Related papers: The \bar\partial_b-complex on decoupled boundaries…

The modern study of singular integral operators on curves in the plane began in the 1970's. Since then, there has been a vast array of work done on the boundedness of singular integral operators defined on lower dimensional sets in…

We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain and with a possibly singular potential. We first focus on the case of homogeneous Dirichlet boundary conditions, and show how to prove the existence…

We give two weighted norm estimates for higher order commutator of classical operators such as singular integral and fractional type operators, between weighted $L^p$ and certain spaces that include Lipschitz, BMO and Morrey spaces. We also…

We prove new pointwise bounds for weighted Bergman kernels in $\mathbb{C}^n$, whenever a coercivity condition is satisfied by the associated weighted Kohn Laplacian on $(0,1)$-forms. Our results extend the ones obtained in $\mathbb{C}$ by…

By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger-Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem involving the nonlinear…

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…

Solution operators for the equation $\bar \partial u=f$ are constructed on general product domains in $\mathbb{C}^n$. When the factors are one-dimensional, the operator is a simple integral operator: it involves specific derivatives of $f$…

The main purpose of this paper is to consider new sandwich pairs and investigate the existence of solution for a new class of fractional differential equations with $p$-Laplacian via variational methods in $\psi$-fractional space…

The fractional Laplacian $(-\Delta)^{\alpha/2}$ is the prototypical non-local elliptic operator. While analytical theory has been advanced and understood for some time, there remain many open problems in the numerical analysis of the…

We study kernel functions, and associated reproducing kernel Hilbert spaces $\mathscr{H}$ over infinite, discrete and countable sets $V$. Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding…

Let $X$ be a pure n-dimensional complex analytic set in $\mathbb{C}^N$ with an isolated singularity at 0. We study the Cauchy-Riemann operator on a deleted neighborhood of the singular point 0 in $X$.

We give a definition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by Caffarelli-Silvestre. While this definition in the compact case is straightforward, in the noncompact setting one…

The main purpose of this paper is to address some questions concerning boundary value problems related to the Laplacian and bi-Laplacian operators, set in the framework of classical $H^s$ Sobolev spaces on a bounded Lipschitz domain of R^N.…

We treat the boundary problem for complex varieties (with isolated singularities) of dimension greater than one, which are contained in a suitable class of strictly pseudoconvex, unbounded domains of C^n.

We study a family of fractional integral operators whose kernels satisfying an non-isotropic dilation have singularity on a coordinate subspace. A characterization is given for these operators bounded from the classical, atom decomposable…

We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven…

In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial n'th Weyl algebra, the polynomial n'th shift algebra, and ZZ^n-graded polynomials in the n'th q-Weyl algebra. The most…

We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterisation of such operators is performed in the Laplace domain it is…

The first part of the paper is a survey of some of the results previously obtained by the authors concerning the $L^p$-dissipativity of scalar and matrix partial differential operators. In the second part we give new necessary and,…

In this paper we study the existence, regularity and geometric properties of an optimal configuration to a free boundary optimization problem governed by the $p$-Laplacian.