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Zbl 0839.35018
Lieberman, Gary M.
Gradient estimates for a new class of degenerate elliptic and parabolic equations.
(English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 21, No.4, 497-522 (1994). ISSN 0391-173X

From the author's introduction: It has been known for a long time that the gradient of the solution of a nondegenerate elliptic or parabolic equation can be estimated in terms of the maximum of the solution and certain structure conditions on the equation. These works consider equations which can be written as \$\$u_t= a^{ij}(x, t, u, Du) D_{ij} u+ b(x, t, u, Du)\$\$ under the basic hypothesis that there is a positive constant \$L\$ such that all eigenvalues of the matrix \$(a^{ij}(x, t, u, Du))\$ are positive and finite when \$|Du|> L\$ and \$Du\$ is finite.\par On the other hand, Mkrtychyan has recently considered problems where, for any positive \$L\$, there is a choice of \$Du\$ with \$|Du|= L\$ which gives \$(a^{ij})\$ a zero eigenvalue. Our goal is to reproduce Mkrtychyan's results in a more general framework. Our proof follows the general outline of Leon Limon's gradient estimate for nondegenerate elliptic equations, which is based on Moser's iteration scheme. Section 4 presents examples to illustrate the variety of equations included in our structure conditions.
[O.John (Praha)]
MSC 2000:
*35B45 A priori estimates
35J70 Elliptic equations of degenerate type
35K65 Parabolic equations of degenerate type
35B65 Smoothness of solutions of PDE

Keywords: degenerate elliptic equation; degenerate parabolic equation; Moser's iteration scheme

Cited in: Zbl 1033.49001

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