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The conormal derivative problem for non-uniformly parabolic equations. (English) Zbl 0707.35077

The author deals with the nonuniformly parabolic equation \(u_ t=div A(x,t,u,Du)+B(x,t,u,Du)\) with the boundary condition \(A(x,t,u,Du)\cdot \gamma +\phi (x,t,u)=0\) where \(\gamma\) is the inner normal of the bounded domain \(\Omega\) under consideration and the initial condition \(u=u_ 0\). Under appropriate structure conditions on A, B, \(\phi\) maximum bounds and gradient estimates are derived in order to establish an existence and uniqueness result. Furthermore, elliptic versions of these results are given which extend previous results obtained by the same author [J. Differ. Equations 49, 218-257 (1983; Zbl 0476.35032)] to equations of nonvariational type. Several examples like generalizations of the mean curvature equation and some uniformly parabolic equations illustrate the allowable behaviour of A, B, \(\phi\).
Reviewer: P.Kröger

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs

Citations:

Zbl 0476.35032
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