Regularity theory

Remarks

Example: let $u \in H^1_{\mathrm{loc}}(\Omega)$ be a weak solution of $-\Delta u = f$ with $f \in L^2_{\mathrm{loc}}(\Omega)$. Then $u \in H^2_{\mathrm{loc}}(\Omega)$, and more generally $f \in H^k_{\mathrm{loc}}$ gives $u \in H^{k+2}_{\mathrm{loc}}$; this is elliptic regularity with $X = H^1_{\mathrm{loc}}$ and $Y = H^2_{\mathrm{loc}}$.

Non-example: for the transport equation $u_t + u_x = 0$ in $\R^2$, the function $u(x, t) = g(x - t)$ is a weak solution for every $g \in L^2_{\mathrm{loc}}(\R)$. So smooth data, here $f = 0$, do not force smooth solutions, and no regularity result of this kind holds; singularities travel along characteristics.

Interior regularity concerns compact subsets of $\Omega$; regularity up to the boundary also needs a smooth boundary and smooth boundary data.