Related papers: Parabolic Boundary Harnack Principles in Domains w…

We use a Harnack-type inequality on exit times and spectral bounds to characterize upper bounds of the heat kernel associated with any regular Dirichlet form without killing part, where the scale function may vary with position. We further…

This paper investigates the Harnack inequality for nonnegative solutions to second-order parabolic equations in double divergence form. We impose conditions where the principal coefficients satisfy the Dini mean oscillation condition in…

We continue our study in \cite{FL} on viscosity solutions to a one-phase free boundary problem for the $p(x)$-Laplacian with non-zero right hand side. We first prove that viscosity solutions are locally Lipschitz continuous, which is the…

We consider linear parabolic equations on a random non-cylindrical domain. Utilizing the domain mapping method, we write the problem as a partial differential equation with random coefficients on a cylindrical deterministic domain.…

We consider degenerate fully nonlinear parabolic equations, which generalize the p-parabolic equation with $p>2$ to nondivergence form operators. We prove an intrinsic Harnack inequality for nonnegative solutions and a weak Harnack…

We develop a method to study the structure of the common part of the boundaries of disjoint and possibly non-complementary time-varying domains in $\mathbb{R}^{n+1}$, $n \geq 2$, at the points of mutual absolute continuity of their…

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…

Regularity of solutions is studied for backward stochastic parabolic Ito equations. An analog of the second energy inequality and the related existence theorem are obtained for domains with boundary.

We prove that if a parabolic Lipschitz (i.e., Lip(1,1/2)) graph domain has the property that its caloric measure is a parabolic $A_\infty$ weight with respect to surface measure (which in turn is equivalent to $L^p$ solvability of the…

We study solutions of parabolic equations with a discontinuous hysteresis operator, described by a free interface boundary. It is established that for spatially transverse initial data from the space $W^{2-2/q}_q$ with $q > 3$, there exists…

We give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary…

We consider boundary Harnack inequalities for regional fractional Laplacian which are generators of censored stable-like processes on G taking \kappa(x,y)/|x-y|^{n+\alpha}dxdy, x,y\in G as the jumping measure. When G is a C^{1,\beta-1} open…

We study an inverse problem involving the restoration of two radiative potentials, not necessarily smooth, simultaneously with initial temperatures in parabolic equations with dynamic boundary conditions. We prove a Lipschitz stability…

We discuss the problem how "bad" may be lower-order coefficients in elliptic and parabolic second order equations to ensure some qualitative properties of solution such as strong maximum principle, Harnack's inequality, Liouville's theorem.…

We prove some sharp regularity results for solutions of classical first order hyperbolic initial boundary value problems. Our two main improvements on the existing litterature are weaker regularity assumptions for the boundary data and…

We discuss a Lipschitz truncation technique for parabolic double-phase problems of $p$-Laplace type in order to prove energy estimates and uniqueness results for the Dirichlet problem. Moreover, we show existence for a non-homogeneous…

We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we…

In this paper we continue the study on intrinsic Harnack inequality for non- homogeneous parabolic equations in non-divergence form initiated by the first author in [1]. We establish a forward-in-time intrinsic Harnack inequality, which in…

In this paper, we establish a parabolic Harnack inequality for positive solutions of the $\phi$-heat equation and prove Gaussian upper and lower bounds for the $\phi$-heat kernel on weighted Riemannian manifolds under lower $N$-Ricci…

We prove that local weak solutions to nonlocal parabolic $p$-Laplace equations are locally Lipschitz continuous in space, uniformly in time for every $1<p<\infty$ and $s \in (0,1)$ whenever $sp > p-1$. Our results hold for symmetric,…