Related papers: On the weighted $\bar\partial$-Neumann problem on …

Let $\mu$ be a non-negative measure defined on bounded $\mathcal F$-hyperconvex domain $\Omega$. We are interested in giving sufficient conditions on $\mu$ such that we can find a plurifinely plurisubharmonic function satisfying $NP (dd^c…

Consider the following $m-$polyharmonic Kirchhoff problem: \begin{eqnarray} \label{ea} \begin{cases} M\left(\int_{\O}|D_r u|^{m} +a|u|^m\right)[\Delta^r_m u +a|u|^{m-2}u]= K(x)f(u) &\mbox{in}\quad \Omega, \\ u=\left(\frac{\partial}{\partial…

This thesis deals with Partial Differential Equations in Several Complex Variables and especially focuses on a general estimate for the $\bar\partial$-Neumann problem on a domain which is $q$-pseudoconvex or $q$-pseudoconcave at a boundary…

We show that compactness of the $\overline{\partial}$-Neumann operator is independent of the metric, and we give a new proof of this independence for subellipticity. We define an abstract obstruction to compactness, namely the common zero…

Let $\Omega$ be either $\mathbb{R}^n$ or a strongly Lipschitz domain of $\mathbb{R}^n$, and $\omega\in A_{\infty}(\mathbb{R}^n)$ (the class of Muckenhoupt weights). Let $L$ be a second order divergence form elliptic operator on $L^2…

Let $\Omega \subset \mathbb{R}^d$ be bounded open and connected. Suppose that $W^{1,2}(\Omega) \subset L^r(\Omega)$ for some $r > 2$. Let $A$ be a pure second-order elliptic differential operator with bounded real measurable coefficients on…

We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p-Laplacian operator and subcritical nonlinearities satisfying…

This paper considers the initial-boundary value problem for the nonlocal Cahn--Hilliard equation $$ \partial_t\varphi + (-\Delta+1)(a(\cdot)\varphi -J\ast\varphi + G'(\varphi)) = 0 \quad \mbox{in}\ \Omega\times(0, T) $$ in an unbounded…

Let $\Omega \subset \mathbb{R}^{n+1}$ be a bounded chord-arc domain, let $\mathcal L=-{\rm div} A\nabla$ be an elliptic operator in $\Omega$ associated with a matrix $A$ having Dini mean oscillation coefficients, and let $1<p\leq 2$. In…

We consider a bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{1,\alpha}$ for some $\alpha\in]0,1[$ and we solve the Neumann problem for the Helmholtz equation both in $\Omega$ and in the exterior of $\Omega$. We look for…

} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $\Delta^2$ on a bounded smooth domain $\Om$ in the Euclidean $n$-space ${\bf R}^n$ ($n\ge2$) and then prove that the corresponding first non-zero…

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^2$ with Lipschitz boundary or a bounded convex domain in $\mathbb{C}^n$ and $\phi\in C(\overline{\Omega})$ such that $H_{\phi}$ is compact on $A^2(\Omega)$. Then $\phi\circ f$ is…

In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $uC_\varphi$ on Bergman type spaces $A_\omega^p $ with double weight $\omega$. Let $X=\{u\in H(D):…

Let $\Omega\subset \mathbb{C}^n$ for $n\geq 2$ be a bounded pseudoconvex domain with a $C^2$-smooth boundary. We study the compactness of composition operators on the Bergman spaces of smoothly bounded convex domains. We give a partial…

We give a sufficient condition for subelliptic estimates for the d-bar-Neumann operator on smoothly bounded, pseudoconvex domains in $\mathbb{C}^n$. This condition is a quantified version of McNeal's condition ($\tilde{P}$) for compactness…

In this paper, we prove the existence of solutions of the Poincar\'e-Lelong equation $\sqrt{-1}\partial\bar{\partial}u=f$ on a strictly convex bounded domain $\Omega\subset\mathbb{C}^n$ $(n\geq1)$, where $f$ is a $d$-closed $(1,1)$ form and…

We consider eigenvalue problems for elliptic operators of arbitrary order $2m$ subject to Neumann boundary conditions on bounded domains of the Euclidean $N$-dimensional space. We study the dependence of the eigenvalues upon variations of…

Let $\Omega$ be a bounded convex Reinhardt domain in $\mathbb{C}^2$ and $\phi\in C(\bar{\Omega})$. We show that the Hankel operator $H_{\phi}$ is compact if and only if $\phi$ is holomorphic along every non-trivial analytic disc in the…

In this paper we study the asymptotic behavior of the principal eigenvalues associated to the Pucci operator in bounded domain $\Omega$ with Neumann/Robin boundary condition i.e. $\partial_n u=\alpha u$ when $\alpha$ tends to infinity. This…

Let $\Omega$ be a bounded convex domain in $\mathbb{C}^{n}$, $n\geq 2$, $1\leq q\leq (n-1)$, and $\phi\in C(\bar{\Omega})$. If the Hankel operator $H^{q-1}_{\phi}$ on $(0,q-1)$--forms with symbol $\phi$ is compact, then $\phi$ is…