Gimbert, F.; Lions, Pierre-Louis Existence and regularity results for solutions of second-order, elliptic, integrodifferential operators. (English) Zbl 0579.45010 Ric. Mat. 33, 315-358 (1984). 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 Cited in 23 Documents MSC: 45K05 Integro-partial differential equations 49L99 Hamilton-Jacobi theories 60J75 Jump processes (MSC2010) Keywords:integrodifferential operators; maximum principle; a priori; estimates; Hamilton-Jacobi-Bellmann-equation; weak solutions; jump-diffusion process; stochastic modelling Citations:Zbl 0292.60122; Zbl 0525.35028; Zbl 0513.45014 PDFBibTeX XMLCite \textit{F. Gimbert} and \textit{P.-L. Lions}, Ric. Mat. 33, 315--358 (1984; Zbl 0579.45010)