Math 6410 • ODE

Fundamental Elliptic Estimate. Let \(\Omega \subset \mathbb {R}^n\) be a bounded smooth domain and let \(H^k(\Omega )\) denote the Sobolev space of functions whose distributional derivatives up to the \(k\)-th order are in \(L^2(\Omega )\). Show that if for some constant \(\lambda \), \(u\in H^1(\Omega )\) is a weak solution to \[ \Delta u+\lambda u=0 \] then \(u\in C^\infty (\Omega )\). [You may assume the fundamental elliptic interior estimate: If \(\Omega '\Subset \Omega \) is a compactly contained subdomain, then there is a finite constant \(C=C(\Omega ,\Omega ',k,n)\) so that \[ \lVert u\rVert _{k+2,\Omega '}\le C\bigl (\lVert \Delta u\rVert _{k;\Omega }+\lVert u\rVert _{0;\Omega }\bigr ) \] for all \(u\in H^{k+2}(\Omega )\). Moreover, if \(f\in H^k(\Omega )\) and \(u\in H^1(\Omega )\) is a weak solution to \(\Delta u=f\), then \(u\in H^{k+2}(\Omega ')\).]