Almost everywhere Hölder continuity of gradients to non-diagonal parabolic systems [PDF]
Manuscripta Math. 144 (2014), no. 1-2, 51-90, 2012 We present a local almost everywhere regularity result for a general nonlinear non-diagonal parabolic system, which main part depends on symmetric part of the gradient.
arxiv
An Introduction to Barenblatt Solutions for Anisotropic $p$-Laplace Equations [PDF]
arXiv, 2020 We introduce Fundamental solutions of Barenblatt type for the equation $u_t=\sum_{i=1}^N \bigg( |u_{x_i}|^{p_i-2}u_{x_i} \bigg)_{x_i}$, $p_i >2 \quad \forall i=1,..,N$, on $\Sigma_T=\mathbb{R}^N \times[0,T]$, and we prove their importance for the regularity properties of the solutions.
arxiv
A mean value formula of sub-p-Laplace parabolic equations on the Heisenberg group [PDF]
arXiv, 2013 We derive two equivalent definitions of the viscosity solutions to the homogeneous sub-p- Laplace parabolic equations on the Heisenberg group, and characterize the viscosity solutions in terms of an asymptotic mean value formula. Moreover, we construct an example to show that these formulae do not hold in non-asymptotic sense.
arxiv
Global higher integrability for parabolic quasiminimizers in metric spaces [PDF]
arXiv, 2013 We prove higher integrability up to the boundary for minimal p-weak upper gradients of parabolic quasiminimizers in metric measure spaces, related to the heat equation. We assume the underlying metric measure space to be equipped with a doubling measure and to support a weak Poincar\'e-inequality.
arxiv
Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds [PDF]
arXiv, 2015 We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.
arxiv
Continuity of the temperature in a multi-phase transition problem. [PDF]
Math Ann, 2022 Gianazza U, Liao N.
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Local regularity for quasi-linear parabolic equations in non-divergence form [PDF]
arXiv, 2018 We consider viscosity solutions to non-homogeneous degenerate and singular parabolic equations of the $p$-Laplacian type and in non-divergence form. We provide local H\"older and Lipschitz estimates for the solutions. In the degenerate case, we prove the H\"older regularity of the gradient.
arxiv
Heterogeneous Diffusion and Nonlinear Advection in a One-Dimensional Fisher-KPP Problem. [PDF]
Entropy (Basel), 2022 Palencia JLD, Rahman SU, Redondo AN.
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A second order gradient flow of p-elastic planar networks [PDF]
arXiv, 2019 We consider a second order gradient flow of the p-elastic energy for a planar theta-network of three curves with fixed lengths. We construct a weak solution of the flow by means of an implicit variational scheme. We show long-time existence of the evolution and convergence to a critical point of the energy.
arxiv
Interior regularity of space derivatives to an evolutionary, symmetric $\varphi$-Laplacian [PDF]
arXiv, 2015 We consider Orlicz-growth generalization to evolutionary $p$-Laplacian and to the evolutionary symmetric $p$-Laplacian. We derive the spatial second-order Caccioppoli estimate for a local weak solution to these systems. The result is new even for the $p$-case.
arxiv