Elliptic Regularity in Sobolev Spaces with Negative Order $(s<0)$

Asked Jul 10 '17 at 20:20

Active Jul 10 '17 at 22:42

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I am looking for an elliptic regularity theorem in the Sobolev Space $H^s(\mathbb R^3)$ for $s<0$. In fact we have the equation $$A\,u=f,$$ where $A$ is an elliptic operator and $f\in H^s(\mathbb R^3)$ with $s<0$.

Can we deduce that $u\in H^{s+2}(\mathbb R^3)$?

I am also looking for a good reference in Elliptic Regularity please.

edited Jul 10 '17 at 22:42

Michael Hardy

asked Jul 10 '17 at 20:20

H. J

Hello and welcome to math.stackexchange. Any assumptions about $A$? Divergence form or not? Smooth coefficients? – Hans Engler Jul 10 '17 at 20:23
$A,u=-\Delta,u+,u,$ – H. J Jul 10 '17 at 20:44
Use the Fourier transform or eigenfunctions. – Hans Engler Jul 10 '17 at 21:13

Elliptic Regularity in Sobolev Spaces with Negative Order $(s<0)$

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