arXiv subjects

Ricardo Duran

Publications and source records attributed to Ricardo Duran.

3 recordsLinked to original sources

arXiv2017

Weighted a priori estimates for elliptic equations

We give a simpler proof of the a priori estimates obtained in the paper by Duran, Sanmartino and Toschi for solutions of elliptic problems in weighted Sobolev norms when the weights belong to the Muckenhoupt class $A_p$. The argument is a generalization to bounded domains of the one used in $\mathbb{R}^n$ to prove the continuity of singular integral operators in weighted norms. In the case of singular integral operators it is known that the $A_p$ condition is also necessary for the continuity. We do not know whether this is also true for the a priori estimates in bounded domains but we are able to prove a weaker result when the operator is the Laplacian or a power of it. We prove that a necessary condition is that the weight belongs to the local $A_p$ class.

math.AP↗

arXiv2010

Weighted a priori estimates for the solution of the homogeneous Dirichlet problem for the powers of the Laplacian Operator

Let $u$ be a weak solution of $ (-Δ)^m u=f $ with Dirichlet boundary conditions in a smooth bounded domain $Ω\subset \mathbb{R}^n$. Then, the main goal of this paper is to prove the following a priori estimate: $$ \|u\|_{W^{2m,p}_ω(Ω)} \le C\, \|f\|_{L^p_ω(Ω)}, $$ where $ω$ is a weight in the Muckenhoupt class $A_p.$

math.CA↗

arXiv2009

Divergence operator and Poincare inequalities on arbitrary bounded domains

Let $Ω$ be an arbitrary bounded domain of $\R^n$. We study the right invertibility of the divergence on $Ω$ in weighted Lebesgue and Sobolev spaces on $Ω$, and rely this invertibility to a geometric characterization of $Ω$ and to weighted Poincaré inequalities on $Ω$. We recover, in particular, well-known results on the right invertibility of the divergence in Sobolev spaces when $Ω$ is Lipschitz or, more generally, when $Ω$ is a John domain, and focus on the case of $s$-John domains.

math.AP↗