Rellich inequalities with weights

Let Ω be a cone in ${\mathbb {R}^{n}}$ with n ≥? 2. For every fixed ${\alpha \in \mathbb {R}}$ we find the best constant in the Rellich inequality ${\int\nolimits_{\Omega}|x|^{\alpha}|\Delta u|^{2}dx \ge C\int\nolimits_{\Omega}|x|^{\alpha-4}|u|^{2}dx}$ for ${u \in C^{2}_{c}(\overline\Omega\setminus\{0\})}$ . We also estimate the best constant for the same inequality on ${C^{2}_{c}(\Omega)}$ . Moreover we show improved Rellich inequalities with remainder terms involving logarithmic weights on cone-like domains.     

Relaxation of signed integral functionals in BV

For integral functionals initially defined for ${u \in {\rm W}^{1,1}(\Omega; \mathbb{R}^m)}$ by $$\int_{\Omega} f(\nabla u) \, {\rm d}x$$ we establish strict continuity and relaxation results in ${{\rm BV}(\Omega; \mathbb{R}^m)}$ . The results cover the case of signed continuous integrands ${f : \mathbb{R}^{m \times d} \to \mathbb{R}}$ of linear growth at infinity. In particular, it is not excluded that the integrands are unbounded below.     

Existence and Multiplicity Results for Some Elliptic Systems in Unbounded Cylinders

We study the following nonlinear elliptic system of Lane–Emden type $$\left\{\begin{array}{ll} -\Delta u = {\rm sgn}(v) |v| ^{p-1} \qquad \qquad \qquad \; {\rm in} \; \Omega , \\ -\Delta v = - \lambda {\rm sgn} (u)|u| \frac{1}{p-1} + f(x, u)\; \; {\rm in}\; \Omega , \\ u = v = 0 \qquad \qquad \qquad \quad \quad \;\;\;\;\; {\rm on}\; \partial \Omega , \end{array}\right.$$ where ${\lambda \in \mathbb{R}}$ . If ${\lambda \geq 0}$ and ${\Omega}$ is an unbounded cylinder, i.e., ${\Omega = \tilde \Omega \times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}}$ , ${N - m \geq 2, m \geq 1}$ , existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if ${\lambda \in \mathbb{R}}$ and ${\Omega}$ is a bounded domain in ${\mathbb{R}^{N}, N \geq 3}$ . In particular, a good finite dimensional decomposition of the Banach space in which we work is given.     

Gamma-convergence of nonlocal perimeter functionals

Given ${\Omega\subset\mathbb{R}^{n}}$ open, connected and with Lipschitz boundary, and ${s\in (0, 1)}$ , we consider the functional $$\mathcal{J}_s(E,\Omega)\,=\, \int_{E\cap \Omega}\int_{E^c\cap\Omega}\frac{dxdy}{|x-y|^{n+s}}+\int_{E\cap \Omega}\int_{E^c\cap \Omega^c}\frac{dxdy}{|x-y|^{n+s}}\,+ \int_{E\cap \Omega^c}\int_{E^c\cap \Omega}\frac{dxdy}{|x-y|^{n+s}},$$ where ${E\subset\mathbb{R}^{n}}$ is an arbitrary measurable set. We prove that the functionals ${(1-s)\mathcal{J}_s(\cdot, \Omega)}$ are equi-coercive in ${L^1_{\rm loc}(\Omega)}$ as ${s\uparrow 1}$ and that $$\Gamma-\lim_{s\uparrow 1}(1-s)\mathcal{J}_s(E,\Omega)=\omega_{n-1}P(E,\Omega),\quad \text{for every }E\subset\mathbb{R}^{n}\,{\rm measurable}$$ where P(E, ??) denotes the perimeter of E in ?? in the sense of De Giorgi. We also prove that as ${s\uparrow 1}$ limit points of local minimizers of ${(1-s)\mathcal{J}_s(\cdot,\Omega)}$ are local minimizers of P(·, ??).     

Young measures generated by sequences in Morrey spaces

Let ${\Omega\subset\mathbb{R}^n}$ be open and bounded. For 1 ≤ p < ∞ and 0 ≤ λ < n, we give a characterization of Young measures generated by sequences of functions ${\{{\bf f}_j\}_{j=1}^\infty}$ uniformly bounded in the Morrey space ${L^{p,\lambda}(\Omega;\mathbb{R}^N)}$ with ${\{\left|{{\bf f}_j}\right|^p\}_{j=1}^\infty}$ equiintegrable. We then treat the case that each f _j = ? u _j for some ${{\bf u}_j\in W^{1,p}(\Omega;\mathbb{R}^N)}$ . As an application of our results, we consider the functional $${\bf u} \mapsto \int\limits_{\Omega}f({\bf x}, {\bf u}({\bf x}), {\bf {\nabla}}{\bf u}({\bf x})){\rm d}{\bf x},$$ and provide conditions that guarantee the existence of a minimizing sequence with gradients uniformly bounded in ${L^{p,\lambda}(\Omega;\mathbb{R}^{N\times n})}$ .     

True Poly-Bergman and Poly-Bergman Kernels for the Complement of a Closed Disk

Let $k$ and $j$ be positive integers. We prove that the action of the two-dimensional singular integral operators $(S_\Omega )^{j-1}$ and $(S_\Omega ^*)^{j-1}$ on a Hilbert base for the Bergman space $\mathcal{A }^2(\Omega )$ and anti-Bergman space $\mathcal{A }^2_{-1}(\Omega ),$ respectively, gives Hilbert bases $\{ \psi _{\pm j , k } \}_{ k }$ for the true poly-Bergman spaces $\mathcal{A }_{(\pm j)}^2(\Omega ),$ where $S_\Omega $ denotes the compression of the Beurling transform to the Lebesgue space $L^2(\Omega , dA).$ The functions $\psi _{\pm j,k}$ will be explicitly represented in terms of the $(2,1)$ -hypergeometric polynomials as well as by formulas of Rodrigues type. We prove explicit representations for the true poly-Bergman kernels and more transparent representations for the poly-Bergman kernels of $\Omega $ . We establish Rodrigues type formulas for the poly-Bergman kernels of $\mathbb{D }$ .     

Boundary behavior of large solutions to elliptic equations with nonlinear gradient terms

In this paper, we mainly study the asymptotic behavior of solutions to the following problems ${\triangle u \pm a(x)| \nabla u|^{q} = b(x)f(u), x \in \Omega, \ u|_{\partial \Omega} = + \infty}$ , where Ω is a bounded domain with a smooth boundary in ${\mathbb{R}^{N} (N \geq 2)}$ , q >  0, ${a \in C^{\alpha}(\bar{\Omega})}$ is positive in Ω, and ${b \in C^{\alpha}(\bar{\Omega})}$ is nonnegative in Ω and may be vanishing on the boundary. We assume that f is Γ-varying at ∞, whose variation at ∞ is not regular. Our analysis is based on the sub-supersolution method and Karamata regular variation theory.     

Sets Computing the Symmetric Tensor Rank

Let ${\nu_{d} : \mathbb{P}^{r} \rightarrow \mathbb{P}^{N}, N := \left( \begin{array}{ll} r + d \\ \,\,\,\,\,\, r \end{array} \right)- 1,}$ denote the degree d Veronese embedding of ${\mathbb{P}^{r}}$ . For any ${P\, \in \, \mathbb{P}^{N}}$ , the symmetric tensor rank sr(P) is the minimal cardinality of a set ${\mathcal{S} \subset \nu_{d}(\mathbb{P}^{r})}$ spanning P. Let ${\mathcal{S}(P)}$ be the set of all ${A \subset \mathbb{P}^{r}}$ such that ${\nu_{d}(A)}$ computes sr(P). Here we classify all ${P \,\in\, \mathbb{P}^{n}}$ such that sr(P) <  3d/2 and sr(P) is computed by at least two subsets of ${\nu_{d}(\mathbb{P}^{r})}$ . For such tensors ${P\, \in\, \mathbb{P}^{N}}$ , we prove that ${\mathcal{S}(P)}$ has no isolated points.     

Fundamental Solutions in the Colombeau Framework: Applications to Solvability and Regularity Theory

In this article we introduce the notion of fundamental solution in the Colombeau context as an element of the dual $\mathcal {L}(\ensuremath {\mathcal {G}_{\mathrm{c}}}(\mathbb {R}^{n}),\widetilde {\mathbb {C}})$ . After having proved the existence of a fundamental solution for a large class of partial differential operators with constant Colombeau coefficients, we investigate the relationships between fundamental solutions in $\mathcal {L}(\ensuremath {\mathcal {G}_{\mathrm{c}}}(\mathbb {R}^{n}),\widetilde {\mathbb {C}})$ , Colombeau solvability and $\ensuremath {\mathcal {G}}$ - and $\ensuremath {\ensuremath {\mathcal {G}}^{\infty}}$ -hypoellipticity respectively.     

A zoo of diffeomorphism groups on \mathbb{R }^{n}

We consider the groups ${\mathrm{Diff }}_\mathcal{B }(\mathbb{R }^n)$ , ${\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)$ , and ${\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)$ of smooth diffeomorphisms on $\mathbb{R }^n$ which differ from the identity by a function which is in either $\mathcal{B }$ (bounded in all derivatives), $H^\infty = \bigcap _{k\ge 0}H^k$ , or $\mathcal{S }$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.     

Spectrum of Functions and Distributions for the Jacobi-Dunkl Transform on {\mathbb{R}}

We establish real Paley-Wiener theorems for the Jacobi-Dunkl transform on ${\mathbb{R}}$ . More precisely, we characterize the functions in the generalized Schwartz space ${\mathcal{S}^{r}_{\alpha , \beta}(\mathbb{R})}$ and in ${L^{p}_{{A}_{\alpha , \beta}} \mathbb{R})}$ whose Jacobi-Dunkl transform has bounded, unbounded, convex and nonconvex support. Finally, we study the spectral problem on the generalized tempered distributions ${\mathcal{S}^{'r}_{\alpha , \beta}(\mathbb{R})}$ .     

On the boundary of the attainable set of the dirichlet spectrum

Denoting by ${\varepsilon\subseteq\mathbb{R}^2}$ the set of the pairs ${(\lambda_1(\Omega),\,\lambda_2(\Omega))}$ for all the open sets ${\Omega\subseteq\mathbb{R}^N}$ with unit measure, and by ${\Theta\subseteq\mathbb{R}^N}$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that ${\partial\varepsilon}$ has horizontal tangent at its lowest point ${(\lambda_1(\Theta),\,\lambda_2(\Theta))}$ .     

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$ with a general function ${\varphi(x, r)}$ defining the Morrey-type norm. Here ${\Pi \subseteq \Omega}$ is an arbitrary subset in Ω including the extremal cases ${\Pi = \{x_0\}, x_0 \in \Omega}$ and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$ we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$ -theorem for the potential operator I ^α . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ${\varphi(x, r)}$ . No monotonicity type condition is imposed on ${\varphi(x, r)}$ . In case ${\varphi}$ has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function ${\varphi}$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces     

The geometric Neumann problem for the Liouville equation

Let Ω denote the upper half-plane ${\mathbb{R}_+^2}$ or the upper half-disk ${D_{\varepsilon}^+\subset \mathbb{R}_+^2}$ of center 0 and radius ${\varepsilon}$ . In this paper we classify the solutions ${v\in\;C^2(\overline{\Omega}\setminus\{0\})}$ to the Neumann problem $$\left\{\begin{array}{lll}{\Delta v+2 Ke^v=0\quad {\rm in}\,\Omega\subseteq \mathbb{R}^2_+=\{(s, t)\in \mathbb{R}^2: t >0 \},}\\ {\frac{\partial v}{\partial t}=c_1e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s >0 \},}\\ {\frac{\partial v}{\partial t}=c_2e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s <0 \},}\end{array}\right.$$ where ${K, c_1, c_2 \in \mathbb{R}}$ , with the finite energy condition ${\int_{\Omega} e^v < \infty}$ As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a finite number of boundary singularities, not assumed a priori to be conical, and constant geodesic curvature along each boundary arc.     

Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Consider a finite dimensional complex Hilbert space ${\mathcal{H}}$ , with ${dim(\mathcal{H}) \geq 3}$ , define ${\mathbb{S}(\mathcal{H}):= \{x\in \mathcal{H} \:|\: \|x\|=1\}}$ , and let ${\nu_\mathcal{H}}$ be the unique regular Borel positive measure invariant under the action of the unitary operators in ${\mathcal{H}}$ , with ${\nu_\mathcal{H}(\mathbb{S}(\mathcal{H}))=1}$ . We prove that if a complex frame function ${f : \mathbb{S}(\mathcal{H})\to \mathbb{C}}$ satisfies ${f \in \mathbb{L}^2(\mathbb{S}(\mathcal{H}), \nu_\mathcal{H})}$ , then it verifies Gleason’s statement: there is a unique linear operator ${A: \mathcal{H} \to \mathcal{H}}$ such that ${f(u) = \langle u| A u\rangle}$ for every ${u \in \mathbb{S}(\mathcal{H}).\,A}$ is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.     

Howe correspondence and Springer correspondence for real reductive dual pairs

We consider a real reductive dual pair (G′, G) of type I, with rank ${({\rm G}^{\prime}) \leq {\rm rank(G)}}$ . Given a nilpotent coadjoint orbit ${\mathcal{O}^{\prime} \subseteq \mathfrak{g}^{{\prime}{*}}}$ , let ${\mathcal{O}^{\prime}_\mathbb{C} \subseteq \mathfrak{g}^{{\prime}{*}}_\mathbb{C}}$ denote the complex orbit containing ${\mathcal{O}^{\prime}}$ . Under some condition on the partition λ′ parametrizing ${\mathcal{O}^{\prime}}$ , we prove that, if λ is the partition obtained from λ by adding a column on the very left, and ${\mathcal{O}}$ is the nilpotent coadjoint orbit parametrized by λ, then ${\mathcal{O}_\mathbb{C}= \tau (\tau^{\prime -1}(\mathcal{O}_\mathbb{C}^{\prime}))}$ , where ${\tau, \tau^{\prime}}$ are the moment maps. Moreover, if ${chc(\hat\mu_{\mathcal{O}^{\prime}}) \neq 0}$ , where chc is the infinitesimal version of the Cauchy-Harish-Chandra integral, then the Weyl group representation attached by Wallach to ${\mu_{\mathcal{O}^{\prime}}}$ with corresponds to ${\mathcal{O}_\mathbb{C}}$ via the Springer correspondence.     

Euclidean hypersurfaces with genuine deformations in codimension two

We classify hypersurfaces of rank two of Euclidean space ${\mathbb{R}^{n+1}}$ that admit genuine isometric deformations in ${\mathbb{R}^{n+2}}$ . That an isometric immersion ${\hat{f}\colon M^n \to \mathbb{R}^{n+2}}$ is a genuine isometric deformation of a hypersurface ${f\colon M^n\to\mathbb{R}^{n+1}}$ means that ${\hat f}$ is nowhere a composition ${\hat f=\hat F\circ f}$ , where ${\hat{F} \colon V\subset \mathbb{R}^{n+1} \to\mathbb{R}^{n+2}}$ is an isometric immersion of an open subset V containing the hypersurface.