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Boundary behavior of nonnegative solutions of fully nonlinear parabolic equations

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Abstract

In this paper we consider fully nonlinear parabolic equations in \({\mathbb{R}^{n+1}}\) of the type

$$F(D^2 u, Du, x, t) = u_{t},\quad \quad \quad {(0.1)}$$

where F satisfies the structural conditions in (2.2) below, along with F(0, 0, x, t) =  0. Viscosity solutions to (0.1) are a subclass of the (viscosity) solutions to the differential inequalities in (2.5) below. Such inequalities involve the parabolic extremal Pucci operators and the class of their viscosity solutions will be denoted by the notation \({\mathcal S(\lambda, \Lambda, a)}\). The primary purpose of the present paper is studying the boundary behavior of nonnegative functions in the class \({\mathcal S(\lambda, \Lambda, a)}\). Since viscosity solutions to (0.1) are in this class, it follows that our results provide corresponding statements for nonnegative viscosity solutions to (0.1) as a special case.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92697-3875, USA

    Agnid Banerjee

  2. Dipartimento di Ingegneria Civile e Ambientale, DICEA, Via Trieste, 63, 35131, Padova, Italy

    Nicola Garofalo

Authors
  1. Agnid Banerjee
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  2. Nicola Garofalo
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Correspondence to Nicola Garofalo.

Additional information

First author supported in part by the second author’s NSF Grant DMS-1001317, and in part by the second author’s Purdue Research Foundation Grant “Gradient bounds, monotonicity of the energy for some nonlinear singular diffusion equations, and unique continuation”, 2012.

Second author supported in part by NSF Grant DMS-1001317.

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Banerjee, A., Garofalo, N. Boundary behavior of nonnegative solutions of fully nonlinear parabolic equations. manuscripta math. 146, 201–222 (2015). https://doi.org/10.1007/s00229-014-0682-x

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