On the Hölder regularity of signed solutions to a doubly nonlinear equation

Bögelein, Verena; Duzaar, Frank; Liao, Naian

doi:10.48550/arxiv.2003.04158

Cited by 7 publications

(19 citation statements)

References 33 publications

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“…As such the final oscillation decay is dominated by that of u o and g near the initial level. The technical realization runs similar to Section 3.5; see also [3,Section 7.1]. We refrain from giving further details, to avoid repetition.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…Let us first observe that, the arguments in Section 3 hinges solely on the energy estimates in Proposition 2.1. Now the test function ζ p (x, t) u(x, t) − k ± against (7.1) 1 and with k satisfying (2.1), is justified modulo an averaging process in the time variable; this can be done as in [3,Proposition 3.1]. No a priori knowledge on the time derivative is needed, because now s → s + H ε (s) is a smooth, increasing function.…”

Section: Uniform Approximationsmentioning

confidence: 99%

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On the logarithmic type boundary modulus of continuity for the Stefan problem

Liao¹

2021

Preprint

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A logarithmic type modulus of continuity is established for weak solutions to a two-phase Stefan problem, up to the parabolic boundary of a cylindrical space-time domain. For the Dirichlet problem, we merely assume that the spatial domain satisfies a measure density property, and the boundary datum has a logarithmic type modulus of continuity. For the Neumann problem, we assume that the lateral boundary is smooth, and the boundary datum is bounded. The proofs are measure theoretical in nature, exploiting De Giorgi's iteration and refining DiBenedetto's approach. Based on the sharp quantitative estimates, construction of continuous weak (physical) solutions is also indicated. The logarithmic type modulus of continuity has been conjectured to be optimal as a structural property for weak solutions to such partial differential equations.

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Section: Proof Of Theorem 11mentioning

confidence: 99%

Section: Uniform Approximationsmentioning

confidence: 99%

On the logarithmic type boundary modulus of continuity for the Stefan problem

Liao¹

2021

Preprint

Self Cite

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“…For more applications of this kind of function in the doubly nonlinear context, we refer to [2,11,17,18].…”

Section: The Energy Estimatementioning

confidence: 99%

“…We denote by t 1 = t 0 − l, t 2 = t 0 and w(x, t) = (u − k) − (x, t) = (k − u) + (x, t). Following [2], for fixed t 1 < l 1 < l 2 < t 2 and ǫ > 0 small enough, we define a Lipschitz continuous cutoff function…”

Section: The Energy Estimatementioning

confidence: 99%

Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic $p$-Laplace equations

Banerjee¹,

Garain²,

Kinnunen³

2021

Preprint

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We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional p-Laplace equations which includes the fractional parabolic p-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case and in the nonlocal elliptic case to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.

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“…as an admissible test function in (2.3). Indeed, for (•) m as defined in (3.13) and following [13,33], the weak subsolution u of (1.1) satisfies the following mollified inequality…”

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confidence: 99%

On the regularity theory for mixed local and nonlocal quasilinear parabolic equations

Garain¹,

Kinnunen²

2021

Preprint

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We consider mixed local and nonlocal quasilinear parabolic equations of p-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, local Hölder continuity of weak solutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.

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On the Hölder regularity of signed solutions to a doubly nonlinear equation

Cited by 7 publications

References 33 publications

On the logarithmic type boundary modulus of continuity for the Stefan problem

On the logarithmic type boundary modulus of continuity for the Stefan problem

Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic $p$-Laplace equations

On the regularity theory for mixed local and nonlocal quasilinear parabolic equations

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