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

Tomasz Klimsiak ¹   ; Andrzej Rozkosz ¹

¹ Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland

Colloquium Mathematicum, Tome 145 (2016) no. 1, pp. 35-67

Résumé

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.

Détail

DOI : 10.4064/cm6466-10-2015

Keywords: mainly concerned equations form lu where operator associated quasi regular possibly nonsymmetric dirichlet form satisfies monotonicity condition mild integrability conditions bounded smooth measure prove general results existence uniqueness regularity probabilistic solutions which expressed terms solutions backward stochastic differential equations applications include equations nonsymmetric divergence form operators gradient perturbations pseudodifferential operators equations ornstein uhlenbeck type operators hilbert spaces briefly discuss existence uniqueness probabilistic solutions where corresponds lower bounded semi dirichlet form