Harnack's Inequality for Degenerate and Singular Parabolic by Emmanuele DiBenedetto PDF

By Emmanuele DiBenedetto

ISBN-10: 1461415837

ISBN-13: 9781461415831

ISBN-10: 1461415845

ISBN-13: 9781461415848

Degenerate and singular parabolic equations were the topic of intensive examine for the final 25 years. regardless of very important achievements, the problem of the Harnack inequality for non-negative strategies to those equations, either one of p-Laplacian and porous medium kind, whereas raised by way of numerous authors, has remained primarily open. lately substantial growth has been made in this factor, to the purpose that, apart from the singular sub-critical diversity, either for the p-laplacian and the porous medium equations, the speculation is fairly complete.

It appeared as a result well timed to track a entire review, that might spotlight the most matters and likewise the issues that also stay open. The authors provide a finished therapy of the Harnack inequality for non-negative strategies to p-laplace and porous medium kind equations, either within the degenerate (p>2 or m>1) and within the singular variety (1<p<2 or 0<m<1), ranging from the thought of resolution and development the entire priceless technical tools.

The publication is self-contained. construction on the same monograph through the 1st writer, the authors of the current e-book concentration completely at the Harnack estimates and on their purposes: certainly a couple of recognized regularity effects are given a brand new evidence, in accordance with the Harnack inequality. it really is addressed to all pros lively within the box, and in addition to complex graduate scholars, drawn to figuring out the most problems with this attention-grabbing study field.

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Extra info for Harnack's Inequality for Degenerate and Singular Parabolic Equations

Example text

However, all our estimates will only involve {m, N, Co , C1 }, while C will appear as an alternative. 3 following them. A positive constant γ depending only on the data is stable as m → 1 if there exists a positive constant γ(1, N, Co , C1 ) such that lim γ(m, N, Co , C1 ) = γ(1, N, Co , C1 ). 9) We will show that all our estimates are stable as m → 1. As a consequence, the classical theory for nondegenerate equations can be recovered from these degenerate and singular equations, by letting m → 1.

E. in (y, s) + Q− ρ (θ) . 2) holds true, or u ≤ μ+ − aξω (y, s) + Q− ρ (θ) . e. 2). 3). We may assume (y, s) = (0, 0) and for n = 0, 1, . . , set ρn = ρ + ρ , 2n Qn = Kn × (−θρpn , 0]. 3) over Kn and Qn to (u − kn )− , for the levels kn = μ− + ξn ω where ξn = aξ + 1−a ξ. 8) 0 ≤ ζ2,t ≤ θ(ρpn 2p(n+1) 1 ≤ . 9) p Cρ 1 + + (Cρ)p |[u < kn ] ∩ Qn | θ(ξω)p−2 ξω 1 1+ |[u < kn ] ∩ Qn |, θ(ξω)p−2 1+ provided ξω ≥ Cρ, and ρ < C −1 , which we assume. 1 of the Preliminaries 3 A DeGiorgi-Type Lemma [(u − kn )− ζ]p N +2 N dx dτ ≤ 31 |D[(u − kn )− ζ]|p dx dτ Qn Qn p N × ≤γ [(u − kn )− ζ(x, t)]2 dx ess sup −θρp n

If u is a weak, local supersolution, the same argument shows that −(u − k)− is a weak, local supersolution. 2) is parabolic. One checks that the assumptions of the lemma are verified for example by equations with principal part ux div A(x, t, u, Du) = |Du|p−2 aij (x, t)uxi + f (x, t) j |Du| xj where f is bounded and nonnegative, and the matrix (aij ) is only measurable and locally positive definite in ET . 3 Dependence on the Parameters {p, N, Co , C1 } and Stability The set of parameters {p, N, Co , C1 } are the data, and we say that a generic positive constant γ = γ(p, N, Co , C1 ) depends only on the data, if it can be quantitatively determined a priori, only in terms of these parameters.

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Harnack's Inequality for Degenerate and Singular Parabolic Equations by Emmanuele DiBenedetto


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