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We study the problem of approximating plurisubharmonic functions on a bounded domain $\Omega$ by continuous plurisubharmonic functions defined on neighborhoods of $\bar\Omega$. It turns out that this problem can be linked to the problem of…

In this article, we establish a general sufficient condition for closed range of the Cauchy-Riemann operator $\bar\partial$ in appropriately weighted $L^2$ and $L^2$-Sobolev spaces on $(0,q)$-forms for a fixed $q$ on domains in…

We establish $L^p$ solvability of the Dirichlet problem, for some finite $p$, in a 1-sided chord-arc domain $\Omega$ (i.e., a uniform domain with Ahlfors-David regular boundary), for elliptic equations of the form \[ Lu=-\text{div}(A\nabla…

We extend results of Caffarelli--Silvestre and Stinga--Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators of integrated families of operators, in…

In this paper, we study the $\bar\partial$-equation on some convex domains of infinite type in $\mathbb C^2$. In detail, we prove that supnorm estimates hold for infinite exponential type domains provided the exponent is less than 1.

A recent result of the first author with Li and Pipher has established the extrapolation of solvability of the $L^p$ parabolic Neumann problem on unbounded graph domains of the form $\Omega=\{(x',x_n):\,x_n>\varphi(x')\}\times\mathbb R$,…

We consider a vector bundle with integrable connection (\cE,\na) on an analytic domain U in the generic fiber \cX_{\eta} of a smooth formal p-adic scheme \cX, in the sense of Berkovich. We define the \emph{diameter} \delta_{\cX}(\xi,U) of U…

In this paper, we study meromorphic functions on a domain $\Omega \subset \mathbb{C}$ whose image has finite spherical area, counted with multiplicity. The paper is composed of two parts. In the first part, we show that the limit of a…

In this paper we discuss the existence and regularity of solutions of strongly indefinite systems involving fractional elliptic operators on a smooth bounded domain $\Omega$ in $\R^n$.

Let $\Omega$ be a product domain in $\mathbb C^n, n\ge 2$, where each slice has smooth boundary. We observe that the canonical solution operator for the $\bar\partial$ equation on $\Omega$ is bounded in $W^{k,p}(\Omega)$, $k\in \mathbb Z^+,…

In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the…

This work is a complement of the study on H\"ormander's solution of the $\bar\partial$ equation initialised by H. Hedenmalm. Let $\varphi$ be a strictly plurisubharmonic function of class C 2 in C n, let $c_\varphi(z)$ be the smallest…

A theoretical analysis of the finite element method for a generalized Robin boundary value problem, which involves a second-order differential operator on the boundary, is presented. If $\Omega$ is a general smooth domain with a curved…

The algorithm for generation of exact solutions of the nonlinear equation in partial derivatives of a divergent type which is included in the formulation of magnetostatics, hydro-and aerodynamics, quantum mechanics (stationary Schr\"odinger…

We extend a recent result of Avelin, Hed, and Persson about approximation of functions $u$ that are plurisubharmonic on a domain $\Omega$ and continuous on $\bar\Omega$, with functions that are plurisubharmonic on (shrinking) neighborhoods…

The goal of this note is to show that, also in a bounded domain $\Omega \subset \mathbb{R}^n$, with $\partial \Omega\in C^2$, any weak solution, $(u(x,t),p(x,t))$, of the Euler equations of ideal incompressible fluid in $\Omega\times (0,T)…

We prove estimates for solutions of the $\bar \partial u=\omega $ equation in a strictly pseudo convex domain $ \Omega $ in ${\mathbb{C}}^{n}.$ For instance if the $ (p,q)$ current $\omega $ has its coefficients in $L^{r}(\Omega )$ with…

Let $\Omega\subset\mathbb{R}^n$ be a bounded $(\varepsilon,\infty)$-domain with $\varepsilon\in(0,1]$, $X(\mathbb{R}^n)$ a ball Banach function space satisfying some extra mild assumptions, and $\{\rho_\nu\}_{\nu\in(0,\nu_0)}$ with…

We consider conformal homeomorphisms $\varphi$ of generalized Jordan domains $U$ onto planar domains $\Omega$ %, possibly {\bf infinitely connected}, that satisfy both of the next two conditions: (1) at most countably many boundary…

We prove that the $\bar\partial$-Neumann solution operator is locally regular in a domain which has compactness estimates, is of finite type outside a curve transversal to the CR directions and for which the holomorphic tangential…