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Edge behavior of the solution of an elliptic problem. (English) Zbl 0639.35008
Let $$\Omega$$ be a plane domain with polygonal boundary $$\Gamma$$. If $$\Omega$$ has just one non-convex corner at $$x=0$$ then is well known that any solution to $$-\Delta v=g\in L^ 2(\Omega),$$ $$v\in H^ 1_ 0(\Omega)$$ can be written as $$v=v_ R+c\cdot s$$ where $$v_ R\in H^ 2(\Omega)$$, $$c\in {\mathbb{R}}$$ and $$s\in H^ 1_ 0(\Omega)\setminus H^ 2(\Omega)$$ is known explicitly. The paper under review is devoted to discussing the analogous singularities in the vicinity of the non-convex edge $$\{0\}\times {\mathbb{R}}$$ on the three-dimensional cylinder $$\Omega\times {\mathbb{R}}$$. Using Fourier transforms in the third variable the author is able to exhibit explicitly a function space S such that any solution u to $$-\Delta u=f\in L^ 2(\Omega),$$ $$u\in H^ 1_ 0(\Omega)$$ can be written as $$u=u_ R+s$$ where $$u_ R\in H^ 2(\Omega)$$ and $$s\in S$$. The results are generalized to an L p-framework. The same technique is applied to the heat equation in the half-cylinder $$Q_+:= \Omega\times{\mathbb{R}}^+$$ with similar results.
Reviewer: N.Weck

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35K05 Heat equation 35A22 Transform methods (e.g., integral transforms) applied to PDEs
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