Hölder regularity for parabolic fractional $p$-Laplacian

Liao, Naian

doi:10.48550/arxiv.2205.10111

Cited by 3 publications

(7 citation statements)

References 23 publications

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“…The Caccioppoli inequality above can be obtained by using the method of Lemma 3.1 in our previous paper [19]. However, we follow the estimates (2.1)-(2.2) in [27] here to get the additional term…”

Section: Auxiliary Lemmasmentioning

confidence: 98%

“…Proof of Theorem 1.2 with 1 < p ≤ 2. The following proof in Sections 4 and 5 is analogous to that in [27], but for the sake of completeness and readability, we present the details here. To streamline, we denote Q ρ (θ) as the backward cylinders and omit the sign "-".…”

Section: If We Enforcementioning

confidence: 99%

“…Notice that due to the change of c, the choice of λ may differ from the one in (27). Hence, we choose λ as…”

Section: The First Step Of Inductionmentioning

confidence: 99%

“…Regarding the inhomogeneous case of ( 5), Ding-Zhang-Zhou [17] proved the local boundedness (1 < p < ∞) and Hölder regularity (2 < p < ∞) for the local weak solutions by using the De Giorgi-Nash-Moser iteration. Very recently, the Hölder regularity of weak solutions of (5) in the full range 1 < p < ∞ was investigated by Liao [27] without using any logarithmic estimate and any comparison principle. At the same time, Adimurthi-Prasad-Tewary [2] obtained the same result by employing the expansion of positivity and provided a new form of isoperimetric inequality.…”

mentioning

confidence: 99%

“…The present paper is a continuation of our previous work [19] to derive the Hölder continuity of weak solutions to (1) for the subquadratic case that 1 < p < 2. Motivated by the ideas developed in a series of papers [15,16,26,27], we will adopt the methods of expansion of positivity with strong geometric characteristics to obtain the regularity results of weak solutions in the full range of exponents 1 < p < ∞. We employ the Caccioppoli type inequality, De Giorgi's iteration without using Logarithmic type estimates and exponential change of variables.…”

mentioning

confidence: 99%

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Hölder regularity for mixed local and nonlocal $ p $-Laplace parabolic equations

Shang¹,

Zhang²

2022

DCDS

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We give a unified proof of Hölder regularity of weak solutions for mixed local and nonlocal <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplace type parabolic equations with the full range of exponents <inline-formula><tex-math id="M3">\begin{document}$ 1<p<\infty $\end{document}</tex-math></inline-formula>. Our proof is based on the expansion of positivity together with the energy estimate and De Giorgi type lemma.

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Section: Auxiliary Lemmasmentioning

confidence: 98%

Section: If We Enforcementioning

confidence: 99%

“…Notice that due to the change of c, the choice of λ may differ from the one in (27). Hence, we choose λ as…”

Section: The First Step Of Inductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Hölder regularity for mixed local and nonlocal $ p $-Laplace parabolic equations

Shang¹,

Zhang²

2022

DCDS

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Existence of variational solutions to nonlocal evolution equations via convex minimization

Harsh

Tewary²

2023

ESAIM: COCV

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We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation      $$ \partial_t u + P.V.\int_{\mathbb{R}^N}& \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qs}} \,dy = 0.$$             The approach of minimization of parameter-dependent convex functionals over space-time trajectories requires only appropriate convexity and coercivity assumptions on the nonlocal operator. As the parameter tends to zero, we recover variational solutions. Under further growth conditions, these variational solutions are global weak solutions. Further, this provides a direct minimization approach to approximation of nonlocal evolution equations.

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Boundedness estimates for nonlinear nonlocal kinetic Kolmogorov-Fokker-Planck equations

Anceschi¹,

Mirco²

2023

Preprint

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We investigate local properties of weak solutions to nonlocal and nonlinear kinetic equations whose prototype is given byWe consider equations whose diffusion part is a (possibly degenerate) integro-differential operator of differentiability order s ∈ (0, 1) and summability exponent p ∈ (1, ∞). Amongst other results, we provide an explicit local boundedness estimate by combining together a suitable Sobolev embedding theorem and a fractional Caccioppoli-type inequality with tail. For this, we introduce in the kinetic framework a new definition of nonlocal tail of a function and of its related tail spaces, also by establishing some useful estimates for the tail of weak solutions. Armed with the aforementioned results we give a precise control of the long-range interactions arising from the nonlocal behaviour of the involved diffusion operator.

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Hölder regularity for parabolic fractional $p$-Laplacian

Cited by 3 publications

References 23 publications

Hölder regularity for mixed local and nonlocal $ p $-Laplace parabolic equations

Hölder regularity for mixed local and nonlocal $ p $-Laplace parabolic equations

Existence of variational solutions to nonlocal evolution equations via convex minimization

Boundedness estimates for nonlinear nonlocal kinetic Kolmogorov-Fokker-Planck equations

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