# Interior smooth regularity

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?

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.

• Thanks! You are right that I forgot to say $f$ is smooth. This is the answer I am looking for! Is this a common technique as dealing with such a problem? – User May 12 '19 at 11:50
• Yes. For another approach in the general case check Sec 10.3.2. of these notes www3.nd.edu/~lnicolae/Lectures.pdf – Liviu Nicolaescu May 12 '19 at 12:12
• This helps me a lot, and thanks for your suggestions about good references! – User May 12 '19 at 12:24

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.

• Thank you! So this result doesn't require any assumption on the boundary of $\Omega?$ – User May 12 '19 at 11:04
• No, you do not need any assumption on the boundary for this result. – Bazin May 12 '19 at 16:43
• Thank you. I will also try to see more general theorems like what you mentioned! – User May 17 '19 at 4:53