Related papers: A stability result for parabolic measures of opera…

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half plane $\mathbb{R}^2_+$. There are several conditions on the behavior of the matrix $A$ in the transversal $t$-direction that yield $\omega\in…

In the present paper, we consider an elliptic divergence form operator in the half-space and prove that its Green function is almost affine, or more precisely, that the normalized difference between the Green function and a suitable affine…

In one-sided Chord-Arc Domains $\Omega$, we demonstrate that the $A_\infty$-absolute continuity of the elliptic measure with respect to the surface measure remains stable under $L^2$ Carleson perturbations. This stability holds provided…

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that if the matrix $A$ is independent in the transversal $t$-direction, then we have $\omega\in A_\infty(\sigma)$. In the present paper we…

We show doubling of the elliptic measure corresponding to the operator with an elliptic principal term and a drift that diverges, on average on Whitney cubes, like the inverse distance to the boundary, with a small constant. Essentially a…

We consider strongly degenerate parabolic operators of the form \[ \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_Y-\partial_t \] in unbounded domains \[ \Omega=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb…

In the present paper, we consider elliptic operators $L=-\textrm{div}(A\nabla)$ in a domain bounded by a chord-arc surface $\Gamma$ with small enough constant, and whose coefficients $A$ satisfy a weak form of the Dahlberg-Kenig-Pipher…

We show small and large Carleson perturbation results for the parabolic Regularity boundary value problem with boundary data in $\dot{L}_{1,1/2}^p$ and small Carelson perturbation results for the Neumann problem with boundary data in $L^p$.…

We prove that the $A_\infty$ property of parabolic measure for operators in certain time-varying domains is equivalent to a Carleson measure property of bounded solutions. Kircheim, Kenig, Pipher, and T. Toro established this criterion on…

We prove an analogue of a perturbation result for the Dirichlet problem of divergence form elliptic operators by Fefferman, Kenig and Pipher, for the degenerate elliptic operators of David, Feneuil and Mayboroda, which were developed to…

Questions concerning quantitative and asymptotic properties of the elliptic measure corresponding to a uniformly elliptic divergence form operator have been the focus of recent studies. In this setting we show that the elliptic measure of…

In the present paper, we consider an elliptic divergence form operator in $\mathbb{R}^n\setminus\mathbb{R}^d$ with $d<n-1$ and prove that its Green function is almost affine, in the sense that the normalized difference between the Green…

We prove that the Dirichlet problem for degenerate elliptic equations $\mathrm{div}(A \nabla u) = 0$ in the upper half-space $(x,t)\in \mathbb{R}^{n+1}_+$ is solvable when $n\geq2$ and the boundary data is in $L^p_\mu(\mathbb{R}^n)$ for…

We consider a parabolic partial differential equation with Dirichlet boundary conditions and measure or $L^1$ data. The key difficulty consists in a presence of a monotone operator~$A$ subjected to a non-standard growth condition,…

We establish $L^p$, $2\le p\le\infty$ solvability of the Dirichlet boundary value problem for a parabolic equation $u_t-\mbox{div}(A\nabla u)=0$ on time-varying domains with coefficient matrix $A=(a_{ij})$ that satisfy a small Carleson…

In the development of controllability and inverse problem results for semi-discrete systems, by using Carleman estimates, it is required to estimate of the discrete operators applied to Carleman weight functions. This work aims to establish…

We study an inverse problem for variable coefficient fractional parabolic operators of the form $(\partial_t -\operatorname{div}(A(x) \nabla_x)^s + q(x,t)$ for $s\in(0,1)$ and show the unique recovery of $q$ from exterior measured data.…

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=\Omega \times(0,T).$ We study problems of the model type \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{\Delta_{p}}u=\mu\qquad\text{in }Q,\\ {u}=0\qquad\text{on…

We study parabolic equations governed by integro-differential operators with nonlocal components in some directions and local components in the remaining directions. The setting contains the purely nonlocal, as well as the purely local…

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), i.e., a set which satisfies the interior Corkscrew and Harnack chain conditions, respectively scale-invariant/quantitative…