)I<g(q)+P(M)<2KI\} O = {( M , q ) ∈ S 2 n × U ∣ ( κ /2 ) I < g ( q ) + P ( M ) < 2 K I } O \mathcal O O F ( M , q ) = log det ( g ( q ) + P ( M ) ) − h ( q ) F(M,q)=\log\det(g(q)+P(M))-h(q) F ( M , q ) = log det ( g ( q ) + P ( M )) − h ( q ) O \mathcal O O ( κ / 2 , 2 K ) (\kappa/2,2K) ( κ /2 , 2 K ) n n n P P P O q \mathcal O_q O q F ∈ C 2 ( O ) F\in C^2(\mathcal O) F ∈ C 2 ( O ) 命題 8.1 (2) によりF F F ( M , q ) ∈ O (M,q)\in\mathcal O ( M , q ) ∈ O ξ ∈ R 2 n \xi\in\R^{2n} ξ ∈ R 2 n 命題 8.1 (2) をκ / 2 , 2 K \kappa/2,2K κ /2 , 2 K N = ξ ⊗ ξ N=\xi\otimes\xi N = ξ ⊗ ξ a ( M , q ) : ξ ⊗ ξ a(M,q):\xi\otimes\xi a ( M , q ) : ξ ⊗ ξ ∣ ξ ∣ 2 / ( 8 K ) |\xi|^2/(8K) ∣ ξ ∣ 2 / ( 8 K ) 2 ∣ ξ ∣ 2 / κ 2|\xi|^2/\kappa 2∣ ξ ∣ 2 / κ §E15.42 命題 3.4 (1) によりF F F O \mathcal O O 1 / ( 8 K ) , 2 / κ 1/(8K),2/\kappa 1/ ( 8 K ) , 2/ κ
n n n X X X ∥ X ∥ \|X\| ∥ X ∥ c c c n n n n c nc n c n n n X , Y X,Y X , Y ∣ tr C ( X Y ) ∣ ≤ n ∥ X ∥ ∥ Y ∥ |\operatorname{tr}_{\C}(XY)|\leq n\|X\|\|Y\| ∣ tr C ( X Y ) ∣ ≤ n ∥ X ∥∥ Y ∥ p , q ∈ B 1 p,q\in B_1 p , q ∈ B 1 g ( q ) − g ( p ) g(q)-g(p) g ( q ) − g ( p ) 2 n Θ ∣ p − q ∣ \sqrt{2n}\,\Theta|p-q| 2 n Θ∣ p − q ∣ ∥ g ( q ) − g ( p ) ∥ ≤ n 2 n Θ ∣ p − q ∣ \|g(q)-g(p)\|\leq n\sqrt{2n}\,\Theta|p-q| ∥ g ( q ) − g ( p ) ∥ ≤ n 2 n Θ∣ p − q ∣ δ = κ / ( 2 n 2 n Θ + 1 ) \delta=\kappa/(2n\sqrt{2n}\,\Theta+1) δ = κ / ( 2 n 2 n Θ + 1 ) p , q ∈ B 1 p,q\in B_1 p , q ∈ B 1 ∣ p − q ∣ < δ |p-q|<\delta ∣ p − q ∣ < δ ∥ g ( q ) − g ( p ) ∥ < κ / 2 \|g(q)-g(p)\|<\kappa/2 ∥ g ( q ) − g ( p ) ∥ < κ /2 g ( q ) + P ( D 2 u ( p ) ) = G ( p ) + ( g ( q ) − g ( p ) ) g(q)+P(D^2u(p))=G(p)+(g(q)-g(p)) g ( q ) + P ( D 2 u ( p )) = G ( p ) + ( g ( q ) − g ( p )) κ / 2 \kappa/2 κ /2 K + κ / 2 K+\kappa/2 K + κ /2 K + κ / 2 < 2 K K+\kappa/2<2K K + κ /2 < 2 K ( D 2 u ( p ) , q ) ∈ O (D^2u(p),q)\in\mathcal O ( D 2 u ( p ) , q ) ∈ O 定理 5.1 条件 (c) が成り立つ。
K \mathcal K K ( M , q ) (M,q) ( M , q ) G = g ( q ) + P ( M ) G=g(q)+P(M) G = g ( q ) + P ( M ) ∥ G − 1 ∥ ≤ 2 / κ \|G^{-1}\|\leq2/\kappa ∥ G − 1 ∥ ≤ 2/ κ 補題 7.1 の一階微分の式により、ξ , η ∈ R 2 n \xi,\eta\in\R^{2n} ξ , η ∈ R 2 n N ∈ S 2 n N\in\mathcal S^{2n} N ∈ S 2 n
∂ ξ F = tr C ( G − 1 ∂ ξ g ) − ∂ ξ h , B ξ : N = − tr C ( G − 1 P ( N ) G − 1 ∂ ξ g ) , \partial_\xi F=\operatorname{tr}_{\C}(G^{-1}\partial_\xi g)-\partial_\xi h,\qquad
B_\xi:N=-\operatorname{tr}_{\C}\bigl(G^{-1}P(N)G^{-1}\partial_\xi g\bigr), ∂ ξ F = tr C ( G − 1 ∂ ξ g ) − ∂ ξ h , B ξ : N = − tr C ( G − 1 P ( N ) G − 1 ∂ ξ g ) , ∂ ξ ∂ η F = − tr C ( G − 1 ∂ η g G − 1 ∂ ξ g ) + tr C ( G − 1 ∂ ξ ∂ η g ) − ∂ ξ ∂ η h \partial_\xi\partial_\eta F=-\operatorname{tr}_{\C}(G^{-1}\partial_\eta g\,G^{-1}\partial_\xi g)+\operatorname{tr}_{\C}(G^{-1}\partial_\xi\partial_\eta g)-\partial_\xi\partial_\eta h ∂ ξ ∂ η F = − tr C ( G − 1 ∂ η g G − 1 ∂ ξ g ) + tr C ( G − 1 ∂ ξ ∂ η g ) − ∂ ξ ∂ η h である。∣ ξ ∣ = ∣ η ∣ = 1 |\xi|=|\eta|=1 ∣ ξ ∣ = ∣ η ∣ = 1 ∂ ξ g \partial_\xi g ∂ ξ g ∂ ξ ∂ η g \partial_\xi\partial_\eta g ∂ ξ ∂ η g 2 n Θ 2n\Theta 2 n Θ ∣ N ∣ ≤ 1 |N|\leq1 ∣ N ∣ ≤ 1 P ( N ) P(N) P ( N ) 1 1 1 ∣ ξ ∣ = ∣ η ∣ = 1 |\xi|=|\eta|=1 ∣ ξ ∣ = ∣ η ∣ = 1 ∣ N ∣ ≤ 1 |N|\leq1 ∣ N ∣ ≤ 1 n , κ , Θ n,\kappa,\Theta n , κ , Θ ∣ D x F ∣ ≤ 2 n max k ∣ ∂ e k F ∣ |D_xF|\leq\sqrt{2n}\max_k|\partial_{e_k}F| ∣ D x F ∣ ≤ 2 n max k ∣ ∂ e k F ∣ ∣ D x 2 F ∣ ≤ 2 n max k , l ∣ ∂ e k ∂ e l F ∣ |D_x^2F|\leq2n\max_{k,l}|\partial_{e_k}\partial_{e_l}F| ∣ D x 2 F ∣ ≤ 2 n max k , l ∣ ∂ e k ∂ e l F ∣ ∣ B ξ ∣ = sup { B ξ : N ∣ ∣ N ∣ ≤ 1 } |B_\xi|=\sup\{B_\xi:N\mid|N|\leq1\} ∣ B ξ ∣ = sup { B ξ : N ∣ ∣ N ∣ ≤ 1 } 定理 5.1 条件 (d) を満たすμ 1 , μ 2 \mu_1,\mu_2 μ 1 , μ 2 n , κ , Θ n,\kappa,\Theta n , κ , Θ F ( D 2 u ( p ) , p ) = log det G ( p ) − h ( p ) = 0 F(D^2u(p),p)=\log\det G(p)-h(p)=0 F ( D 2 u ( p ) , p ) = log det G ( p ) − h ( p ) = 0 定理 5.1 を次元2 n 2n 2 n 1 / ( 8 K ) , 2 / κ 1/(8K),2/\kappa 1/ ( 8 K ) , 2/ κ K 0 K_0 K 0 δ \delta δ μ 1 \mu_1 μ 1 μ 2 \mu_2 μ 2 ▨