# Regularity results for elliptic problems with measure

Abstract : In this work, we study the solution of the Laplace equation: $-\Delta u =g \delta_\sigma\;\;\;\hbox{ in } Q\subseteq \mathbb{R}^3,$ where $$\delta_\sigma$$ is the Dirac mass on a crack $$\sigma$$ of $$Q$$ and $$g\in L^2(\sigma)$$. First, we discuss the existence and the uniqueness of a solution in $$W^{1,p}(Q)$$ for $$p<2$$ (due to the Dirac mass, the right-hand side is not in $$H^{-1}(Q)$$). Then, we prove the regularity of the solution and a priori estimates in weighted Sobolev spaces.
Document type :
Poster communications
Domain :

https://hal-uphf.archives-ouvertes.fr/hal-03142664
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Submitted on : Tuesday, February 16, 2021 - 11:11:35 AM
Last modification on : Wednesday, July 6, 2022 - 3:06:24 PM

### Identifiers

• HAL Id : hal-03142664, version 1

### Citation

Sadjiya Ariche, Colette De Coster, Serge Nicaise. Regularity results for elliptic problems with measure. Nord Pas-de-Calais/Belgium congress of mathematics, Oct 2013, Valenciennes, France. ⟨hal-03142664⟩

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