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It is known that if $D$ is a bounded domain in $\mathbb R^n$, $n \geq 2$, with $\partial D \in C^2$ then the Dirichlet problem $$ \begin{array}{rl} \Delta u & = 0 \quad \text{in $D$}, \tag{1}\\ u|_{\partial D} & = \varphi \end{array} $$ has a classical solution for any $\varphi \in C(\partial D)$, i.e. there exists $u \in C^2(D) \cap C(\overline D)$, satisfying (1).

Consider now the following Dirichlet problem: $$ \begin{array}{rl} \Delta u + v(x)u & = 0 \quad \text{in $D$,} \\ u|_{\partial D} & = \varphi, \tag{2} \end{array} $$ where $v \in C(\overline D)$. Is it true that problem (2) has a classical solution for any $\varphi \in C(\partial D)$? If not, what is the minimal requirement on smoothness of $v$ for existence of classical solution? Is the requirement $v \in C^\infty(\overline D)$ sufficient?

Andrews
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Appliqué
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  • Very nice question, I just spent 45 minutes going back over some pages in Brezis' "Functional Analysis, Sobolev Spaces and Partial Differential Equations" and have concluded that I'm too tired to be doing regularity of PDEs; however the good news is the feeling I get is that as long as $v \in L^2$ (or maybe $L^\infty$) there is (seems to me the fact that you're still looking at a homogenous equation leaves things in a pretty nice place). If I have time tomorrow I'll revisit the problem. – DanZimm Jan 12 '15 at 09:32
  • According to Gilbarg-Trudinger section 6.3, if $v\leq 0$ and $v\in C^{0,\alpha}(D)$, then there is a unique classical (in your sense) solution for boundary values $\varphi\in C(\partial D)$. The proof they give is the Perron method of constructing solutions, for which the maximum principle is an essential tool, so I don't see a straightforward way of generalizing to the case of arbitrary $v$. – Jose27 Jan 12 '15 at 10:15
  • @Jose27 We could also require that $v \in C^{0,\alpha}_c(D)$ and $|v|_C(D) \ll 1$. Then the solution is given by the (successive approximation)-solution of a corresponding integral equation. – Appliqué Jan 12 '15 at 10:31
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    I don't think that this is true for $v$ only continuous (take a look here). On the other hand, if $v\in C^{0,\alpha}$ and $|v|_\infty<\lambda_1$, where $\lambda_1$ is the first Dirichlet eigenvalue then, it seems true. Also, take a look in Theorem 9.35. in Brezis Functional Analysis book. – Tomás Jan 12 '15 at 16:09

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