## Differential and Integral Equations

- Differential Integral Equations
- Volume 8, Number 2 (1995), 333-352.

### On homeomorphisms for an elliptic equation in domains with corners

#### Abstract

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.

#### Article information

**Source**

Differential Integral Equations, Volume 8, Number 2 (1995), 333-352.

**Dates**

First available in Project Euclid: 20 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1369083473

**Mathematical Reviews number (MathSciNet)**

MR1296128

**Zentralblatt MATH identifier**

0829.35026

**Subjects**

Primary: 35J25: Boundary value problems for second-order elliptic equations

Secondary: 35A20: Analytic methods, singularities 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

#### Citation

Aziz, A. K.; Kellogg, R. B. On homeomorphisms for an elliptic equation in domains with corners. Differential Integral Equations 8 (1995), no. 2, 333--352. https://projecteuclid.org/euclid.die/1369083473