Related papers: The Stokes operator in two-dimensional bounded Lip…

We consider the mixed problem for $L$ the Lam\'e system of elasticity in a bounded Lipschitz domain $ \Omega\subset\reals ^2$. We suppose that the boundary is written as the union of two disjoint sets, $\partial\Omega =D\cup N$. We take…

This paper studies the Neumann boundary value problems for the Stokes equations in a convex domain in $\mathbb{R}^d$. We obtain nontangential-maximal-function estimates in $L^p$ and $W^{1, p}$ estimates for $p$ in certain ranges depending…

Given a Lipschitz domain $\Omega $ in ${\mathbb R} ^N $ and a nonnegative potential $V$ in $\Omega $ such that $V(x)\, d(x,\partial \Omega)^2$ is bounded in $\Omega $ we study the fine regularity of boundary points with respect to the…

Let $\Omega\subset\dR^d$ be a bounded or an unbounded Lipschitz domain. In this note we address the problem of continuation of functions from the Sobolev space $H^1(\Omega)$ up to functions in the Sobolev space $H^1(\dR^d)$ via a linear…

We study the operator $L=-\Delta+q$ on a bounded domain $\Omega\subset\mathbb R^n$, where $q(x)$ is a distributional potential. We find sufficient conditions for $q(x)$ which guarantee that $L$ is well--defined with Dirichlet and…

We study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain $\Omega\subset\mathbb{R}^{3}$ of class $\mathcal{C}^{1,1}$. We prove existence, uniqueness of weak and strong…

We prove that the Stokes semigroup is a bounded analytic semigroup on $L^{\infty}_{\sigma}$ of angle $\pi/2$ for two-dimensional exterior domains. This result is an end point case of the $L^{p}$-boundedness of the semigroup for $p\in…

We settle the issue of well-posedness for the Dirichlet problem for a higher order elliptic system ${\mathcal L}(x,D_x)$ with complex-valued, bounded, measurable coefficients in a Lipschitz domain $\Omega$, with boundary data in Besov…

In this paper, we consider the characterizations of the Lipschitz spaces and homogeneous Lipschitz spaces associated to the biharmonic operator $\Delta^2.$ With this characterizations, we prove the boundedness of the Bessel potentials,…

The aim of this paper is to show the existence of $\mathcal{R}$-bounded solution operator families for two-phase Stokes resolvent equations in $\dot\Omega=\Omega_+\cup\Omega_-$, where $\Omega_\pm$ are uniform $W_r^{2-1/r}$ domains of…

If $\Omega\subset\R^n$ is a bounded domain, the existence of solutions ${\bf u}\in H^1_0(\Omega)^n$ of ${div} {\bf u} = f$ for $f\in L^2(\Omega)$ with vanishing mean value, is a basic result in the analysis of the Stokes equations. In…

Using Maz'ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in $R^n$. For $n\ge 8$, combined with a result in \cite{S2}, these estimates lead to the…

We use the scale $B^s_{\tau}(L_\tau(\Omega))$, $1/\tau=s/d+1/2$, $s>0$, to study the regularity of the stationary Stokes equation on bounded Lipschitz domains $\Omega\subset\mathbb{R}^d$, $d\geq 3$, with connected boundary. The regularity…

On bounded domains $\Omega \subset \mathbb{R}^d , d \geq 2$, reaching far beyond the scope of Lipschitz domains, we consider an elliptic system of order $2 m$ in divergence form with complex $\mathrm{L}^{\infty}$-coefficients complemented…

As it was shown by Shen, the Riesz transforms associated to the Schr\"odinger operator $L=-\Delta + V$ are not bounded on $L^p(\mathbb{R}^d)$-spaces for all $p, 1<p<\infty$, under the only assumption that the potential satisfies a reverse…

Let (SD_\Omega) be the Stokes operator defined in a bounded domain \Omega of R^3 with Dirichlet boundary conditions. We prove that, generically with respect to the domain \Omega with C^5 boundary, the spectrum of (SD_\Omega) satisfies a non…

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two…

In this article, we give an overview on known as well as new results on the boundedness of the $H^{\infty}$-calculus of the Stokes operator in rough as well as in unbounded (smoother) domains. We present a special case of an abstract…

We prove the solvability of the Dirichlet problem for the variable exponent $p$-Laplacian with boundary data in $W^{1,p(x)}(\Omega)$ on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$. Our main focus will be on an a.e. finite…

In the present paper we establish the solvability of the Regularity boundary value problem in domains with (flat and Lipschitz) lower dimensional boundaries for operators whose coefficients exhibit small oscillations analogous to the…