Question

I asked a question before where I wanted a simple example where regularity up to the boundary fails for a linear elliptic PDE. I was presented an example with $\Omega = B(0,1) \backslash {0}$ (ball minus a point) which is nice but I would like something less pathalogical. I would like an example where my domain is at least Lipschitz (so a rectangle is an example). In the case where we subtract off a point, a boundary value problem doesn't really even make sense in a weak formulation to begin with.

More precisely,

Question: I would like an example where an elliptic PDE with smooth coefficients satisfies $Lu = f$ in $\Omega$ for smooth $f$ (or zero) and $u = 0$ on $\partial \Omega$ (which is of Lipschitz class) but where $u$ somehow fails to be "regular" up to the boundary (I'm being vague on purpose here as any failure of regularity will do for the most part).

My guesses have been try try looking at the upper right quadrant $[{ (x,y) : x, y > 0}]$ but nothing has come from this so far. Any suggestions/ideas are welcome and appreciated.

Answer 1

You might want to consider the function $v(z)=\Im(z^{\pi/\alpha})=r^{\pi/\alpha}\sin(\pi\theta/\alpha)$ in the sector $0<\theta<\alpha$ of the complex plane. It is harmonic in the interior, continuous up to the boundary and vanishes there, but has a "singularity" at $0$ if $\alpha$ is not of the form $\pi/n$. More precisely, the derivative of order $k$ along the ray $\theta=\alpha/2$ explodes at $0$ when $k>\pi/\alpha$ and $\pi/\alpha$ is not an integer.

More generally, you can take $v(z)=\Im(F(z))$, where $F$ is [edit: the inverse of] a Schwarz-Cristoffel mapping, taking the interior $\Omega$ of a polygon in the complex plane to the upper half plane $\Im w >0$.

Answer 2

$-\Delta u=1$ on the unit disk excluding a solid sector (i.e. a round cake with piece taken) with homogeneous Dirichlet boundary condition. There is an explicit characterization of regularity loss on polygonal domains in e.g. Grisvard's book. Maybe you can work out the case of e.g. the plane with a quadrant missing.