Gradient Hölder regularity for degenerate parabolic systems

Bögelein, Verena; Duzaar, Frank; Liao, Naian; Scheven, Christoph

doi:10.1016/j.na.2022.113119

Cited by 9 publications

(3 citation statements)

References 64 publications

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“…In the case of constant p, the second-order regularity of the weak solutions to equations or systems of parabolic equations of the type (1.4) u t = ∆ p u + f, p > 1, were studied by many authors. We refer to [15] for the global estimates (1.3) and to [15,21,22,16,11,20,29,1] for local results, see also references therein. The works [21,22] deal with the homogeneous equation (1.4).…”

Section: Introductionmentioning

confidence: 99%

Existence and global second-order regularity for anisotropic parabolic equations with variable growth

Arora

Shmarev

2023

Journal of Differential Equations

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Section: Introductionmentioning

confidence: 99%

Existence and global second-order regularity for anisotropic parabolic equations with variable growth

Arora

Shmarev

2023

Journal of Differential Equations

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“…A motivation for studying PDEs of the type (1.1) can be found in gas filtration problems taking into account the initial pressure gradient (see [4], [6], [7] and the references therein). * Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II", Via Cintia, 80126 Napoli, Italy.…”

Section: Introductionmentioning

confidence: 99%

Regularity results for a class of widely degenerate parabolic equations

Ambrosio,

Passarelli di Napoli

2023

Advances in Calculus of Variations

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Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE u t - div ⁡ ( ( | D ⁢ u | - ν ) + p - 1 ⁢ D ⁢ u | D ⁢ u | ) = f in ⁢ Ω T = Ω × ( 0 , T ) , u_{t}-\operatorname{div}\Bigl{(}(\lvert Du\rvert-\nu)_{+}^{p-1}\frac{Du}{% \lvert Du\rvert}\Bigr{)}=f\quad\text{in }\Omega_{T}=\Omega\times(0,T), where Ω is a bounded domain in ℝ n {\mathbb{R}^{n}} for n ≥ 2 {n\geq 2} , p ≥ 2 {p\geq 2} , ν is a positive constant and ( ⋅ ) + {(\,\cdot\,)_{+}} stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue–Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative u t {u_{t}} . The main novelty here is that the structure function of the above equation satisfies standard growth and ellipticity conditions only outside a ball with radius ν centered at the origin. We would like to point out that the first result obtained here can be considered, on the one hand, as the parabolic counterpart of an elliptic result established in [L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations [corrected version of mr2584740], J. Math. Pures Appl. (9) 93 2010, 6, 652–671], and on the other hand as the extension to a strongly degenerate context of some known results for less degenerate parabolic equations.

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“…We also refer to [18,1,20] for everywhere C 1,α -regularity results for elliptic systems with p-growth. For the parabolic p-Laplace system, DiBenedetto and Friedman [10,11] (see also the monograph [12]) proved Hölder continuity of Du when 2n n+2 < p < ∞ and we refer to [5,37,6,8,9,4] for further related results for parabolic p-Laplace systems.…”

Section: Introductionmentioning

confidence: 99%

Boundary partial regularity for minimizers of discontinuous quasiconvex integrals with general growth

Ok,

Scilla,

Stroffolini

2022

CPAA

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We prove the partial Hölder continuity on boundary points for minimizers of quasiconvex non-degenerate functionals<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \mathcal{F}({\mathbf u}) \colon = \int_{\Omega} f(x, {\mathbf u}, D{\mathbf u})\, \mathrm{d}x\, , \end{equation*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> satisfies a uniform VMO condition with respect to the <inline-formula><tex-math id="M2">\begin{document}$ x $\end{document}</tex-math></inline-formula>-variable, is continuous with respect to <inline-formula><tex-math id="M3">\begin{document}$ {\mathbf u} $\end{document}</tex-math></inline-formula> and has a general growth with respect to the gradient variable.

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Gradient Hölder regularity for degenerate parabolic systems

Cited by 9 publications

References 64 publications

Existence and global second-order regularity for anisotropic parabolic equations with variable growth

Existence and global second-order regularity for anisotropic parabolic equations with variable growth

Regularity results for a class of widely degenerate parabolic equations

Boundary partial regularity for minimizers of discontinuous quasiconvex integrals with general growth

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