Related papers: Composition operator into the space of function of…

The purpose of this paper is to provide a detailed description of the spaces that can be specified as $L^2$ domains for the operators of a first order elliptic complex on a compact manifold with conical singularities. This entails an…

We show that for any regular bounded domain $\Omega\subseteq \mathbb R^n$, $n=2,3$, there exist infinitely many global diffeomorphisms equal to the identity on $\partial \Omega$ which solve the Eikonal equation. We also provide explicit…

We consider general symmetric systems of first order linear partial differential operators on domains $\Omega \subset \mathbb{R}^d$, and we seek sufficient conditions on the coefficients which ensure essential self-adjointness. The…

We establish the existence of positive solutions for a system of coupled fourth-order partial differential equations on a bounded domain $\Omega \subset \mathbb{R}^n$\begin{align*} \left\{\begin{array}{l} \Delta^2u_1 +\beta_1 \Delta…

Let $f:\Omega\to\IR^2$ be a mapping of finite distortion, where $\Omega\subset\IR^2 .$ Assume that the distortion function $K(x,f)$ satisfies $e^{K(\cdot, f)}\in L^p_{loc}(\Omega)$ for some $p>0.$ We establish optimal regularity and area…

Given a strictly increasing sequence $\Lambda=(\lambda_n)$ of nonegative real numbers, with $\sum_{n=1}^\infty \frac{1}{\lambda_n}<\infty$, the M\"untz spaces $M_\Lambda^p$ are defined as the closure in $L^p([0,1])$ of the monomials…

We use localized topologies to prove existence and optimal regularity results for the divergence equation $\mathrm{div} (v) = F$ in critical cases $v \in L_1(\Omega;\mathbb{R}^m)$ or $v \in C_0(\Omega;\mathbb{R}^m)$, i.e. we characterize…

In this paper we investigate the following problem: when a bounded analytic function $\phi$ on the unit disk $\mathbb{D}$, fixing 0, is such that $\{\phi^n : n = 0, 1, 2, . . . \}$ is orthogonal in $\mathbb{D}$?, and consider the problem of…

We prove local boundedness of local minimizers of scalar integral functionals $\int_\Omega f(x,\nabla u(x))\,dx$, $\Omega\subset\mathbb R^n$ where the integrand satisfies $(p,q)$-growth of the form \begin{equation*} |z|^p\lesssim…

In this article we investigate $L^p$ boundedness of the spherical maximal operator $\mathfrak{m}^\alpha$ of (complex) order $\alpha$ on the $n$-dimensional hyperbolic space $\mathbb{H}^n$, which was introduced and studied by El Kohen. We…

We show that if a reproducing kernel Hilbert space $H_K,$ consisting of functions defined on ${\bf E},$ enjoys Double Boundary Vanishing Condition (DBVC) and Linear Independent Condition (LIC), then for any preset natural number $n,$ and…

If $f$ is in the Eremenko-Lyubich class (transcendental entire functions with bounded singular set) then $\Omega= \{ z: |f(z)| > R\}$ and $f|_\Omega$ must satisfy certain simple topological conditions when $R$ is sufficiently large. A model…

We study the variational problem $$\inf \{\lambda_k(\Omega): \Omega\ \textup{open in}\ \R^m,\ |\Omega| < \infty, \ \h(\partial \Omega) \le 1 \},$$ where $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in…

We prove that if a weight is a Bekoll\'{e}-Bonami weight for some $q$ and it satisfies another simple condition that depends on $0 < p < \infty$, then the operator taking a function to its harmonic conjugate is bounded on the harmonic…

Let $\Omega $ be an open subset of $\mathbb{R}^{N}$, and let $p,\, q:\Omega \rightarrow \left[ 1,\infty \right] $ be measurable functions. We give a necessary and sufficient condition for the embedding of the variable exponent space…

A necessary and sufficient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into $L^\infty(\mathbb R^n)$ is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to…

We consider a shape optimization problem written in the optimal control form: the governing operator is the $p$-Laplacian in the Euclidean space $\R^d$, the cost is of an integral type, and the control variable is the domain of the state…

We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain $\Omega\subset \mathbb{R}^n$, $\alpha>0$ and $0<A<|\Omega|$, find a subset $D\subset \Omega$ of area $A$…

The composition operator $C_{\phi_a}f=f\circ\phi_a$ on the Hardy-Hilbert space $H^2(\mathbb{D})$ with affine symbol $\phi_a(z)=az+1-a$ and $0<a<1$ has the property that the Invariant Subspace Problem for complex separable Hilbert spaces…

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two…