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 $p>1$.
Highlights
Let E be a separable locally compact metric space and let m be a Radon measure on E such that supp[m] = E
In the present paper we study semilinear equations of the form
In this subsection we give an equivalent definition of solution of (3.1) using stochastic equations involving a Hunt process X associated with the Dirichlet operator A
Summary
Let E be a separable locally compact metric space and let m be a Radon measure on E such that supp[m] = E. In the present paper we study semilinear equations of the form where μ is a Borel measure on E, f : E × R → R is a measurable function such that f (·, u) = 0, u ≤ 0, and f is nonincreasing and continuous with respect to u. As for the operator A, we assume that it is a negative definite self-adjoint Dirichlet operator on L2(E; m). Saying that A is a Dirichlet operator we mean that (Au, (u − 1)+) ≤ 0, u ∈ D(A). Communicated by H. Brezis.
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