Electron. J. Diff. Eqns., Vol. 1999(1999), No. 24, pp. 1-20.

Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations

M. G. Crandall, M. Kocan, P. L. Lions, & A. Swiech

Abstract:
We study existence of continuous weak (viscosity) solutions of Dirichlet and Cauchy-Dirichlet problems for fully nonlinear uniformly elliptic and parabolic equations. Two types of results are obtained in contexts where uniqueness of solutions fails or is unknown. For equations with merely measurable coefficients we prove solvability of the problem, while in the continuous case we construct maximal and minimal solutions. Necessary barriers on external cones are also constructed.

Submitted May 20, 1999. Published July 1, 1999.
Math Subject Classification: 35J25, 35J60, 35J65, 35K20, 35K55, 35K60, 49L25.
Key Words: Uniformly elliptic and parabolic equations, viscosity solutions, good solutions, exterior cone condition, barrier functions.

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M. G. Crandall
Department of Mathematics, University of California, Santa Barbara
Santa Barbara, CA 93106, USA
e-mail: crandall@math.ucsb.edu

M. Kocan
Mathematics Institute, University of Cologne
50923 Cologne, Germany
e-mail: mkocan@mi.uni-koeln.de

P. L. Lions
Ceremade, Universite Paris-Dauphine
Place de Lattre de Tassigny, 75775
Paris 16, France

A. Swiech
School of Mathematics, Georgia Institute of Technology
Atlanta, GA 30332, USA
e-mail: swiech@math.gatech.edu

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