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Dirichlet principle: Let $\Omega \subset R^n$ be a compact set with $C^1$ boundary. Then, there exists a unique solution $f$ satisfying $\Delta f = 0$ in $\Omega$ and $f=g$ on $\partial \Omega$.

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

$$E(u)=\int_{\Omega}|\nabla u|^2$$

then it is easy to see for $u_1, u_2$ is in perron set

$$E(\sup (u_1,u_2))\geq \max\{E(u_1),E(u_2)\}$$

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 at a proof.

But when I was a freshman in undergraduate school and I did not know the method of perron lifting, I tried something I named discrete-to-continuous approach to try to solve the problem. It is always a puzzle in my mind iff we can solve the Dirichlet principle in this way, roughly speaking, it is divided into two parts:

  1. Investigate the discretization of harmonic function in smaller and smaller scale. The discretization I consider is just $\Omega \cap \epsilon \mathbb Z^2$, i.e., the $\epsilon$-lattice in $\Omega$, and discretization Laplace operator $$\Delta_{\epsilon} u(x_1,\dots,x_n) =\sum_{i_1,\dots,i_n\in\{-1,1\}}\frac{u(x_1+i_1,\dots,x_n+i_n)}{2^n}-u(x_1,\dots,x_n)$$ 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 $\hat f_{\epsilon}$ of $\epsilon$ discretization and we do a extension $\Omega\cap \epsilon \mathbb Z^2$ to $\Omega$ by take value of a small tube by the center of the tube, where the value have a definition by $\hat f_{\epsilon}$, and now we get $f_{\epsilon}$.

  2. The second step is to proof the solution $f_{\epsilon}$ with $\epsilon$-discretization problem will coverage to the solution of original problem;i.e. we want to proof a $L^{\infty}$ estimate;i.e. $\forall \delta>0$, $\exists \epsilon>0, \forall 0<\epsilon_1,\epsilon_2<\epsilon$ we have $\forall x\in \Omega$, $|f_{\epsilon_1}(x)-f_{\epsilon_2}(x)|<\delta$. and by Albano-Ascoli theorem to construct $f$. 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 $f$ satisfied mean-value property for every ball in $\Omega$.

Here is my first question, which is focus on rigorous the above sketch.

Question 1: How to prove the $L^{\infty}$ estimate and the MVP of $\epsilon$-discretization will coverage to the MVP in $R^n$ case occor in second step?

My attempt to the $L^{\infty}$ 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?

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    $\begingroup$ Yes, you can in general prove existence in this way. In place of step 2, all you need is that $|f_\epsilon|$ is uniformly bounded (which you can get from the discrete maximum principle). Then you can get convergence via the Barles-Souganidis framework: Barles and Souganidis. Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Analysis, vol. 4, no. 3, pp. 271-283, 1991 $\endgroup$ – Jeff Nov 24 '17 at 19:55

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