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## Regularity results for elliptic problems with measure

(1) , (1) , (1)
1
Colette De Coster
Serge Nicaise
• Function : Author
• PersonId : 1080829

#### 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.

#### Domains

Mathematics [math]

### Dates and versions

hal-03142664 , version 1 (16-02-2021)

### Identifiers

• HAL Id : hal-03142664 , version 1

### Cite

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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