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

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.

References in corpus (2)

• Dirichlet forms and semilinear elliptic equations with measure data
• Transition Semigroups of Banach Space Valued Ornstein-Uhlenbeck Processes

Cited by in corpus (6)

• Reflected BSDEs on Filtered Probability Spaces
• Reduced measures for semilinear elliptic equations involving Dirichlet operators
• Backward problems for stochastic differential equations on the Sierpinski gasket
• Right Markov processes and systems of semilinear equations with measure data
• Nonlinear elliptic equations with integro-differential divergence form operators and measure data under sign condition on the nonlinearity
• Nonlinear parabolic SPDEs involving Dirichlet operators