Reduced measures for semilinear elliptic equations involving Dirichlet operators. (English) Zbl 1357.35140
In this paper, the author studies semilinear equations of the form
\[
Au = f (x, u) + \mu,
\]
where \(E\) is a separable locally compact metric space, \(\mu\) is a Borel measure on \(E\), \(f : E\times \mathbb R \to \mathbb R\) is a measurable function such that \(f (\cdot, u) = 0\) for \(u\leq 0\), and f is nonincreasing and continuous with respect to \(u\). While \(A\) is a negative definite and self-adjoint Dirichlet operator on \(L^2(E;m)\). Here, \(m\) is a Radon measure on \(E\) such that \(\operatorname{supp}[m] = E\).
Along the paper, the author introduces the definition of probabilistic solution for the problem under consideration. In this context, he proves some comparison results, a priori estimates, some regularity results of solutions, the so-called inverse maximum principle and Kato’s type inequality.
Along the paper, the author introduces the definition of probabilistic solution for the problem under consideration. In this context, he proves some comparison results, a priori estimates, some regularity results of solutions, the so-called inverse maximum principle and Kato’s type inequality.
Reviewer: Raffaella Servadei (Arcavata di Rende)
MSC:
| 35J61 | Semilinear elliptic equations |
| 35J75 | Singular elliptic equations |
| 60J45 | Probabilistic potential theory |
| 35B51 | Comparison principles in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35B45 | A priori estimates in context of PDEs |
Keywords:
semilinear equations; Dirichlet operators; good measure; reduced measure; probabilistic solution; comparison results; a priori estimates; regularity; inverse maximum principle; Kato’s type inequalityReferences:
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