Advances in Differential Equations

Gradient estimates for solutions of parabolic differential equations degenerating at infinity

Alberto Favaron and Alfredo Lorenzi

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Abstract

For $p\in (1,+\infty)$ we derive a weighted $L^p$ estimate for the (spatial) gradient of the solution $u$ of a degenerate parabolic differential equation. Here the underlying domain $\Omega\subset\mathbf{R}^n$, $n\ge 2$, is unbounded and the equation may degenerate only at infinity along some unbounded branch of $\Omega$. Our estimate is strictly related with the still-open problem of giving a concrete characterization of the interpolation space between $W^{2,p}(\Omega)$ and $L^p(\Omega)$ to which the (spatial) gradient of $u$ belongs.

Article information

Source
Adv. Differential Equations Volume 12, Number 4 (2007), 435-460.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867458

Mathematical Reviews number (MathSciNet)
MR2305875

Zentralblatt MATH identifier
1152.35013

Subjects
Primary: 35K65: Degenerate parabolic equations
Secondary: 35B45: A priori estimates 35B65: Smoothness and regularity of solutions

Citation

Favaron, Alberto; Lorenzi, Alfredo. Gradient estimates for solutions of parabolic differential equations degenerating at infinity. Adv. Differential Equations 12 (2007), no. 4, 435--460. https://projecteuclid.org/euclid.ade/1355867458.


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