## $$L^ 2$$-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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