Dirichlet 原理：从离散调和函数到连续调和函数

旧博客原文

原题：A discret to continuous approach to the Dirichlet principle.

Direchlet principle:
is a compact set with $C^1$ boundary. then there exists unique solution $f$ satisfied $\Delta f=0$ in $\Omega$, $f=g$ on .

Perron lifting and barrier function

We know the standard approach of the Dirichlet principle is perron lifting and construction of barrier function on the boundary.

The key point is if we define the variation energy , then it is easy to see for is in perron set, . So we can begin from a maximization sequence to construct a Cauchy sequence by perron lifting and by the involve of barrier function to make the solution compatible with the boundary condition then arrive a proof.

But when I was a freshman in undergraduate school and I do not know the method of perron lifting I try something I name it from discret to continuous approach to try to solved the problem. It is always a puzzle in my mind iff we can solve the Dirichlet principle in this way, roughly speaking, it is divid into two part:

1. Investigate the discretization of harmonic function in smaller and smaller scale. The discretization I consider is just i.e. the -latties in $\Omega$,and discretization Laplace operator . Some result is much easier to arrive with the discretization thing, you know ,such as the existence of solution is just come from simple linear algebra. and we can deduce harneck inequality, gradient estimate, even green function. So we get a solution of discretization and we do a extension to by take value of a small tube by the center of the tube, where the value have a definition by , and now we get .

2. The second step is to proof the solution with -discretization problem will coverage to the solution of original problem;i.e. we want to proof a estimate;i.e. , we have , . and by Albano-Ascoli theorem to construct . Then we need to proof $f$ is the harmonic function we find, to verify this information we use the mean-value property. So we need to prove satisfied mean-value property for every ball in .

Here is my first question,
> **Question 1:** How to prove the estimate and the MVP of -discretization will coverage to the MVP in case occor in second step?

My attempt to the estimate is by renomelazation which seems could work, but the annoying thing is to proof the mean-value property will coverage to the real one, I try to use some result of random walk, but it seem not works…

My second question is:
> **Question 2:** Are this approach a universal phenomenon? At least could we use this approach to establish the existence of solution for linear elliptic and parabolic equation?

The Third question is:
> **Question 3:** If we consider some inverse problem, that is to say, form a MVP instead of a PDE to derive a solution, could this always be possible? some example is, if we change the mean value property for harmonic function from the average of ball to cube or triangle or elliptic or something else, what happen? Is there always a solution satisfied the news MVP point-wise? If not, Is there some counterexample? on another hand, if yes, are them came from some PDE?

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。

Dirichlet 原理说：给定边界值，调和函数是 Dirichlet energy 的极小者。除了 Perron 方法和 barrier function，也可以从离散问题逼近连续问题。

1. 连续问题

设 $\Omega$ 有光滑边界，给定 $g$，要求

$$\Delta u=0\quad\text{in }\Omega,\qquad u|_{\partial\Omega}=g.$$

变分形式是极小化

$$E(u)=\int_\Omega |\nabla u|^2\,dx.$$

Euler-Lagrange 方程正是 Laplace 方程。

2. 离散化

取 $\varepsilon$-lattice 上的区域 $\Omega_\varepsilon$，定义离散 Laplacian

$$\Delta_\varepsilon u(x)=\frac1{\varepsilon^2}\sum_{y\sim x}(u(y)-u(x)).$$

离散 Dirichlet 问题是一个有限维线性代数问题，因此存在唯一解。

3. 离散估计

离散调和函数仍有最大值原理、Harnack inequality、Green function 和能量估计。这些估计若能与 $\varepsilon$ 无关，就可以取极限。

4. 延拓与紧性

把格点函数延拓为分片常数或分片线性函数。若能得到统一的 Holder 或 Sobolev 控制，就能通过 compactness 取出收敛子列。极限函数满足弱形式

$$\int_\Omega \nabla u\cdot\nabla\varphi=0.$$

5. 边界条件

最细的是边界兼容性。Perron 方法中 barrier function 正是为处理边界极限；离散方法也需要离散 barrier，保证延拓后的极限真正取到给定边界值。