Reduced measures for semilinear elliptic equations involving Dirichlet operators

Abstract

We consider elliptic equations of the form (E) $-Au=f(x,u)+\mu$, where $A$ is a negative definite self-adjoint Dirichlet operator, $f$ is a function which is continuous and nonincreasing with respect to $u$ and $\mu$ is a Borel measure of finite potential. We introduce a probabilistic definition of a solution of (E), develop the theory of good and reduced measures introduced by H. Brezis, M. Marcus and A.C. Ponce in the case where $A=\Delta$ and show basic properties of solutions of (E). We also prove Kato's type inequality. Finally, we characterize the set of good measures in case $f(u)=-u^p$ for some $p>1$.

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• Inverse Problem for a Curved Quantum Guide
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• Dirichlet forms and semilinear elliptic equations with measure data
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• Nonlinear elliptic equations with integro-differential divergence form operators and measure data under sign condition on the nonlinearity
• Obstacle problem for evolution equations involving measure data and operator corresponding to semi-Dirichlet form