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Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms

Volume 145 / 2016

Tomasz Klimsiak, Andrzej Rozkosz Colloquium Mathematicum 145 (2016), 35-67 MSC: 35J61, 35R06; Secondary 60H30. DOI: 10.4064/cm6466-10-2015 Published online: 14 April 2016

Abstract

We are mainly concerned with equations of the form $-Lu=f(x,u)+\mu $, where $L$ is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, $f$ satisfies the monotonicity condition and mild integrability conditions, and $\mu $ is a bounded smooth measure. We prove general results on existence, uniqueness and regularity of probabilistic solutions, which are expressed in terms of solutions to backward stochastic differential equations. Applications include equations with nonsymmetric divergence form operators, with gradient perturbations of some pseudodifferential operators and equations with Ornstein–Uhlenbeck type operators in Hilbert spaces. We also briefly discuss the existence and uniqueness of probabilistic solutions in the case where $L$ corresponds to a lower bounded semi-Dirichlet form.

Authors

  • Tomasz KlimsiakFaculty of Mathematics and Computer Science
    Nicolaus Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland
    e-mail
  • Andrzej RozkoszFaculty of Mathematics and Computer Science
    Nicolaus Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland
    e-mail

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