Elliptic regularity on bad domain

Question

Let $\Omega \subset R^n$ be an open bounded set. Consider the Dirichlet problem $$ -\triangle u = 0, x \in \Omega, \quad u = \varphi, x \in \partial \Omega. $$ If $\varphi$ is a continuous function, then the problem is solvable for rather general $\Omega$. In fact, it suffices to assume that every point on the boundary $\partial \Omega$ is a endpoint of a segment, the other point of the segment lies outside $\Omega$. See the book 'Complex analysis' by Ahlfors.

Now we turn to the Possion equation $$ -\triangle u = \varphi, x \in \Omega, \quad u = 0, x \in \partial \Omega. $$ If $\Omega$ is a cube, we have the regularity result: $u$ is bounded in $W^{2,p}(\Omega)$ if $\varphi$ is bounded in $L^p(\Omega)$ with $ 1< p < \infty$.

This result can be extended to any $C^{1,1}$ domain $\Omega$. With some calculations, it seems that the result also holds for any convex $\Omega$. My questions are

(i)Does the regularity holds for more general $\Omega$, just like Dirichlet problem?

(ii)I have seen a paper which says that bad domain problem and bad coefficient problem has some relationship, could someone explain it more precisely?

Answer 1

The paper Besov Regularity for Elliptic Boundary Value Problems by Dahlke and DeVore discusses regularity results for these problems on Lipschitz domains. I reproduce their theorem 4.1 below:

Theorem 4.1 Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^d$. Then, there is an $0<\epsilon < 1$ depending only on the Lipschitz character of $\Omega$ such that whenever $u$ is a solution to

$$\begin{align} - \Delta u &= f ~ on ~ \Omega \subset \mathbb{R}^d, \\ u &= 0 ~ on ~ \partial \Omega \end{align}$$

with $f \in B_p^{\lambda-2}(L_p(\Omega))$, $\lambda:=\frac{d}{d-1} (1+\frac{1}{p})$, $1 < p<2+\epsilon$, then $u \in B_\tau^\alpha(L_\tau(\Omega))$, $\tau=(\alpha/d+1/p)^{-1}$, for all $0 < \alpha < \lambda$.

The proofs therein use an interesting technique of expanding the solution via a wavelet multiresolution analysis, and then using the connection between the decay rate of a functions wavelet coefficients and the smoothness space that function lies in. It turns out that Besov spaces are the proper smoothness spaces for elliptic PDE solutions on "bad" domains, or with rough boundary data.

For some background on Besov spaces and wavelets, see the book Theory of Function Spaces II by Triebel, and the excellent expository paper Wavelets by DeVore and Lucier.

A more classical treatment is given by Grisvard in his book, Elliptic Problems in Nonsmooth Domains, though the results are not quite as good as the newer Besov results.

Answer 2

What do you mean by "$u$ is bounded in $W^{2,p}(\Omega)$ if $\phi$ is bounded in $L^p(\Omega)$"? Bounded with respect to which variable? Is there a time dependence which you have not written down? Or do you actually mean: "$u$ is in $W^{2,p}(\Omega)$ if $\phi$ is in $L^p(\Omega)$"? If so, for $p=2$ the assertion is true for any bounded open set, by Lax-Milgram.