The maximum principles and symmetry results for viscosity solutions of fully nonlinear equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:258

Article

Lu, Guozhen^1,2  Zhu, Jiuyi³ 

[1] Beijing Normal Univ, Sch Math Sci, Beijing 100875, Peoples R China

[2] Wayne State Univ, Dept Math, Detroit, MI 48202 USA

[3] Johns Hopkins Univ, Dept Math, Baltimore, MD 21218 USA

关键词: Viscosity solutions;    Fully nonlinear equation;    Pucci's extremal operators;    Maximum principle;    Radial symmetry;    Punctured ball;   

DOI  :  10.1016/j.jde.2014.11.022

来源: Elsevier

【 摘 要 】

This paper is concerned about maximum principles and radial symmetry for viscosity solutions of fully nonlinear partial differential equations. We obtain the radial symmetry and monotonicity properties for nonnegative viscosity solutions of F(D(2)u) + u(P) = 0 in R-n (0.1) under the asymptotic decay rate u = o(vertical bar x vertical bar-2/P-1) at infinity, where p > 1 (Theorem 1, Corollary 1). As a consequence of our symmetry results, we obtain the nonexistence of any nontrivial and nonnegative solutions when F is the Pucci extremal operators (Corollary 2). Our symmetry and monotonicity results also apply to Hamilton-Jacobi-Bellman or Isaacs equations. A new maximum principle for viscosity solutions to fully nonlinear elliptic equations is established (Theorem 2). As a result, different forms of maximum principles on bounded and unbounded domains are obtained. Radial symmetry, monotonicity and the corresponding maximum principle for fully nonlinear elliptic equations in a punctured ball are shown (Theorem 3). We also investigate the radial symmetry for viscosity solutions of fully nonlinear parabolic partial differential equations (Theorem 4). (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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