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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?

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A quick remark on the solvability of Laplace's equation: We need $n = 2$ for the segment endpoint criterion to work, I think because complex analysis gives a nice way to construct a barrier. In higher dimensions, an exterior cone condition suffices; for a weaker condition see "Wiener criterion" in e.g. Gilbarg-Trudinger. –  Connor Mooney Aug 29 '12 at 12:28
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(In the above comment I mean solve the Dirichlet problem for Laplace with solution continuous up to the boundary). –  Connor Mooney Aug 29 '12 at 12:31
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2 Answers 2

up vote 1 down vote accepted

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.

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Your answer is very helpful, thanks. –  Wang Ming Oct 20 '12 at 12:54
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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.

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No time varialbe get involved. Could you give some precise references ? –  Wang Ming Aug 29 '12 at 13:59
    
§6.1 and §6.2 in Evans' Partial Differential Equations book, for instance. –  Delio Mugnolo Aug 29 '12 at 15:23
    
Only interior regularity is obtained there, this is a weak type estimate, it need not any assumption on the domain of course. –  Wang Ming Aug 30 '12 at 13:07
    
I do not understand your problem yet. If your conjecture actually is "$u$ is in $W^{2,p}(\Omega)$ if $\phi$ is in $L^p(\Omega)$", then for $p=2$ this is what Lax-Milgram says. Are you asking about boundary regularity or some kind of boundedness of $u$? If the former is correct, then I can also understand your reference to $C^{1,1}$-boundaries. –  Delio Mugnolo Aug 30 '12 at 14:32
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