\(L^ 2\)-integrability of second order derivatives for Poisson’s equation in nonsmooth domains. (English) Zbl 0761.35018

Summary: We define a certain class of domains with corners directed outwards only, thus being natural extensions of convex domains. We show that such a domain \(\Omega\) can be approximated with smooth domains \(\Omega_ m\) of the same type. Using a technique based on integration by parts we derive an a priori estimate \[ \| u\|_{H^ 2(\Omega_ m)}\leq C(\Omega)\|\Delta u\|_{L^ 2(\Omega_ m)}\quad\text{for}\quad u\in H^ 2(\Omega_ m)\cap H^ 1_ 0(\Omega_ m) \] , where \(C(\Omega)\) is independent of \(m\). This enables us to obtain a solution \(u\) in \(H^ 2(\Omega)\) of the Dirichlet problem \[ \Delta u=f\in L^ 2(\Omega),\quad \gamma u=0. \] Here \(\gamma\) is the trace operator on the boundary of \(\Omega\).


35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B65 Smoothness and regularity of solutions to PDEs
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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