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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 |
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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 $C^\infty$. |
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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. |
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