Annali di Matematica Pura ed Applicata

, Volume 170, Issue 1, pp 57–87 | Cite as

Existence and regularity results for relaxed Dirichlet problems with measure data

  • Annalisa Malusa
  • Luigi Orsina


We study the following relaxed Dirichlet problem
$$\left\{ \begin{gathered} Lu + \mu u = vin\Omega , \hfill \\ u = 0on\partial \Omega , \hfill \\ \end{gathered} \right.$$
where Ω is a bounded open subset ofRN,Lu=−div(A∇u) is an elliptic operator, μ is a positive Borel measure on Ω not charging polar sets, and v is a measure with bounded variation on Ω. We give a definition of solution for such a problem, and then prove existence and regularity results. As a consequence, the Green function for relaxed Dirichlet problems can be defined, and some of its properties are proved, including the standard representation formula for solutions.


Measure Data Open Subset Green Function Dirichlet Problem Elliptic Operator 
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© Fondazione Annali di Matematica Pura ed Applicata 1996

Authors and Affiliations

  • Annalisa Malusa
    • 1
  • Luigi Orsina
    • 2
  1. 1.Facoltà di ArchitetturaIstituto di MatematicaNapoliItaly
  2. 2.Dipartimento di MatematicaUniversità di Roma IRomaItaly

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