L<sup>p</sup>- Theory for fully nonlinear uniformly parabolic equations

Crandall, Michael G.; Kocan, Maciej; Święch, Andrzej

doi:10.1080/03605300008821576

Cited by 140 publications

(181 citation statements)

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“…Since then this has been a central subject of research. Wang in [15,16] proves Harnack inequality and C 1+α, 1+α 2 estimates for fully nonlinear parabolic equations, and Crandall et al in [2] develop an L p -viscosity theory. Krylov in [6,7] obtains C 2+α, 2+α 2 estimates for solutions to u t − F(D 2 u) = 0, under convexity assumptions, and Caffarelli and Stefanelli in [1] exhibit solutions to uniform parabolic equations that are not C 2,1 .…”

Section: Introductionmentioning

confidence: 99%

Sharp regularity estimates for second order fully nonlinear parabolic equations

Silva

Teixeira

2016

Math. Ann.

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ABSTRACT. We prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the formwhere F is elliptic with respect to the Hessian argument and f ∈ L p,q (Q 1 ). The quantity κ(n, p,q) := n p + 2 q determines to which regularity regime a solution of (Eq) belongs. We prove that when 1 < κ(n, p,q) < 2 − ε F , solutions are parabolic-Hölder continuous for a sharp, quantitative exponent 0 < α(n, p,q) < 1. Precisely at the critical borderline case, κ(n, p,q) = 1, we obtain sharp Log-Lipschitz regularity estimates. When 0 < κ(n, p,q) < 1, solutions are locally of class C 1+σ, 1+σ 2 and in the limiting case κ(n, p,q) = 0, we show C 1,Log-Lip regularity estimates provided F has "better" a priori estimates.

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Section: Introductionmentioning

confidence: 99%

Sharp regularity estimates for second order fully nonlinear parabolic equations

Silva

Teixeira

2016

Math. Ann.

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“…At the parabolic boundary, we require that the solution equals the function u ε given by 1. We are now in a situation where the C 1,α -theory of [6] applies. All the requirements of Theorem 9.3 of [6] are satisfied for the problem on the cylinder, and this implies the existence of a unique viscosity solutionū ∈ C 1,ᾱ loc for allᾱ ∈ (0, 1).…”

Section: Proof Of Lemma 32mentioning

confidence: 99%

“…We are now in a situation where the C 1,α -theory of [6] applies. All the requirements of Theorem 9.3 of [6] are satisfied for the problem on the cylinder, and this implies the existence of a unique viscosity solutionū ∈ C 1,ᾱ loc for allᾱ ∈ (0, 1). Uniqueness implies that u = u ε on the cylinder, and since x, r was arbitrary, u ε ∈ C 1,ᾱ loc (Q T ).…”

Section: Proof Of Lemma 32mentioning

confidence: 99%

“…In general, degenerate parabolic equations do not have smooth solutions, not even when the initial data is smooth. However, regularizing effects have been studied in the subclasses of uniformly parabolic equations [6,14,17,18,19], and first-order Hamilton-Jacobi equations [15,16,7,4,2]. In the first case (for convex equations), the solutions typically become continuously differentiable, twice in x and once in t. In the second case (for strictly convex Hamiltonians), the solutions typically become Lipschitz continuous and x-semiconcave.…”

Section: Introductionmentioning

confidence: 99%

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$W^{2,\infty }$ regularizing effect in a nonlinear, degenerate parabolic equation in one space dimension

Jakobsen

2004

Proc. Amer. Math. Soc.

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Abstract. In this paper we provide and analyze a nonlinear degenerate parabolic equation in one space dimension with the following smoothing property: If the initial data is only uniformly continuous, at positive times, the solution has bounded second derivatives (it belongs to W 2,∞ ). We call this surprising phenomenon a W 2,∞ regularizing effect. So far, such phenomena have only been observed in uniformly parabolic equations.

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“…Moreover assume that for some m ∈ N we have 1 [5] and [24]. In order to use these results, we make a change of variable to transform the BSB equation (which is never uniformly parabolic) into another one.…”

Section: The Bsb Equationmentioning

confidence: 99%