Quasilinear elliptic equations with signed measure

Neil S. Trudinger; Xu-Jia Wang

doi:10.3934/dcds.2009.23.477

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2009, Volume 23, Issue 1&2: 477-494. Doi: 10.3934/dcds.2009.23.477

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Quasilinear elliptic equations with signed measure

Neil S. Trudinger^1, and
Xu-Jia Wang^2,

1.
Centre for Mathematics and Its Applications, the Australian National University, Canberra, ACT 0200
2.
Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200

Received: October 31, 2007

Revised: February 29, 2008

Published: April 2009

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Abstract

This paper treats quasilinear elliptic equations indivergence form whose inhomogeneous term is a signed measure. Wefirst prove the existence and continuity of generalized solutions tothe Dirichlet problem. The main result of this paper is a weakconvergence result, extending previous work of the authors forsubharmonic functions and non-negative measures. We also prove auniqueness result for uniformly elliptic operators and for operatorsof $p$-Laplacian type.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 35D05.

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