Regularity of weak solutions of elliptic and parabolic equations with some critical or supercritical potentials

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Li, Zijin^1,2  Zhang, Qi S.³ 

[1] Nanjing Univ, Dept Math, Nanjing 210093, Jiangsu, Peoples R China

[2] Nanjing Univ, IMS, Nanjing 210093, Jiangsu, Peoples R China

[3] Univ Calif Riverside, Dept Math, Riverside, CA 92521 USA

关键词: Weak solutions;    Elliptic;    Parabolic;    Holder continuity;    Critical;    Supercritical potential;   

DOI  :  10.1016/j.jde.2017.02.029

来源: Elsevier

【 摘 要 】

We prove Holder continuity of weak solutions of the uniformly elliptic and parabolic equations partial derivative(i)(a(ij)(x)partial derivative(j)u(x)) - A/vertical bar x vertical bar(2+beta)u(x) = 0 (A > 0, beta >= 0), (0.1) partial derivative(i)(a(ij)(x, t)partial derivative(j)u(x, t)) - A/vertical bar x vertical bar(2+beta)u(x, t) - partial derivative(t)u(x, t) = 0 (A > 0, beta >= 0), (0.2) with critical or supercritical 0-order term coefficients which are beyond De Giorgi-Nash-Moser's Theory. We also prove, in some special cases, weak solutions are even differentiable. Previously P. Baras and J. A. Goldstein [3] treated the case when A < 0, (a(ij)) = I and beta = 0 for which they show that there does not exist any regular positive solution or singular positive solutions, depending on the size of vertical bar A vertical bar. When A > 0, beta = 0 and (a(jj)) = I, P.D. Milman and Y.A. Semenov [7,8] obtain bounds for the heat kernel. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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