The Uniform Hopf Inequality for discontinuous coefficients and optimal regularity in BMO for singular problems

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:437

Article

El Berdan, Nada¹  Ildelfonso Diaz, Jesus²  Rakotoson, Jean Michel¹ 

[1] Univ Poitiers, Lab Math & Applicat, CNRS, UMR 7848,SP2MI, Blvd Marie & Pierre Curie Teleport 2, F-86962 Futuroscope, France

[2] Univ Complutense Madrid, Dept Matemat Aplicada, Inst Matemat Interdisciplinar, Plaza Ciencias 3, E-28040 Madrid, Spain

关键词: Uniform Hopf Inequality;    BMO and Campanato spaces;    Elliptic operator with discontinuous coefficients;    Singular problems;   

DOI  :  10.1016/j.jmaa.2015.11.065

来源: Elsevier

【 摘 要 】

We consider some singular second order semilinear problems which include, among many other special cases, the boundary layer equations such as they were treated by O.A. Oleinik in her pioneering works. We consider diffusion linear operator with possible discontinuous coefficients and prove an optimal criterion to get a quantitative strong maximum principle that we call Uniform Hopf Inequality UHI. Since the solutions of the singular semilinear problems under consideration are not Lipschitz continuous we carry out a careful study of the regularity of solutions when the coefficients of the diffusion matrix are merely in the vmo space and bounded. We prove that the gradient of the solution is still p-integrable, in absence of any continuity assumption on the spatial potential coefficient in the singular term. To this end, the UHI property is used several times. We also apply and improve previous a priori estimates due to S. Campanato in 1965. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free