一类椭圆偏微分方程解的水平集的高斯曲率估计
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摘要
凸性作为一个重要的几何特征,长期以来一直是椭圆偏微分方程研究中的重要主题.本文的主要研究对象是椭圆偏微分方程解的水平集的凸性.利用经典的极大值原理,本文给出了p-调和函数水平集高斯曲率的最佳正下界估计,也给出了Rn：极小曲面水平集高斯曲率的最佳正下界估计和一类半线性方程解的水平集高斯曲率的正下界估计.另一方面,本文还研究了p-调和函数水平集的高斯曲率关于函数高度的凹性.具体地说.本文的主要结果如下
Ⅰ.p-调和函数水平集高斯曲率的正下界估计定理0.0.1.设Ω(?)Rn(n≥2)是一个有界光滑区域,u∈C4(Ω)∩C2(Ω)是定义在Q上的p-调和函数,即u满足p-调和方程div(|▽u|p-2▽u)=0in Ω.设1 情形1:若n≥2,1 情形2:若n=2,1 情形3:若杀n:2,3/2≤p≤3;n=3,2≤p<+∞或n≥4,p=n=1/2则函数K边界上取到最小值.
根据定理0.0.1,我们可以得到p-调和函数水平集高斯曲率的正下界估计推论0.0.2.设Ω0和Ω1是Rn(n≥2)中有界光滑凸区域,并且Ω1(?)Ω0.设u满足下述Dirichlet(?)问题其中1 特别地,对于调和函数,我们有下面的命题.命题0.0.3.设Ω是Rn(n≥2)中的区域,u是定义在Ω上的调和函数,并且u在Ω内没有临界点.记u的水平集的高斯曲率为K.定义函数ψ=|▽u|-1K.设u的水平集相对于梯度▽u的方向是严格凸的.那么,在模掉梯度项▽Ψ的意义下函数Ψ是Ω上的上调和函数,即成立下面的微分不等式△ψ≤C|▽ψ|inΩ,其中正常数C依赖于n和||u||C3(Ω).
Ⅱ.极小曲面方程解的水平集的高斯曲率正下界估计定理0.0.4.设Ω是Rn(n≥2)中的有界光滑区域,u∈C4(Ω)∩C2(Ω)(?)茜足下述极小曲面方程设在Ω上|▽u|≠0.记u的水平集的高斯曲率为K.若u的水平集相对于梯度▽u的方向是严格凸的,那么我们有下面的结论.最小值.
类似地,我们可以得到极小曲面方程解的水平集的高斯曲率正下界估计推论0.0.5.设Ω0和Q1是Rn(n≥2)中的有界光滑凸区域,并且Ω1(?) Ω0.记Ω=Ω0\Ω1.设u满足Dirichle(?)问题记u的水平集的高斯曲率为K.那么,我们有下述估计
Ⅲ.半线性方程解的水平集的高斯曲率正下界估计定理0.0.6.设Ω是Rn(n≥2)中的有界光滑区域,u∈C4(Ω)∩C2(Ω)；满足半线性方程△u=f(x,u,▽u) inΩ,其中f≥0,f∈C2(Ω×R×Rn)设在Ω上|▽u|≠0.记u的水平集的高斯曲率为K.设u的水平集相对于梯度Vu的方向是严格凸的.为表述方便,我们记下述两个断言分别为(A1)和(A2),即(A1)函数|▽u|-2K在边界上取到最小值(A2)函数|▽u|n-1K在边界上取到最小值.
那么我们有如下结论.情形1:f=f(u).当fu≥0时,(A1)成立;当fu≤0时,(A2)成立.情形2:f=f(x).如果映射F:(0,+∞)×Ω→R,(t,x)→t3f(x)
是凸的(当f>0时,等价于f-1/2是凹的),那么(A2)成立.情形3:f=f(x,u)设对每一个固定的u∈(0.1),映射Fu:(0,+∞)×Ω→R,(t,x)→t3f(x,u)
是凸的.如果fu≤0,那么(A2)成立情形4:f=f(u,▽u)设对每一个固定的u∈(0,1),映射Fu:(0,+∞)×Sn-1→R,(t,p)→t3f(u,p/t)
是凸的.当.九≥0时,(A1)成立;当fu≤0时,(A2)成立,情形Jf=f(x,u,▽u)设对每一个固定的u∈(0,1),映射Fu:(0,+∞)×Ω×Sn-1→R,(t,x,p)→t3f(x,u,p/t)
是凸的.当fu≤0时,(A2)成立.
推论0.0.7.设Ω0和Ω1是Rn(n≥2)中的有界光滑凸区域,并且Ω1(?)Ω0.记Ω=Ωu\Ω1设u满足Dirichlet边值问题这里f∈C2(R),单调递增,并且f(0)=0.记u的水平集的高斯曲率为K.那么,我们有下述估计
Ⅳ.p-调和函数水平集的高斯曲率关于函数高度的凹性定理0.0.8.设u满足Dirichlet(?)司题
其中Ω0和Ω1是Rn(n≥2)中的有界光滑严格凸区域,并且Ω1(?)Ω0,1Green函数时,相应的f(t)是仿射函数.
As one of the most important geometric properties, convexity is an issue in the study of elliptic partial differential equations for a long time. In the present paper, we are concerned on the convexity of level sets of solutions to elliptic partial differential equations. By the classical maximum principle, we obtain the sharp positive lower bound estimates of the Gaussian curvature of the convex level sets of p-harmonic functions and minimal graphs in R". Under certain structural con-ditions, we also give the positive lower bound estimates of the Gaussian curvature of the convex level sets of solutions to some semi-linear elliptic equations. On the other hand, we establish the sharp concavity of the Gaussian curvature of the level sets of p-harmonic functions with respect to the height of the functions. More precisely, the main results of the paper are as follows.
I. Gaussian curvature estimates of level sets of jy-harmonic functions Theorem0.0.1. Let Q be a bounded smooth domain in M.n(n≥2) and u∈C4(Ω)∩C2(Ω) be a p-harmonic function, i.e. div(|▽u|p-2▽u)=0in Ω. Assume1gradient▽u, then we have the following statements.
Case1; For n≥2,1 Case2: For n=2,1 Case3:For n=2,2/3≤p≤3; n=3,2≤p<+∞or n≥4, p=n+1/2, the function K attains its minimum on the boundary.
By Theorem0.0.1, we can obtain the Gaussian curvature estimates of level sets of p-harmonic functions. Corollary0.0.2. Let u satisfy the following Dirichlet problem where1g estimates.
Case1a: For1 Case1b: Forn+1/2 Case2: For n=2,1 Case3:Forn=2,3/2≤p≤3; n=3,2≤p<+∞or n≥4, p=n+1/2,we have Ωmin K≥δΩmin K.
In particular, for the harmonic functions we have the following proposition. Proposition0.0.3. Let Ω Q be a domain in Rn(n≥2).Let u be a harmonic function defined in Q. Assume|▽u|≠0in Ω. Let K be the Gaussian curvature of the level sets. Let ψ=|▽u|-K.If the level sets of u are strictly convex with respect to the direction of gradient▽u, then the function ip is a super-harmonic function modulo the gradient terms▽ψ. Namely, we have the following differential inequality ε▽ψ≤C|▽ψ|in Ω, where the positive constant C depends on n and||u||c3(Ω)
Ⅱ. Gaussian curvature estimates of level sets of minimal surfaces Theorem0.0.4. Let Ω1be a smooth bounded domain in Rn(n≥2). Let u∈C4(Ω)(?)C2(Ω1) satisfy the following minimal surface equation Assume|▽u|≠0on Ω. Let K be the Gaussian curvature of the level sets. If the level sets of u are strictly convex with respect to the direction of gradient Vu, then we have the following results.(ⅰ) For n=2, the function K attains its minimum and max-imum on the boundary.(ⅱ) For n≥3, the function attains its minimum on the boundary for
In a similar way, we have the Gaussian curvature estimates of level sets of minimal surfaces.
Corollary0.0.5. Let u satisfy the following Dirichlet problem
where Ω0Ω1are bounded smooth convex domains in Rn(n≥2),Ω1(?)Ω0. Let K be the Gaussian curvature of the level sets. Then we have the following estimates
Ⅲ. Gaussian curvature estimates of level sets of solutions to semi-linear elliptic equations Theorem0.0.6. Let Ωbe a bounded smooth domain in Rn(n≥2). Let u∈C4(Ω)∩C2(Ω1) satisfy the following semi-linear elliptic equation▽u=f(x,u,▽u) in Ω,
where f≥0,f∈C2(Ω×R×Rn). Assume|▽u|≠0on Ω. Let K be the Gaussian curvature of the level sets. Assume the level sets of u are strictly convex with respect to the direction of gradient▽u. Denote the following two assertions by (A1) and (A2), i.e.
(A1) The function|▽u|-2K attains its minimum on the boundary;
(A2) The function|▽u|n-1K attains its minimum on the boundary. Then we have the following facts.
Case1: f=f(u). If fu≥0, then (A1) is valid; if fu≤0, then (A2) is valid.
Case2: f=f(x). If the map F:(0,+∞)×Ω→R,(t, x) h→t3f(x)
is convex or f-1/2is concave for positive f), then (A2) is valid. Case3:f=f(x,u). Assume for every choice of u∈(0,1), the map Fu:(0,+∞)×Ω×R,(t, x)→t3(x, u)
is convex. If fu≤0, then (A2) is valid. Case4:f=f(it,▽u). Assume/or e^/ery choice of u∈(0,1), the map Fu:(0,+∞)×Sn-1→R,(t, p)→t3f(u,p/t)
is convex. If fu≥0, then (A1) is valid; if fu≤0, then (A2) is valid. Case5:f=f(x,u,▽u).Assume for every choice of u∈(0,1), t/je map Fu:(0,+∞)×Ω×Sn-1→R,(t, x,p)→t3f(x,u,p/t)
is convex. If fu≤0, i/ien (A2) z
Corollary0.0.7. Lei u satisfy the following Dirichkt problem
where Ω0and Ω1are bounded smooth convex domains in Rn(n≥2), Ω1×Ω0and
f∈C2(R), nondecreasing,f(0)=0. Let K be the Gaussian curvature of the level
sets. Then we have the following estimates Ⅳ. Concavity of the Gaussian curvature of level sets ofp-harmonic
functions with respect to the height of the functions
Theorem0.0.8. Let u satisfy the following Dmchlet problem Xll
where Ω0and Ω1are bounded smooth strictly convex domains in Rn(n≥2),Ω1(?) Ω0and1 Then f(t) is a concave function for t∈[0,1], Equivalently, for any point x∈Ωt, we have the following estimates (|▽u|n+1-2pK)1/n-1(x)≥(1-t)δΩ0max(|▽u|n+1-2pK)1/n-1=tδΩ1max(|▽u|n+1-2pK)1/n-1.
Furthermore, the function f(t) is an affine function of the height t when the p-harmonic function is the p-Green function on the ball.
Ⅰ.p-调和函数水平集高斯曲率的正下界估计定理0.0.1.设Ω(?)Rn(n≥2)是一个有界光滑区域,u∈C4(Ω)∩C2(Ω)是定义在Q上的p-调和函数,即u满足p-调和方程div(|▽u|p-2▽u)=0in Ω.设1
根据定理0.0.1,我们可以得到p-调和函数水平集高斯曲率的正下界估计推论0.0.2.设Ω0和Ω1是Rn(n≥2)中有界光滑凸区域,并且Ω1(?)Ω0.设u满足下述Dirichlet(?)问题其中1
Ⅱ.极小曲面方程解的水平集的高斯曲率正下界估计定理0.0.4.设Ω是Rn(n≥2)中的有界光滑区域,u∈C4(Ω)∩C2(Ω)(?)茜足下述极小曲面方程设在Ω上|▽u|≠0.记u的水平集的高斯曲率为K.若u的水平集相对于梯度▽u的方向是严格凸的,那么我们有下面的结论.最小值.
类似地,我们可以得到极小曲面方程解的水平集的高斯曲率正下界估计推论0.0.5.设Ω0和Q1是Rn(n≥2)中的有界光滑凸区域,并且Ω1(?) Ω0.记Ω=Ω0\Ω1.设u满足Dirichle(?)问题记u的水平集的高斯曲率为K.那么,我们有下述估计
Ⅲ.半线性方程解的水平集的高斯曲率正下界估计定理0.0.6.设Ω是Rn(n≥2)中的有界光滑区域,u∈C4(Ω)∩C2(Ω)；满足半线性方程△u=f(x,u,▽u) inΩ,其中f≥0,f∈C2(Ω×R×Rn)设在Ω上|▽u|≠0.记u的水平集的高斯曲率为K.设u的水平集相对于梯度Vu的方向是严格凸的.为表述方便,我们记下述两个断言分别为(A1)和(A2),即(A1)函数|▽u|-2K在边界上取到最小值(A2)函数|▽u|n-1K在边界上取到最小值.
那么我们有如下结论.情形1:f=f(u).当fu≥0时,(A1)成立;当fu≤0时,(A2)成立.情形2:f=f(x).如果映射F:(0,+∞)×Ω→R,(t,x)→t3f(x)
是凸的(当f>0时,等价于f-1/2是凹的),那么(A2)成立.情形3:f=f(x,u)设对每一个固定的u∈(0.1),映射Fu:(0,+∞)×Ω→R,(t,x)→t3f(x,u)
是凸的.如果fu≤0,那么(A2)成立情形4:f=f(u,▽u)设对每一个固定的u∈(0,1),映射Fu:(0,+∞)×Sn-1→R,(t,p)→t3f(u,p/t)
是凸的.当.九≥0时,(A1)成立;当fu≤0时,(A2)成立,情形Jf=f(x,u,▽u)设对每一个固定的u∈(0,1),映射Fu:(0,+∞)×Ω×Sn-1→R,(t,x,p)→t3f(x,u,p/t)
是凸的.当fu≤0时,(A2)成立.
推论0.0.7.设Ω0和Ω1是Rn(n≥2)中的有界光滑凸区域,并且Ω1(?)Ω0.记Ω=Ωu\Ω1设u满足Dirichlet边值问题这里f∈C2(R),单调递增,并且f(0)=0.记u的水平集的高斯曲率为K.那么,我们有下述估计
Ⅳ.p-调和函数水平集的高斯曲率关于函数高度的凹性定理0.0.8.设u满足Dirichlet(?)司题
其中Ω0和Ω1是Rn(n≥2)中的有界光滑严格凸区域,并且Ω1(?)Ω0,1
As one of the most important geometric properties, convexity is an issue in the study of elliptic partial differential equations for a long time. In the present paper, we are concerned on the convexity of level sets of solutions to elliptic partial differential equations. By the classical maximum principle, we obtain the sharp positive lower bound estimates of the Gaussian curvature of the convex level sets of p-harmonic functions and minimal graphs in R". Under certain structural con-ditions, we also give the positive lower bound estimates of the Gaussian curvature of the convex level sets of solutions to some semi-linear elliptic equations. On the other hand, we establish the sharp concavity of the Gaussian curvature of the level sets of p-harmonic functions with respect to the height of the functions. More precisely, the main results of the paper are as follows.
I. Gaussian curvature estimates of level sets of jy-harmonic functions Theorem0.0.1. Let Q be a bounded smooth domain in M.n(n≥2) and u∈C4(Ω)∩C2(Ω) be a p-harmonic function, i.e. div(|▽u|p-2▽u)=0in Ω. Assume1
Case1; For n≥2,1
By Theorem0.0.1, we can obtain the Gaussian curvature estimates of level sets of p-harmonic functions. Corollary0.0.2. Let u satisfy the following Dirichlet problem where1
Case1a: For1
In particular, for the harmonic functions we have the following proposition. Proposition0.0.3. Let Ω Q be a domain in Rn(n≥2).Let u be a harmonic function defined in Q. Assume|▽u|≠0in Ω. Let K be the Gaussian curvature of the level sets. Let ψ=|▽u|-K.If the level sets of u are strictly convex with respect to the direction of gradient▽u, then the function ip is a super-harmonic function modulo the gradient terms▽ψ. Namely, we have the following differential inequality ε▽ψ≤C|▽ψ|in Ω, where the positive constant C depends on n and||u||c3(Ω)
Ⅱ. Gaussian curvature estimates of level sets of minimal surfaces Theorem0.0.4. Let Ω1be a smooth bounded domain in Rn(n≥2). Let u∈C4(Ω)(?)C2(Ω1) satisfy the following minimal surface equation Assume|▽u|≠0on Ω. Let K be the Gaussian curvature of the level sets. If the level sets of u are strictly convex with respect to the direction of gradient Vu, then we have the following results.(ⅰ) For n=2, the function K attains its minimum and max-imum on the boundary.(ⅱ) For n≥3, the function attains its minimum on the boundary for
In a similar way, we have the Gaussian curvature estimates of level sets of minimal surfaces.
Corollary0.0.5. Let u satisfy the following Dirichlet problem
where Ω0Ω1are bounded smooth convex domains in Rn(n≥2),Ω1(?)Ω0. Let K be the Gaussian curvature of the level sets. Then we have the following estimates
Ⅲ. Gaussian curvature estimates of level sets of solutions to semi-linear elliptic equations Theorem0.0.6. Let Ωbe a bounded smooth domain in Rn(n≥2). Let u∈C4(Ω)∩C2(Ω1) satisfy the following semi-linear elliptic equation▽u=f(x,u,▽u) in Ω,
where f≥0,f∈C2(Ω×R×Rn). Assume|▽u|≠0on Ω. Let K be the Gaussian curvature of the level sets. Assume the level sets of u are strictly convex with respect to the direction of gradient▽u. Denote the following two assertions by (A1) and (A2), i.e.
(A1) The function|▽u|-2K attains its minimum on the boundary;
(A2) The function|▽u|n-1K attains its minimum on the boundary. Then we have the following facts.
Case1: f=f(u). If fu≥0, then (A1) is valid; if fu≤0, then (A2) is valid.
Case2: f=f(x). If the map F:(0,+∞)×Ω→R,(t, x) h→t3f(x)
is convex or f-1/2is concave for positive f), then (A2) is valid. Case3:f=f(x,u). Assume for every choice of u∈(0,1), the map Fu:(0,+∞)×Ω×R,(t, x)→t3(x, u)
is convex. If fu≤0, then (A2) is valid. Case4:f=f(it,▽u). Assume/or e^/ery choice of u∈(0,1), the map Fu:(0,+∞)×Sn-1→R,(t, p)→t3f(u,p/t)
is convex. If fu≥0, then (A1) is valid; if fu≤0, then (A2) is valid. Case5:f=f(x,u,▽u).Assume for every choice of u∈(0,1), t/je map Fu:(0,+∞)×Ω×Sn-1→R,(t, x,p)→t3f(x,u,p/t)
is convex. If fu≤0, i/ien (A2) z
Corollary0.0.7. Lei u satisfy the following Dirichkt problem
where Ω0and Ω1are bounded smooth convex domains in Rn(n≥2), Ω1×Ω0and
f∈C2(R), nondecreasing,f(0)=0. Let K be the Gaussian curvature of the level
sets. Then we have the following estimates Ⅳ. Concavity of the Gaussian curvature of level sets ofp-harmonic
functions with respect to the height of the functions
Theorem0.0.8. Let u satisfy the following Dmchlet problem Xll
where Ω0and Ω1are bounded smooth strictly convex domains in Rn(n≥2),Ω1(?) Ω0and1
Furthermore, the function f(t) is an affine function of the height t when the p-harmonic function is the p-Green function on the ball.
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[62]Xu, Lu A microscopic convexity theorem of level sets for solutions to elliptic equa-tions, Cal. Var. 40, (2011), 51-63.