Open Access
1995 On homeomorphisms for an elliptic equation in domains with corners
A. K. Aziz, R. B. Kellogg
Differential Integral Equations 8(2): 333-352 (1995). DOI: 10.57262/die/1369083473


Our purpose is to construct a space of distribution solutions in a simple case for which the data is not smooth. We consider the mixed boundary value problem for the equation $-\Delta u=f$ in a domain $\Omega$ with polygonal boundary $\Gamma$. On each side of $\Gamma$ we impose either Dirichlet or Neumann boundary conditions. This is a ``corner problem'' whose solution contains corner singularities that are well understood [2, 3]. We are therefore in a position to construct a dual theory of distribution solutions for this mixed problem. In this paper we make this construction for distribution solutions $u \in L_2(\Omega)$; that is, the case when the solution is one step below the energy space in regularity. For this, we must give a careful definition of the data space associated with the mixed problem, and the "trace space" associated with a function $u \in L_2(\Omega)$. We find that there is always a distribution solution to the mixed boundary value problem, but the solution may not be unique; there may be distribution solutions to the homogeneous problem constructed with the help of the corner singular functions.


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A. K. Aziz. R. B. Kellogg. "On homeomorphisms for an elliptic equation in domains with corners." Differential Integral Equations 8 (2) 333 - 352, 1995.


Published: 1995
First available in Project Euclid: 20 May 2013

zbMATH: 0829.35026
MathSciNet: MR1296128
Digital Object Identifier: 10.57262/die/1369083473

Primary: 35J25
Secondary: 35A20 , 47F05

Rights: Copyright © 1995 Khayyam Publishing, Inc.

Vol.8 • No. 2 • 1995
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