## Real Analysis Exchange

- Real Anal. Exchange
- Volume 39, Number 2 (2014), 345-362.

### Weighted a Priori Estimates for the Solution of the Dirichlet Problem in Polygonal Domains in \(\mathbb{R}^2\)

Marcela Sanmartino and Marisa Toschi

#### Abstract

Let \(\Omega\) be a polygonal domain in \(\mathbb{R}^2\) and let \(U\) be a weak solution of \( -\Delta u=f\) in \( \Omega\) with Dirichlet boundary condition, where \(f\in L^p_\omega(\Omega)\) and \(\omega\) is a weight in \(A_p(\mathbb{R}^2)\), \(1<p<\infty\). We give some estimates of the Green function associated to this problem involving some functions of the distance to the vertices and the angles of \(\Omega\). As a consequence, we can prove an a priori estimate for the solution \(u\) on the weighted Sobolev spaces \(W^{2,p}_\omega(\Omega)\), \(1<p<\infty\).

#### Article information

**Source**

Real Anal. Exchange, Volume 39, Number 2 (2014), 345-362.

**Dates**

First available in Project Euclid: 30 June 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1435670000

**Mathematical Reviews number (MathSciNet)**

MR3365379

**Zentralblatt MATH identifier**

1330.35111

**Subjects**

Primary: 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]

Secondary: 35J08: Green's functions

**Keywords**

Dirichlet problem Green function Weighted Sobolev spaces

#### Citation

Sanmartino, Marcela; Toschi, Marisa. Weighted a Priori Estimates for the Solution of the Dirichlet Problem in Polygonal Domains in \(\mathbb{R}^2\). Real Anal. Exchange 39 (2014), no. 2, 345--362. https://projecteuclid.org/euclid.rae/1435670000