Related papers: Solving the $\partial \bar{\partial}$ with prescri…

In this paper we investigate solvability of a partial integral equation in the space $L_2(\Omega\times\Omega),$ where $\Omega=[a,b]^\nu.$ We define a determinant for the partial integral equation as a continuous function on $\Omega$ and for…

Let $\Omega\subset\mathbb{C}^n$ be a product of one-dimensional open bounded domains with $C^{1,\alpha}$ boundary, where $0<\alpha<1$. Using methods from complex analysis in one variable, we construct an integral operator that solves…

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 $\R^n$.

In this paper we consider the $\bar\partial$-problem in fiber bundles (fibers biholomorphic to $\mathbb C^k$, $k\geq 1$), namely the equation $\bar\partial\sigma =\omega$ for $(0,1)$-forms $\omega$ which decrease along the fibers. The order…

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the…

In this paper we study the solvability of the Cauchy-Riemann equation with prescibed support in different spaces of forms. The unbounded Hartogs triangle in $\mathbb C^2$ and the Hartogs domains in $\mathbb C\mathbb P^2$ provide us new…

We develop and analyze a parametric registration procedure for manifolds associated with the solutions to parametric partial differential equations in two-dimensional domains. Given the domain $\Omega \subset \mathbb{R}^2$ and the manifold…

We generalize to intersection of strictly $c$ -convex domains in Stein manifold, $ L^{r}-L^{s}$ and Lipschitz estimates for the solutions of the $ \bar \partial $ equation done by Ma and Vassiliadou for domains in $ {\mathbb{C}}^{n}.$ For…

Let $M$ be a strongly pseudoconvex complex $G$-manifold with compact quotient $M/G$. We provide a simple condition on forms $\alpha$ sufficient for the regular solvability of the equation $\square u=\alpha$ and other problems related to the…

We prove regularity of solutions of the $\bar\partial$-problem in the H\"older-Zygmund spaces of bounded, strongly $\mathbf C$-linearly convex domains of class $C^{1,1}$. The proofs rely on a new, analytic characterization of said domains…

In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in \cite{GX} into the case of general domains with partial umbilical boundary in space forms.…

Weakly harmonic maps from a domain $\Omega$ (the upper half-space $\Rd$ or a bounded $C^{1,\alpha}$ domain, $\alpha\in (0,1]$) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes…

In this paper, we solve a logarithmic $\bar{\partial}$-equation on a compact K\"ahler manifold associated to a smooth divisor by using the cyclic covering trick. As applications, we discuss the closedness of logarithmic forms, injectivity…

Let $(X, g^+)$ be an asymptotically hyperbolic manifold and $(M, [\hat{h}])$ its conformal infinity. Our primary aim in this paper is to introduce the prescribed fractional scalar curvature problem on $M$ and provide solutions under various…

In this paper we study the obstacle problems for the fractional Lapalcian of order $s\in(0,1)$ in a bounded domain $\Omega\subset\mathbb R^n$, under mild assumptions on the data.

In this article, we use a Berndtsson-Andersson operator and the Bergman metric in order to solve the $\bar\partial$ equation on convex domains of finite type for forms satisifying a Carleson condition and get norm estimates of the solution…

In this paper we study the obstacle problems for the Navier (spectral) fractional Laplacian $\left(-\Delta_\Omega\right)^{\!s}$ of order $s\in(0,1)$, in a bounded domain $\Omega\subset\mathbb R^n$.

We consider the problem of finding a net that supports prescribed forces applied at prescribed points, yet avoids certain obstacles, with all the elements of the net under compression (strut net) or under tension (cable web). In the case of…

We study the $\overline{\partial}$-Neumann problem using the Sobolev space inner product. We show that the problem can be solved on any smoothly bounded, pseudoconvex domain. We further formulate estimates and the basic results of a Sobolev…

Let $X$ be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of $\overline{\partial}$-equation on $X$ and prove a Dolbeault-Grothendieck lemma. We obtain fine sheaves $\mathcal{A}_X^q$ of $(0,q)$-currents, so…