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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