Interior smooth regularity

Question

I recently read the PDE book of L. Evans, and in its chapter 6 some kinds of regularities of second-order elliptic equations were discussed. My question is about its proof of interior smooth regularity (thm 3 in Ch. 6.3.1).

The theorem asserts that if a second-order elliptic PDE $Lu=f$ has smooth coefficients and admits an $H^1$ weak solution $u$ on a bounded domain $U\subseteq \mathbb R^n,$ then $u$ is smooth.

In the statement of the theorem, no condition is assumed about $\partial U,$ but in the book, the author would like to apply the general Sobolev inequality to assure $u\in C^\infty(U)$ if $u\in H^m_{\text{loc}}(U)$ for all $m\in\mathbb N,$ which can be obtained by the theorem 6 in its chapter 5.6.3, but with $\partial U$ is $C^1.$

My problem is whether the regular assumption on the boundary is necessary. However, without the condition $\partial U$ being $C^1,$ I only know some Sobolev inequality about $W^{k,p}_0(U).$ I am not sure if these are sufficient to derive the desired conclusion (since I don't know the behavior of $u$ near the boundary a priori). Perhaps there are alternatives to this problem, or the conclusion is just wrong without the boundary assumption?

Thanks in advance!

Answer 1

I assume that you require $f\in C^\infty(U)$. You do not need regularity of the boundary of $U\subset \mathbb{R}^N$. The condition $u\in H^m_{loc}(U)$ is equivalent with $\widetilde{\phi u}\in H^m(\mathbb{R}^N)$, for any $\phi\in C^\infty_0(U)$. Here $\widetilde{g}$ denotes the extension of the function $g:U\to\mathbb{R}$ by zero outside $U$. Apply the Sobolev embedding theorems to $\widetilde{\phi u}$.

I strongly recommend opening Brezis' book on functional anaylsis and pde's.

Answer 2

If I understand your question correctly, you speak about interior regularity. Let me quote a classical result for linear elliptic equations with $C^\infty$ coefficients, even true for pseudo-differential equations.

Let $P$ be an elliptic differential operator with $C^\infty$ coefficients in an open subset $\Omega$ of $\mathbb R^N$. Then for $u$ a distribution on $\Omega$, $$ Pu\in C^\infty(\Omega)\Longrightarrow u\in C^\infty(\Omega). $$ You may refine that result in the Sobolev scale with $$ Pu\in H^s_{loc}(\Omega)\Longrightarrow u\in H^{s+m}_{loc}(\Omega), $$ where $m$ is the order of $P$. If you like the wave-front-set, you have $$WF(Pu)\subset WF(u)\subset WF(Pu)\cup \text{char}P, $$ and in the elliptic case $\text{char}P=\emptyset$. You can also formulate a result on the $H^s$ wave-front-set.