Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form

Abstract

We generalize the notion of renormalized solution to semilinear elliptic and parabolic equations involving operator associated with general (possibly nonlocal) regular Dirichlet form and smooth measure on the right-hand side. We show that under mild integrability assumption on the data a quasi-continuous function $u$ is a renormalized solution to an elliptic (or parabolic) equation in the sense of our definition iff $u$ is its probabilistic solution, i.e. $u$ can be represented by a suitable nonlinear Feynman-Kac formula. This implies in particular that for a broad class of local and nonlocal semilinear equations there exists a unique renormalized solution.

References in corpus (5)

• Inverse Problem for a Curved Quantum Guide
• Renormalized solutions of nonlinear parabolic equations with general measure data
• Diffuse measures and nonlinear parabolic equations
• Dirichlet forms and semilinear elliptic equations with measure data
• Obstacle problem for semilinear parabolic equations with measure data

Cited by in corpus (7)

• Reduced measures for semilinear elliptic equations involving Dirichlet operators
• Dirichlet problem for semilinear partial integro-differential equations: the method of orthogonal projection
• Renormalized solutions of semilinear elliptic equations with general measure data
• On semilinear elliptic equations with diffuse measures
• Nonlocal operators in divergence form and existence theory for integrable data
• Smooth measures and capacities associated with nonlocal parabolic operators
• On approximation of the Dirichlet problem for divergence form operator by Robin problems