Related papers: Optimal Sobolev regularity for the Stokes equation…

In this paper we show that the concept of maximal $L^p$-regularity is stable under a large class of unbounded perturbations, namely Staffans-Weiss perturbations. To that purpose, we first prove that the analyticity of semigroups is…

In this paper we consider parabolic problems with stress tensor depending only on the symmetric gradient. By developing a new approximation method (which allows to use energy-type methods typical for linear problems) we provide an approach…

We establish well-posedness and maximal regularity estimates for linear parabolic SPDE in divergence form involving random coefficients that are merely bounded and measurable in the time, space, and probability variables. To reach this…

In this article, the Hodge decomposition for any degree of differential forms is investigated on the whole space $\mathbb{R}^n$ and the half-space $\mathbb{R}^n_+$ on different scale of function spaces namely homogeneous and inhomogeneous…

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\mathbb{R}) \times W^{1,q}(\mathbb{R}) \to W^{1,r}(\mathbb{R})$ with $1 <p,q < \infty$ and $r\geq 1$, boundedly and…

A new iteration method is represented to study the interior $L_{p}$ regularity for Stokes systems both in divergence form and in non-divergence form. By the iteration, we improve the integrability of derivatives of solutions for Stokes…

We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of…

We firstly describe a maximal inequality for dual Sobolev spaces W^{-1,p}. This one corresponds to a "Sobolev version" of usual properties of the Hardy-Littlewood maximal operator in Lebesgue spaces. Even in the euclidean space, this one…

We establish functional analytic properties of the Stokes operator with bounded measurable coefficients on $L^p_{\sigma} (\mathbb{R}^d)$, $d \geq 2$, for $\lvert 1 / p - 1 / 2 \rvert < 1 / d$. These include optimal resolvent bounds and the…

In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the…

We announce a well-posedness result for the Laplace equation in weighted Sobolev spaces on polyhedral domains in $\RR^n$ with Dirichlet boundary conditions. The weight is the distance to the set of singular boundary points. We give a…

We are concerned with local regularity of the solutions for the Stokes and Navier-Stokes equations near boundary. Firstly, we construct a bounded solution but its normal derivatives are singular in any $L^p$ with $1<p$ locally near…

We establish optimal L^p bounds for the nontangential maximal function of the gradient of the solution to a second order elliptic operator in divergence form, possibly non-symmetric, with bounded measurable coefficients independent of the…

We establish quantitative second-order Sobolev regularity for functions having a $2$-integrable $p$-Laplacian in bounded RCD spaces, with $p$ in a suitable range. In the finite-dimensional case, we also obtain Lipschitz regularity under the…

We consider the well-known following shape optimization problem: $$\lambda_1(\Omega^*)=\min_{\stackrel{|\Omega|=a} {\Omega\subset{D}}} \lambda_1(\Omega), $$ where $\lambda_1$ denotes the first eigenvalue of the Laplace operator with…

In this paper, we study the optimal regularity of the stationary sonic-subsonic solution to the unipolar isothermal hydrodynamic model of semiconductors with sonic boundary. Applying the comparison principle and the energy estimate, we…

This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations (*) $Pu+\partial_tu=f$ on $\Omega\times I $, where $P$ is a nonlocal operator, and $\Omega \subset R^n$,…

We derive regularity estimates for viscosity solutions to the parabolic normalized p-Laplace. By using approximation methods and scaling arguments for the normalized p-parabolic operator, we show that the gradient of bounded viscosity…

We study the perturbed Sobolev spaces ${H^{s,p}_\alpha(\mathbb{R}^d)}$, associated with singular perturbation $\Delta_\alpha$ of Laplace operator in Euclidean space of dimensions 2 and 3. We extend the $L^2$ theory of perturbed Sobolev…

We obtain uniqueness and existence of a solution $u$ to the following second-order stochastic partial differential equation (SPDE) : \begin{align} \label{abs eqn} du= \left( \bar a^{ij}(\omega,t)u_{x^ix^j}+ f \right)dt + g^k dw^k_t, \quad t…