Related papers: $\overline{\partial}$-Estimates on the product of …

In this paper we investigate elliptic partial differential equations on Lipschitz domains in the plane whose coefficient matrices have small (but possibly nonzero) imaginary parts and depend only on one of the two coordinates. We show that…

In this paper we obtain sharp weighted estimates for solutions of the $\partial$-equation in a lineally convex domains of finite type. Precisely we obtain estimates in spaces of the form L p ({\Omega},$\delta$ $\gamma$), $\delta$ being the…

We consider the Dirichlet problem for positive solutions of the equation $-\Delta_p (u) = f(u)$ in a convex, bounded, smooth domain $\Omega \subset\R^N$, with $f$ locally Lipschitz continuous. \par We provide sufficient conditions…

In this paper, we are interested in the periodic homogenization of quasilinear elliptic equations. We obtain error estimates $O(\varepsilon^{1/2})$ for a $C^{1,1}$ domain, and $O(\varepsilon^\sigma)$ for a Lipschitz domain, in which…

We consider the Stokes resolvent problem in a two-dimensional bounded Lipschitz domain $\Omega$ subject to homogeneous Dirichlet boundary conditions. We prove $\mathrm{L}^p$-resolvent estimates for $p$ satisfying the condition $\lvert 1 / p…

We construct homotopy formulas for the $\overline\partial$-equation on convex domains of finite type that have optimal Sobolev and H\"older estimates. For a bounded smooth finite type convex domain $\Omega\subset\mathbb C^n$ that has…

We construct homotopy formulae $f=\overline\partial\mathcal H_qf+\mathcal H_{q+1}\overline\partial f$ for $(0,q)$ forms on the product domain $\Omega_1\times\dots\times\Omega_m$, where each $\Omega_j$ is either a bounded Lipschitz domain in…

Let $\Omega$ be a Lipschitz domain in $\mathbb R^n$ $n\geq 2,$ and $L=\mbox{div} (A\nabla\cdot)$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in…

We consider nonhomogeneous fractional $p$-Laplace equations defined on a bounded nonsmooth domain which goes beyond the Lipschitz category. Under a sufficient flatness assumption on the domain in the sense of Reifenberg, we establish…

The purpose of this paper is to study H\"older estimates for the $\bar\partial$ problem for $(p,q)$ forms on products of general planar domains. As indicated by an example of Stein and Kerzman, solutions to the $\bar\partial$ problem on…

The aim of this paper is to establish $W^2_p$ estimate for non-divergence form second-order elliptic equations with the oblique derivative boundary condition in domains with small Lipschitz constants. Our result generalizes those in [14,…

We consider the complement value problem for a class of second order elliptic integro-differential operators. Let $D$ be a bounded Lipschitz domain of $\mathbb{R}^d$. Under mild conditions, we show that there exists a unique bounded…

We decompose $p$ - integrable functions on the boundary of a simply connected Lipschitz domain $\Omega \subset \mathbb C$ into the sum of the boundary values of two, uniquely determined holomorphic functions, where one is holomorphic in…

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain $\Omega$is equivalent to the solvability of the Dirichlet problem for the biharmonic equationin $\Omega$. As a result, we prove that for any given bounded…

We consider the Robin boundary value problem $\mathrm{div} (A \nabla u) = \mathrm{div} \mathbf{f}+F$ in $\Omega$, $\mathcal{C}^1$ domain, with $(A \nabla u - \mathbf{f})\cdot \mathbf{n} + \alpha u = g$ on $\Gamma$, where the matrix $A$…

In this paper we study the Riemann-Liouville fractional integral of order $\alpha>0$ as a linear operator from $L^p(I,X)$ into itself, when $1\leq p\leq \infty$, $I=[t_0,t_1]$ (or $I=[t_0,\infty)$) and $X$ is a Banach space. In particular,…

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential…

We consider the dyadic paraproducts $\pi_\f$ on $\T$ associated with an $\M$-valued function $\f.$ Here $\T$ is the unit circle and $\M$ is a tracial von Neumann algebra. We prove that their boundedness on $L^p(\T,L^p(\M))$ for some…

Let $\Omega$ be a strictly pseudoconvex domain in $\mathbb{C}^n$ with $C^{k+2}$ boundary, $k \geq 1$. We construct a $\overline\partial$ solution operator (depending on $k$) that gains $\frac12$ derivative in the Sobolev space $H^{s,p}…

In this paper, we study a Dirichlet problem of a fractional Laplace equation in a bounded Lipschitz domain in $ \R, n \geq 2$. Our main result is that for the given data $F \in \dot H^s(\Om^c), 0 < s<1$, we find the function which satisfies…