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



Poisson Equation:Why the boundary regularity of the domain is important for the regularity of the solution?
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 $C^\infty$.


