Poisson Equation:Why the boundary regularity of the domain is important for the regularity of the solution? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T03:48:06Z http://mathoverflow.net/feeds/question/83176 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83176/poisson-equationwhy-the-boundary-regularity-of-the-domain-is-important-for-the-r Poisson Equation:Why the boundary regularity of the domain is important for the regularity of the solution? warsaga 2011-12-11T12:37:16Z 2011-12-11T17:45:47Z <p>Dear all,</p> <p>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?</p> <p>Thank you,</p> <p>Sebastian</p> http://mathoverflow.net/questions/83176/poisson-equationwhy-the-boundary-regularity-of-the-domain-is-important-for-the-r/83177#83177 Answer by Piero D'Ancona for Poisson Equation:Why the boundary regularity of the domain is important for the regularity of the solution? Piero D'Ancona 2011-12-11T13:13:56Z 2011-12-11T13:13:56Z <p>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 $C^\infty$.</p> http://mathoverflow.net/questions/83176/poisson-equationwhy-the-boundary-regularity-of-the-domain-is-important-for-the-r/83191#83191 Answer by Yuri Bakhtin for Poisson Equation:Why the boundary regularity of the domain is important for the regularity of the solution? Yuri Bakhtin 2011-12-11T17:45:47Z 2011-12-11T17:45:47Z <p>I am not sure if this helps when teaching a basic PDE class, but this is certainly a useful understanding:</p> <p>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.</p> <p>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.</p> <p>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.</p>