# Some elliptic problems with singularities

Abstract : In this dissertation, we first study removable singularity results for second order elliptic PDEs, the model case being the following: $- \Delta u + cu \geq f$ in $\Omega \backslash K$, with $u \geq 0$ and $(\rm cap)_2((K))=0$. We also prove a strong maximum principle for the operator $- \Delta + a(x)$, with a potential $a \in L^1$. These results rely on some variants of the standard Kato's inequality. We next present an estimate in the spirit of Poincare's inequality. The motivation for our result comes from a new characterization of the Sobolev spaces. We also study topological singularities of maps $g \in W^(1,1)(S^2;S^1)$; we compute, for instance, the relaxed energy and the total variation of the Jacobian of $g$. Finally, we consider several properties of the distributions of the form $\sum_j((\delta_(p_j) - \delta_(n_j)))$, defined on a complete metric space.
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https://tel.archives-ouvertes.fr/tel-00009043
Contributor : Augusto Ponce <>
Submitted on : Sunday, April 17, 2005 - 11:32:35 PM
Last modification on : Wednesday, December 9, 2020 - 3:06:02 PM
Long-term archiving on: : Friday, April 2, 2010 - 9:53:58 PM

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• HAL Id : tel-00009043, version 1

### Citation

Augusto Ponce. Some elliptic problems with singularities. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2004. English. ⟨tel-00009043⟩

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