# Poisson Equation:Why the boundary regularity of the domain is important for the regularity of the solution?

Dear all,

giving a support class for PDE lecture i am wondering is there an easy argument for : Why the boundary regularity of the domain important for the regularity of the solution of the weak form of the Poisson equation with Dirichlet boundary conditions?

Thank you,

Sebastian

-
First, note that for an elliptic PDE, the interior regularity does not require any assumptions on the boundary regularity of the domain or the solution. If, however, you want to study the regularity of the solution at the boundary, then assumptions are needed. If the boundary is smooth, then it can locally be transformed into the upper half space of $R^n$ and it is possible to define the tangential and normal derivatives of the solution at the boundary and study their properties using calculus. If the boundary is not smooth, then these are not well defined. – Deane Yang Dec 11 '11 at 16:47

You might start by looking at the book by Grisvard (Elliptic problems in nonsmooth domains). For instance, in Theorem 3.1.1.1 he proves a very precise identity which shows basically the following: if you want to estimate ANY second derivative of a function $u$ defined on a domain $\Omega$ in terms of the laplacian $\Delta u$ (i.e., if you want to prove regularity of $u$ from the regularity of $f=\Delta u$), then you can if the boundary is $C^2$, but with a constant depending on the negative part of the curvature of the boundary. Even in the simplest case when $\Omega$ is a nonconvex polygon, you can construct $u$ not in $H^2(\Omega)$ such that $\Delta u$ is in $C^\infty$.

-

I am not sure if this helps when teaching a basic PDE class, but this is certainly a useful understanding:

Elliptic problems can be interpreted via diffusion processes. The solution at a point $x$ can be written as expectation of the boundary condition at the (random) exit point for the diffusion emitted from $x$ and associated to the elliptic operator.

If the boundary is smooth, then as $x\to x_0\in\partial \Omega$ the exit distribution converges to the Dirac measure at $x_0$, hence regularity of the solution.

If the boundary is bad, then the diffusion initiated at a boundary point $x_0$ can, with positive probability, hit the boundary next time at a completely different place, and the exit distribution can be very far from the Dirac measure, hence there is a problem.

-