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Given two elliptic operators L and M in nondivergence form, with coefficients A_L(x), A_M(x) and drift terms b_L(x), b_M(x), respectively, satisfying a Carleson measure disagreement condition in a Lipschitz domain Omega in R^{n+1}, then…

Let $\mathcal{L}$ be a second-order linear elliptic operator with complex coefficients. We show that if the $L^p$ Dirichlet problem for the elliptic system $\mathcal{L}(u)=0$ in a fixed Lipschitz domain $\Omega$ in $\mathbb{R}^d$ is…

Let $\Sigma$ be a closed subset of $\mathbb{R}^ {n+1}$ which is parabolic Ahlfors-David regular and assume that $\Sigma$ satisfies a 2-sided corkscrew condition. Assume, in addition, that $\Sigma$ is either time-forwards Ahlfors-David…

The theory of second order complex coefficient operators of the form $\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the assumption of $p$-ellipticity. In particular, if the matrix $A$ is $p$-elliptic, the solutions…

In this article, we present a simpler and alternative proof of the solvability of the regularity problem - that is, the Dirichlet problem with boundary data in $\dot W^{1,p}$ - for uniformly elliptic operators on $\mathbb{R}^n_+$ under a…

We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}=-\mbox{div}\, A(X,t)\nabla,$$ in $\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We…

We provide very mild sufficient conditions for space-time domains (non-necessarily cylindrical) which ensure that the continuous Dirichlet problem and the H\"older Dirichlet problem are well-posed, for any parabolic operator in divergence…

In the paper arXiv:1708.02289 we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domains above a Lipschitz graph, satisfying $L^p$-boundary data for $p$ near $2$. The main novel…

This paper studies the regularity of the free boundary for viscosity solutions to a parabolic Bernoulli-type free boundary problem with variable coefficients. The main result is that Lipschitz free boundaries are $C^1$ with a normal vector…

It is well-known that solvability of the $\mathrm{L}^{p}$-Dirichlet problem for elliptic equations $Lu:=-\mathrm{div}(A\nabla u)=0$ with real-valued, bounded and measurable coefficients $A$ on Lipschitz domains…

We establish the existence of Lipschitz continuous solutions to the Cauchy Dirichlet problem for a class of evolutionary partial differential equations of the form $$ \partial_tu-\text{div}_x \nabla_\xi f(\nabla u)=0 $$ in a space-time…

This paper examines the solvability of the equation $\mathrm{div} \ \mathbf{u} = f$ with a zero Dirichlet boundary condition for $\mathbf{u}$. A classical result establishes that for a bounded domain $\Omega \subset \mathbb{R}^N$ with a…

Let $\Omega$ be a Lipschitz domain in $\mathbb R^d$, and let $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$ be a strongly elliptic operator on $\Omega$. We suppose that $\varepsilon$ is small and the function $A$ is…

Let ${\mathscr{L}}=-\text{div}A\nabla$ be a uniformly elliptic operator on $\mathbb{R}^n$, $n\ge 2$. Let $\Omega$ be an exterior Lipschitz domain, and let ${\mathscr{L}}_D$ and ${\mathscr{L}}_N$ be the operator ${\mathscr{L}}$ on $\Omega$…

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in…

The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with $t$-independent complex bounded measurable coefficients ($t$ being the transversal direction to the boundary). To be…

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $\Omega\subset \reals^n$, $n\geq2$, with boundary that is decomposed as $\partial\Omega=D\cup N$, $D$ and $N$ disjoint. We let…

We study positive solutions to the problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$ in $\mathbb{R}^N_+$ with the zero Dirichlet boundary condition, where $p>1$, $\gamma>0$, $0<q\le p$, $\vartheta\ge0$ and…

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…

In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $\Omega\subset\mathbb{R}^d$ with time-dependent Dirichlet boundary condition for the temperature $\vartheta=\vartheta(x,t)$, $\vartheta=g$ on…