2020/10/14 by Tomasz Klimsiak, Klimsiak, Tomasz, Tomasz Komorowski +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2010.07181
openalex publication_date 2020/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the present paper we prove estimates on subsolutions of the equation\n-Av+c(x)v=0, x\∈ D, where D\⊂ bbRd is a domain (i.e. an open and\nconnected set) and A is an integro-differential operator of the Waldenfels\ntype, whose differential part satisfies the uniform ellipticity condition on\ncompact sets. In general, the coefficients of the operator need not be\ncontinuous but only bounded and Borel measurable. Some of our results may be\nconsidered "quantitative" versions of the Hopf lemma, as they provide the lower\nbound on the outward normal directional derivative at the maximum point of a\nsubsolution %on a boundary of a domain in terms of its value at the point. We\nshall also show lower bounds on the subsolution around its maximum point by the\nprincipal eigenfunction associated with A and the domain. Additional results,\namong them the weak and strong maximum principles, the weak Harnack inequality\nare also proven.\n