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Mediterranean Journal of Mathematics

, Volume 5, Issue 2, pp 173–185 | Cite as

Nonlinear Elliptic Problems with L 1 Data: an Approach via Symmetrization Methods

  • Angelo Alvino
  • Anna Mercaldo
Article

Abstract.

We consider nonlinear elliptic problems whose prototype is
$$\left\{ \begin{array}{ll} -\Delta_p u = -{\rm{div}}(|\nabla u|^{p - 2} \nabla u) = f, & \rm{in}\,\Omega, \\ u = 0, & \rm{on}\,\partial \Omega, \end{array} \right.$$
, with Ω bounded open subset of \({\mathbb{R}}^{N}\) and p > 1. When \(f \in L^{1}(\Omega)\) several notions of solutions have been introduced; we refer to distributional solutions which can be obtained by an approximation procedure and point out that the question can be faced by a new method which uses symmetrization techniques. In this way we prove both a priori estimates and a continuity with respect to data result which allow us to deduce existence and uniqueness of the solution.

Keywords.

Symmetrization methods Isoperimetrical estimates L1 data 

Mathematics Subject Classification (2000).

Primary 35J25 Secondary 35J60 

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Copyright information

© Birkhaueser 2008

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”Universitá degli Studi di Napoli “Federico II”NapoliItaly

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