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\(L^ 2\)-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 \(\Omega\).

MSC:

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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