Related papers: Carleson-type removability for $p$-parabolic equat…

We consider a second-order parabolic equation in $\bR^{d+1}$ with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally H\"older continuous in the space variables.…

Motivated by recent results on the (possibly conditional) regularity for time-dependent hypoelliptic equations, we prove a parabolic version of the Poincar\'e inequality, and as a consequence, we deduce a version of the classical Moser…

We announce some new results for proving H\"older continuity of weak solutions to quasilinear parabolic equations whose prototype takes the form $$u_t - div (|\nabla u|^{p-2}\nabla u)= 0 \qquad \text{or} \qquad u_t - div…

We give a necessary and sufficient condition for the two weight $L^p$-estimates for paraproducts in non-homogeneous settings, $1<p<\infty$. We are mainly interested in the case $p\ne 2$, since the case $p=2$ is a well-known and easy…

This paper studies H\"older regularity property of bounded weak solutions to a class of strongly coupled degenerate parabolic systems.

We demonstrate two proofs for the local H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{q-1}u\big)-\Delta_p u=0,\quad p>2,\quad 0<q<p-1. \]…

We consider degenerate Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative. It is well known that these equations admit global solutions when epsilon is small enough, and that these solutions…

The present paper, along with its companion [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space.…

This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate as a weight belonging to the $A_{1+\frac{1}{n}}$ class of Muckenhoupt…

We consider radial solutions of a general elliptic equation involving a weighted $p$-Laplace operator with a subcritical nonlinearity. By a shooting method we prove the existence of solutions with any prescribed number of nodes. The method…

A general modulus of continuity is quantified for locally bounded, local, weak solutions to nonlocal parabolic equations, under a minimal tail condition. H\"older modulus of continuity is then deduced under a slightly stronger tail…

We establish the global existence of a class of strongly coupled parabolic systems. The necessary apriori estimates will be obtained via our new approach to the regularity theory of parabolic scalar equations with integrable data and new…

We analyze $p$-Laplace operators with degenerate elliptic coefficients. This investigation includes Gru\v{s}in type $p$-Laplace operators. We describe a \emph{separation phenomenon} in elliptic and parabolic $p$-Laplace type equations,…

The existence of strong solutions to general class of strongly coupled parabolic systems will be discussed. These systems can be degenerate or singular as boundedness of theirs solutions are unavailable and not assummed. The results greatly…

We prove the necessary and sufficient condition for the removability of the fundamental singularity, and equivalently for the unique solvability of the singular Dirichlet problem for the heat equation. In the measure-theoretical context,…

We study local boundedness and H\"older continuity of a parabolic equation involving the fractional $p$-Laplacian of order $s$, with $0<s<1$, $2\leq p < \infty$, with a general right hand side. We focus on obtaining precise H\"older…

In this note we give an upper bound on the Hausdorff dimension of removable setsfor elliptic and canceling homogeneous differential operators with constant coefficients in the class of bounded functions, using a simple extension of…

We consider p-evolution equations, for $p\geq2$, with complex valued coefficients. We prove that a necessary condition for $H^\infty$ well-posedness of the associated Cauchy problem is that the imaginary part of the coefficient of the…

We prove local H\"older continuity for non negative, locally bounded, local weak solutions to the class of doubly nonlinear parabolic equations $\partial_t (u_q) - \text{div} (|Du|^{p-2} Du) = 0$ for $p > 2$, $ 0 < q < p-1$. The proof…

We establish a new regularity property for weak solutions of parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally H{\"o}lder continuous Lp valued functions…