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Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations. (English) Zbl 0927.35029

The authors study the existence of continuous weak (viscosity) solutions of Dirichlet and Cauchy-Dirichlet problems for fully nonlinear uniformly elliptic and parabolic equations.
To illustrate this, it was considered the Isaacs’ equation in a bounded open domain coupled with a Dirichlet condition in two situations.
In both cases, a fully nonlinear equation \(F=0\) including the Isaacs’ equation as a special case is treated. The equation \(F=0\) is approximated by better equations \(F^\varepsilon= 0\) for which the Dirichlet problem is uniquely solvable, the solutions of the approximate problems are uniformly bounded and equicontinuous, and the approximations were set up so that available results guarantee that the original problem is solved by uniform limits of solutions of the approximate problems. This last step uses the appropriate result from viscosity solutions theory, which varies between the two cases.
In the continuous coefficient case, the additional structure to bracket the original equation by approximations \(F_\varepsilon\leq F\leq F^\varepsilon\) which are monotone in the parameter \(\varepsilon\) is exploited. This automatically constructs maximal and minimal solutions of the original Dirichlet problem.
In the measurable case, approximation is provided by a simple mollification of the equation in the independent variables. Uniqueness in general remains an interesting issue for the continuous coefficient case of Isaacs’ equation and fails even for the linear equation in the measurable coefficient case.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35J60 Nonlinear elliptic equations
35K20 Initial-boundary value problems for second-order parabolic equations
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