## Real Analysis Exchange

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

#### 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

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