Real Analysis Exchange

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

Marcela Sanmartino and Marisa Toschi

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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


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