ntrinsic Harnack estimates for non-negative solutions of singular, quasi-linear, parabolic equations are established, including the prototype p-Laplacian equation. For p in the supercritical range 2N/(N + 1) < p < 2, the Harnack inequality is shown to hold in a parabolic form, both forward and backward in time, and in an elliptic form at fixed time. These estimates fail for the heat equation (p → 2). It is shown by counterexamples that they fail for p in the subcritical range 1 < p ≤ 2N/(N + 1). Thus the indicated supercritical range is optimal for a Harnack estimate to hold. The novel proofs are based on measure-theoretical arguments, as opposed to comparison principles, and are sufficiently flexible to hold for a large class of singular parabolic equations including the porous medium equation and its quasi-linear versions.

Intrinsic Harnack estimates estimates for quasi linear singular parabolic differential equations / E. DI BENEDETTO ; U. GIANAZZA ; V. VESPRI.. - In: ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI LINCEI. MATEMATICA E APPLICAZIONI. - ISSN 1120-6330. - STAMPA. - 18:(2007), pp. 359-364. [10.4171/RLM/502]

Intrinsic Harnack estimates estimates for quasi linear singular parabolic differential equations

VESPRI, VINCENZO
2007

Abstract

ntrinsic Harnack estimates for non-negative solutions of singular, quasi-linear, parabolic equations are established, including the prototype p-Laplacian equation. For p in the supercritical range 2N/(N + 1) < p < 2, the Harnack inequality is shown to hold in a parabolic form, both forward and backward in time, and in an elliptic form at fixed time. These estimates fail for the heat equation (p → 2). It is shown by counterexamples that they fail for p in the subcritical range 1 < p ≤ 2N/(N + 1). Thus the indicated supercritical range is optimal for a Harnack estimate to hold. The novel proofs are based on measure-theoretical arguments, as opposed to comparison principles, and are sufficiently flexible to hold for a large class of singular parabolic equations including the porous medium equation and its quasi-linear versions.
2007
18
359
364
E. DI BENEDETTO ; U. GIANAZZA ; V. VESPRI.
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/337068
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