分类： Elliptic equation

非线性椭圆方程平均值性质的几何直觉

旧博客原文

原题：Geometric intuition of mean value property of nonlinear elliptic equation

I wish to gain some understanding of the MVP of nonlinear elliptic equation by geometric intuition.

Linear elliptic equation case

First of all, I have a very good geometric explain of the MVP of Laplace equation, i.e.

MVP of laplace equation

$\Delta u=0$ in $\Omega$ , $\forall B(x_0,r)\subset \Omega$ is a Ball, we have following identity:

$\frac{1}{\mu(\partial(B))}\int_{\partial B}u(x)dx=u(x_0)$

I need to point out first, this property is not difficult to proof by standard integral by part method, but the following method have more geometric intuition. And in some sense explain why this property holds.

The proof is not very difficult to explain by mathematic formula, but I wish to divide the proof into two part, explain one part by graph and literal interpretation.

part 1 of the proof:

we consider a 1-parameter group of foliation, and consider the integral identity with this foliation.

$\int_{v\in S_{n-1}}\int_{\gamma_v} \frac{\partial \partial_{v}u}{\partial t}dt=\int_{B(x_0,r)}\partial_{n}u-\int_{S_{n-1}}\partial_{v}udv=\int_{B(x_0,r)}\partial_{n}u$

Part 2 of the proof:

and we have:

$\int_{v\in S_{n-1}}\int_{\gamma_v} \frac{\partial \partial_{v}u}{\partial t}dt=0$

by the pointwise equation $\Delta u=0$ , one key point is $\partial_{-v-v}u=\partial_{vv}u, \forall v\in S_{n-1}$ .

This approach cloud easily to transform to the general elliptic equation case and it seems a little difficult to transform to Possion equation, the non-hemomorphism case.

Nonlinear elliptic equation case

A-B-P estimate for general nonlinear uniformly elliptic equation

ABP estimate is the most basic estimate in fully nonliear elliptic equation.
The ABP maximum principle states (roughly) that, if

$a^{ij} \partial _i \partial _j u \geq f, in \ \Omega \subset \mathbb{R}^n (a^{ij} \geq C Id >0),$

Then (assuming sufficient regularity of the coefficients),

\sup _{\Omega} u \leq \sup _{\partial \Omega} u + C (\int _{\Omega} \vert f \vert^n )^{1/n} ………. (*)

I will give an intuitive explanation of the proof of (*) .
Usually, in order to prove maximum principles, the key idea is to use that at a local max the second derivative is negative-definite, then choose a good basis and get some identity of 1-order drivative and inequality for 2-order’s. This process is used in such like the proof of the Hopf lemma, and some inter gradient estimate, consider some flexiable function like $e^{Au}$ or sometihng else anyway.

But in the proof of ABP we need more geometric intution and more trick.

First we do a rescaling:
if $a^{ij} \partial _i \partial _j u \geq 1, in B_1 \subset \mathbb{R}^n (a^{ij} \geq C Id >0),u|_{\partial B_1}\geq 0.$
then:

$|inf_{B_1}u| \leq C |A|^{1/n} ....(**)$

And then We explain what is the contact set. It is the subset $\Gamma^{+} of \Omega$ such that u agree with it convex envelop. i.e. $\Gamma^{+}=\{x|u=convex \ evolap \ of \ u\ at \ x\}$ . The geometric meaning is it has at least one lower support plane. So what is $\Gamma^{+}$ it is just the set that u is very low on it. Or in another way of view you consider $-u$ as a lot of mountains then $\Gamma^+$ is the place near the tops of which mountain can see every thing (locally).
Then we look at every point in $\Gamma^{+}$ , then determination of the hessian matrix $det(u_{ij})$ at this point have a control due to the PDE $a^{ij} \partial _i \partial _j u \geq 1$ , and the uniformly elliptic property.

The determination of hessian matrix could be view as a determination of Jacobe matrix of the map $(u_1,...,u_n)\to (e_1,...,e_n)$ .and by Area formula we have:

$\int_{\Phi(\Omega)} f( \Phi^{-1}(y)) dy =\int_{\Omega}f(x)|J({\Phi(x)})| dx,$

It is easy to see for a constant $c$ , $B_{c|sup_{\Omega}|u||}(0)\subset \Phi(\Omega)$ (Base on the PDE on every point, the geometric intution is just the function u could not be very narrow cone at every point). So we have , take $f=\chi_{\Gamma^{+}}$ ,

$|B_{c\sup_{\Omega}|u|}(0)|^{n}\leq \int_{\Gamma^{+}}\chi_{\Gamma_{+}}(x)|J_{\Phi}(x)|dx$

so we have:

$|B_{c\sup_{\Omega}|u|}(0)|\lesssim |{\Gamma_{+}}|^{\frac{1}{n}}...(***)$

and the classical matrix inequality for every positive definite matrix A we have
:

$det(AB)\leq (\frac{tr(AB)}{n})^n...(****) .$

combine (***),(****) ,we have:

$sup_{\Omega}|u|\lesssim ||\frac{a^{ij}u_{ij}}{D^*}||_{L^n({\Gamma^+})} .$

graph

General approach to get MVP for elliptic equation which is come from geometry

MVP for K-hessian equation, with geometric explanation

MVP for K-curvature equation, with geometric explanation

MVP for p-Laplace equation, with geometric explanation

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

调和函数的平均值性质是椭圆方程最漂亮的几何现象之一。在线性情形，它来自球面对称和分部积分；在非线性椭圆方程中，真正的替代物往往不是严格平均值公式，而是 ABP estimate、contact set 和凸包几何。

线性调和函数有球面平均值性质，非线性椭圆方程则用 ABP estimate 和接触集合几何替代。

1. Laplace 方程的平均值性质

若 $\Delta u=0$，则对球 $B_r(x)$，

$$u(x)=\frac1{|\partial B_r|}\int_{\partial B_r(x)}u\,d\sigma.$$

证明可用 Green identity，也可以看成沿同心球 foliation 的通量守恒。

2. 几何解释

令

$$A(r)=\frac1{|\partial B_r|}\int_{\partial B_r(x)}u.$$

对 $r$ 求导后，$\Delta u=0$ 让通量项消失，于是 $A(r)$ 为常数。球面对称和散度结构共同产生平均值性质。

3. 非线性情形的问题

对一般 fully nonlinear elliptic equation

$$F(D^2u,x)=f,$$

没有线性叠加，也没有简单的球面平均公式。我们需要用比较原理和接触几何替代平均值。

4. ABP estimate

ABP 最大值原理粗略地说给出

$$\sup_\Omega u\le \sup_{\partial\Omega}u+C\|f\|_{L^n(\Omega)}.$$

其证明看 $u$ 的凸包和 contact set。梯度映射的面积公式把函数的最大值和右端 $f$ 的积分联系起来。

5. 一条直觉

线性平均值性质说：每个点由周围平均决定。非线性椭圆理论说：最大值由接触集合和 Monge-Ampere 型几何控制。二者背后的共同点是椭圆性阻止信息沿单一方向集中，迫使解受到周围区域的整体约束。

2017年12月11日
Schauder estimate 与 Sobolev inequality：椭圆方程正则性的两种语言

旧博客原文

原题：Schauder estimate and Sobelov inequality

In this note we discuss the Schauder theory for uniformly elliptic linear equations and Sobelov inequality.

the three main topics ars a priori estimate in Holder norms,regularity of arbitrary solutions and the solvability of the Dirichlet problem.Among these topics,a priori estimates are the most fundamental and the basis of the follows two.we will discuss both the interior Schauder estimate and global Schauder estimate.

-Schauder Theory-

1. Interior Schauder Theory

${\Omega}$ be a domain in ${R^n}$ ,bounded most of the time.
${a_{ij},b_i,c}$ be defined in ${\Omega}$ ,with ${a_{ij}=a_{ji}}$ .where ${1\leq i,j\leq n}$ .
we consider the operator ${L}$ given by,

$\displaystyle Lu=a_{ij}\partial_{ij}u+b_i\partial_iu+c,in \ \Omega.$

easy to see ${Lu}$ is defined for any ${u\in C^2(\Omega)}$ .
the operator ${L}$ is always be assumed to be strictly elliptic in ${\Omega}$ ;namely,
$\displaystyle a{ij}\xi_i\xi_j \geq \lambda|\xi|^2$

for any ${\xi\in R^n,x\in \Omega}$ ,where ${\lambda}$ is a positive constant.
1.1. Interior Schauder Estimate

define the weighted ${C^{k,\alpha}}$ norm,

$\displaystyle |u|^*_{C^{k,\alpha}(B_R)}=\sum_{i=0}^k R^i|D^iu|_{L^{\infty}(B_R)}+R^{k+\alpha}[D^ku]_{C^{\alpha}(B_R)}$

easy to see ${R}$ come from a scaling.
consider the PDE.
$\displaystyle Lu=a_{ij}\partial_{ij}u+b_i\partial_iu+c=f,in \ \Omega.$

we want to proof this type estimate,
$\displaystyle |u|_{C^{2,\alpha}(A)} \leq C(|u|_{L^{\infty}(\Omega)}+|f|_{C^{\alpha}(\Omega)})$

where ${A\subset \Omega }$
we first deal with a easy case, ${a_{ij}}$ is constant. in this case we proof the estimate:

Lemma 1 ${f \in C^{\alpha}(B_R)}$ ,for some ${\alpha \in (0,1)}$ ,and ${(a_{ij})}$ be a constant symmetric ${n\times n}$ matrix satisfying
$\displaystyle \lambda |\xi|^2 \leq a_{ij}\xi_i\xi_j \leq \Lambda|\xi|^2$

${\exists \lambda,\Lambda >0,\forall \xi \in R^n}$ . suppose ${u\in C^2(B_R)}$ satisfies:
$\displaystyle a_{ij}\partial_{ij}u=f, in \ B_R$

then , ${u \in C^{2,\alpha}(B_{\frac{R}{2}})}$ ,moreover,
$\displaystyle |u|^*_{C^{2,\alpha}(B_{\frac{R}{2}})}\leq C[|u|_{L^{\infty}(B_R)}+R^2|f|^*_{C^{\alpha}(B_R)}]$

Proof: $\Box$ to continue,we prove an interpolation inequality for Holder continuous functions.

Lemma 2 Let ${\alpha,\mu \in (0,1)}$ and ${B_R}$ be a ball of radius ${R}$ in ${R^n}$ ,then, (1)for any ${u\in C^{1,\alpha}(\overline B_R)}$ ,
$\displaystyle \mu^{\alpha}R^{\alpha}[u]_{C^{\alpha}(B_R)}\leq C[\mu R |\nabla u|_{L^{\infty}(B_R)}+|u|_{L^{\infty}(B_1)}]$

(2)for any ${u\in C^{1,\alpha}(\overline B_R)}$ ,
$\displaystyle \mu R |\nabla u|_{L^{\infty}(B_R)} \leq C[\mu^{1+\alpha}R^{1+\alpha}|\nabla u|_{C^{\alpha}(B_R)}+|u|_{L^{\infty}(B_1)}]$

(3)for any ${u\in C^2(\overline B_R)}$ ,
$\displaystyle \mu R|\nabla u|_{L^{\infty}(B_R)}\leq C[\mu^2R^2|\nabla^2 u|_{L^{\infty}(B_R)}+|u|_{L^{\infty}(B_R)}]$

where ${C}$ is a positive constant depending on n and ${\alpha}$ .
Proof: $\Box$

Corollary 3 Let ${\alpha,\mu\in (0,1)}$ and ${B_R}$ be a ball of radius ${R}$ in ${R^n}$ .Then,for any ${u\in C^{2,\alpha}(\overline B_R)}$ ,
$\displaystyle \sum_{i=0}^2(\mu R)^i|\nabla^i u|_{L^{\infty}(B_R)}+\sum_{i=0}^1(\mu R)^{i+\alpha}[\nabla^i u]_{C^{\alpha}(B_R)}\leq C[(\mu R)^{2+\alpha}[\nabla^2 u]_{C^{\alpha}(B_R)}+|u|_{L^{\infty}(B_R)}]$

Proof: $\Box$

Now we are ready to prove an interior estimate for ${C^{2,\alpha}}$ -norms of solutions of uniformly elliptic equations.The trick is to freeze coefficients.

Lemma 4
2. Global Schauder Theory

-Sobelov inequality-

Theorem 5
$\displaystyle W_0^{1,p}(\Omega)\longrightarrow L^{\frac{np}{n-p}}(\Omega),1\leq p <n$

moreover,we have: ${\exists C=C(n,p)}$ , ${\forall u\in W^{1,p}_0(\Omega)}$ ,
$\displaystyle ||u||_{\frac{np}{n-p}} \leq C||Du||_p,1\leq p<n$

Proof:

$\displaystyle p=1$

suffice to proof:
$\displaystyle ||u||_{\frac{n}{n-1}}\leq C||Du||_1$

obvious we have:
$\displaystyle |u(x)|\leq \int_{-\infty}^{\infty}|Du(x)|dx$

so ${\int_{\Omega} |u|^{\frac{n}{n-1}}\leq \int_{\Omega} \Pi_{i=1}^n(\int_{-\infty}^{\infty}|D_iu(x)|dx)^{\frac{1}{n-1}}}$ .
so ${||u||_{\frac{n}{n-1}}\leq (\int_{\Omega}\Pi_{i=1}^n(\int_{-\infty}^{\infty}|D_iu|)^{\frac{1}{n-1}})^{\frac{n-1}{n}}\leq \int_{\Omega} \Pi_{i=1}^n(\int_{-\infty}^{\infty}|D_iu|)^{\frac{1}{n}} \leq \int_{\Omega} \frac{1}{n} \sum_{i=1}^n(\int_{-\infty}^{\infty}|D_iu|)\leq C||Du||_1}$
$\displaystyle 1<p<n$

use the similar argument as ${p=1}$ to prove the situation ${1<p<n}$ .
suffice to prove ${||u||_{\frac{np}{n-p}}\leq C||Du||_p}$ .
obvious we have: ${|u(x)|^p\leq \int_{-\infty}^{\infty}p|u|^{p-1}|Du|}$ .
${(\int_{\Omega}|u(x)|^{\frac{np}{n-p}})^{\frac{n-p}{np}}}$
${\leq (\int_{\Omega} \Pi_{i=1}(\int_{-\infty}^{\infty} p|u|^{p-1}|D_iu| )^{\frac{1}{n-p}})^{\frac{n-p}{np}} }$
${\leq C\int_{\Omega} \Pi_{i=1}^n(\int_{-\infty}^{\infty}p|u|^{p-1}|D_iu|)^{\frac{1}{np}}}$
${\leq\frac{c}{n}\sum_{i=1}^n\int_{\Omega}(\int_{-\infty}^{\infty}p|u|^{p-1}|D_iu|)^{\frac{1}{p}}}$
${\leq \frac{c}{n}\sum_{i=1}^n\tilde C p[(\int_{\Omega} (|u|^{p-1})^{\frac{p}{p-1}})^{\frac{p-1}{p}}+(\int_{\Omega} |D_iu|^p)^{\frac{1}{p}}]^{\frac{1}{p}} }$
${\leq C||Du||_p}$ . Q.E.D. $\Box$
4.

$\displaystyle W_0^{1,p}(\Omega)\longrightarrow C(\bar\Omega),n<p$

moreover,we have: ${\exists C=C(n,p)}$ , ${\forall u\in W^{1,p}_0(\Omega)}$ ,
$\displaystyle sup_{\Omega}|u| \leq C|\Omega|^{\frac{1}{n}-\frac{1}{p}}||Du||_p,p>n$

${\mu\in (0,1]}$ ,

$\displaystyle (V_{\mu}f)(x)=\int_{\Omega}|x-y|^{n(\mu-1)}f(y)dy$

then ${V_{\mu}: L^1(\Omega) \longrightarrow L^1(\Omega) }$ is well-defined by the following lemma:
Lemma 6 ${V_{\mu}:L^p \longrightarrow L^q}$ continously for any q, ${1\leq q \leq \infty}$ satisfy ${0\leq \delta=\delta(p,q)=\frac{1}{p}-\frac{1}{q} \leq \mu}$ .
furthermore,for any ${f\in L^p(\Omega)}$
$\displaystyle ||V_{\mu}f||_q \leq (\frac{1-\delta}{\mu -\delta})^{1-\delta}w_n^{1-\mu}|\Omega|^{\mu-\delta}||f||_p$

Proof: ${h(x-y)=|x-y|^n(\mu-1)}$ directly calculate follows that :

$\displaystyle ||h||_r \leq (\frac{1-\delta}{\mu -\delta})^{1-\delta} w_n^{1-\mu}|\Omega|^{\mu-\delta}$

now follows young inequality and this priori estimate we have:
${||V_{\mu}f||_q=(\int_{\Omega}(\int_{\Omega}|x-y|^{n(\mu-1)}f(y)dy)dx)^{\frac{1}{q}}}$
${\leq (\int_{\Omega}(\int_{\Omega}h^{\frac{r}{q}}h^{r(1-\frac{1}{p})}|f|^{\frac{p}{q}}|f|^{p\delta})^qdx)^{\frac{1}{q}}}$
${\leq (\int_{\Omega}(\int (h^r|f|^p)^{\frac{1}{q}}(\int h^r)^{1-\frac{1}{p}}(\int f^p)^{\delta})^{q})^{\frac{1}{q}}}$
${\Longrightarrow}$
$\displaystyle ||V_{\mu}f||_q \leq sup_{x \in \Omega} \{\int h^r(x-y)dy\}^{\frac{1}{r}}||f||_p$

and by the priori estimate,we have:
$\displaystyle ||V_{\mu}f||_q \leq (\frac{1-\delta}{\mu -\delta})^{1-\delta}w_n^{1-\mu}|\Omega|^{\mu-\delta}||f||_p$

Q.E.D. $\Box$
Lemma 7 ${f\in L^p(\Omega)}$ , ${g=V_{\mu}f}$ .
${\Longrightarrow}$ ${\exists c_1,c_2}$ constant depend only on ${n,p}$ ,such that
$\displaystyle \int_{\Omega} exp[\frac{g}{c_1||f||_p}]^{p^`}dx\leq c_2|\Omega|,p^`=\frac{p}{p-1}$

Proof: we have

$\displaystyle ||g||_q \leq q^{1-\frac{1}{p}+\frac{1}{q}}w_n^{1-\frac{1}{p}}|\Omega|^{\frac{1}{q}}||f||_p$

${\Longrightarrow}$
$\displaystyle \int_{\Omega} |g|^{p^`q}dx \leq p^`q(w_np^`q||f||_p^{p^`})^q|\Omega|$

${\Longrightarrow}$
$\displaystyle \int_{\Omega}\sum_{N_0}^{N}\frac{1}{k!}(\frac{|g|}{c_1||f||_p})^{p^`k}\leq p^`|\Omega|\sum(\frac{p^`w_n}{c_1^p})^k\frac{k^k}{(k-1)!}$

then take ${c_1,c_2}$ suffice large. Q.E.D. $\Box$
Lemma 8 let ${u\in W^{1,1}_0(\Omega)}$
$\displaystyle u(x)=\frac{1}{nw_n} \int_{\Omega} \frac{(x_i-y_i)D_iu(y)}{|x-y|^n}$

a.e. in ${\Omega}$ .
Proof: frist zero extended ${u}$ to whole space.and we have ${u(x)=\int_{-\infty}^xD_iu(x)}$ .

$\displaystyle u(x)=\int_0^{\infty}D_ru(x+rw)dr$

forall ${w\in \partial B_1(0)}$ ,so
$\displaystyle u(x)=-\frac{1}{nw_n}\int_0^{\infty}\int_{|w|=1}D_ru(x+rw)drdw=\frac{1}{nw_n}\int_{\Omega}\frac{(x_i-y_i)D_iu(y))}{|x-y|^ndy}$

Q.E.D. $\Box$
Theorem 9 let ${u\in W^{1,n}_0(\Omega)}$ ,then there exists constant ${c_1,c_2}$ such that
$\displaystyle \int_{\Omega}exp[\frac{|u|}{c_1||Du||_n}]^{\frac{n}{n-1}}dx\leq c_2|\Omega|$

Proof: a $\Box$

Theorem 10 ${u\in W_0^{1,p}(\Omega),p>n}$ ,then ${u\in C^{\gamma}(\Omega)}$ , ${\gamma=1-\frac{n}{p}}$ .
moreover ${\forall ball B=B_R}$
$\displaystyle osc_{\Omega \cap B_R}u \leq C R^{\gamma} ||Du||_p$

Proof: a $\Box$

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

Schauder 理论和 Sobolev 理论是椭圆方程正则性的两套基本语言。前者追踪 Holder 范数，适合系数和右端足够连续的情形；后者追踪积分可积性，适合弱解和变分方法。

Schauder 估计追踪 Holder 正则性，Sobolev 不等式追踪积分正则性，两者共同构成椭圆方程的基本正则性工具。

1. 线性一致椭圆算子

考虑

$$Lu=a^{ij}(x)\partial_{ij}u+b^i(x)\partial_i u+c(x)u=f.$$

一致椭圆性是指存在 $\lambda>0$，使

$$a^{ij}(x)\xi_i\xi_j\ge \lambda|\xi|^2.$$

这是所有先验估计的起点。

2. Interior Schauder estimate

若系数和 $f$ 都在 $C^\alpha$ 中，则局部有

$$\|u\|_{C^{2,\alpha}(B_{1/2})}\le C\bigl(\|u\|_{C^0(B_1)}+\|f\|_{C^\alpha(B_1)}\bigr).$$

常系数情形可以先通过 Newton potential 或 Fourier 方法得到，再用冻结系数和 perturbation 推广到变系数情形。

3. Scaling 与 weighted norms

Schauder 估计的形状由 scaling 决定。若把球 $B_r$ 缩放到单位球，二阶导数带来 $r^{-2}$，Holder seminorm 还会多出 $r^{-\alpha}$。weighted norm 正是为了把这些尺度因子记录清楚。

4. Sobolev inequality

Sobolev 不等式给出

$$\|u\|_{L^{p^\ast}}\le C\|\nabla u\|_{L^p},\qquad p^\ast=\frac{np}{n-p}.$$

它不直接给出经典二阶 Holder 正则性，但能建立弱解存在性、能量估计和 bootstrapping。

5. 两种理论的关系

Schauder 理论适合光滑数据的 classical solution，Sobolev 理论适合弱解和变分框架。椭圆正则性常常先用 Sobolev 得到弱解，再通过 De Giorgi-Nash-Moser 或 Schauder 估计提升正则性。

2017年10月4日