The maximum principles and symmetry results for viscosity solutions of fully nonlinear equations
نویسندگان
چکیده
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 ( D2u ) + u = 0 in R (0.1) under the asymptotic decay rate u = o(|x|− 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). © 2014 Elsevier Inc. All rights reserved. ✩ Research of this work was partly supported by NNSF grant of China (No. 11371056) and a US NSF grant DMS#1301595. * Corresponding author at: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. E-mail addresses: [email protected] (G. Lu), [email protected] (J. Zhu). http://dx.doi.org/10.1016/j.jde.2014.11.022 0022-0396/© 2014 Elsevier Inc. All rights reserved. G. Lu, J. Zhu / J. Differential Equations 258 (2015) 2054–2079 2055 MSC: 35B50; 35B53; 35B06; 35D40
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تاریخ انتشار 2013