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


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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