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Existence and regularity results for solutions of second-order, elliptic, integrodifferential operators. (English) Zbl 0579.45010
Let \(\Omega\) be a bounded smooth domain in \({\mathbb{R}}^ N\). Given an elliptic second order partial differential operator, A, a non local perturbation in form of an integrodifferential-(Lévy)-operator, B, is considered, which is arranged to be relatively A-small in some sense. For the resulting operator \(L=A-B\) the classical Dirichlet problem is studied with the aid of an analogue of the classical maximum principle (extended to Sobolev-spaces) and related pointwise a priori estimates. The dependence of existence and regularity of strong and weak solutions upon the particular nature of the measures determining the operator B is studied in detail. The method given is very general in that it can be paraphrased for other boundary conditions and extended to cover non- linear equations like the Hamilton-Jacobi-Bellman equation for the controlled jump-diffusion process; as a byproduct, the class of problems tractable with this theory includes some differential-delay operators.
For the stochastic modelling the author refers to D. Strook [Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 209-244 (1975; Zbl 0292.60122)], the extension of the maximum principle is related to the second author [Proc. Am. Math. Soc. 88, 503-508 (1983; Zbl 0525.35028)]; the paper is a continuation and generalization of S. Lenhart’s article in Appl. Math. Optimization 9, 177-191 (1982; Zbl 0513.45014).
Reviewer: G.Leugering

45K05 Integro-partial differential equations
49L99 Hamilton-Jacobi theories
60J75 Jump processes (MSC2010)