Related papers: Scale invariant elliptic operators with singular c…

In this paper we study perturbed Ornstein-Uhlenbeck operators \begin{align*}[\mathcal{L}_{\infty} v](x)=A\triangle v(x)+\langle Sx,\nabla v(x)\rangle-B v(x),\,x\in\mathbb{R}^d,\,d\geqslant 2,\end{align*} for simultaneously diagonalizable…

We shall first present an explicit realization of the simple $N=4$ superconformal vertex algebra $L_{c} ^{N=4}$ with central charge $c=-9$. This vertex superalgebra is realized inside of the $ b c \beta \gamma $ system and contains a…

A method to construct in explicit form the generators of the simple roots of an arbitrary finite-dimensional representation of a quantum or standard semisimple algebra is found. The method is based on general results from the global theory…

We consider properties of second-order operators $H = -\sum^d_{i,j=1} \partial_i \, c_{ij} \, \partial_j$ on $\Ri^d$ with bounded real symmetric measurable coefficients. We assume that $C = (c_{ij}) \geq 0$ almost everywhere, but allow for…

We consider the operators \[ \nabla_X\cdot(A(X)\nabla_X),\ \nabla_X\cdot(A(X)\nabla_X)-\partial_t,\ \nabla_X\cdot(A(X)\nabla_X)+X\cdot\nabla_Y-\partial_t, \] where $X\in \Omega$, $(X,t)\in \Omega\times \mathbb R$ and $(X,Y,t)\in…

The Stokes equation on a domain $\Omega \subset R^n$ is well understood in the $L^p$-setting for a large class of domains including bounded and exterior domains with smooth boundaries provided $1<p<\infty$. The situation is very different…

We study the existence of solutions to the fractional elliptic equation (E1) $(-\Delta)^\alpha u+\epsilon g(|\nabla u|)=\nu $ in a bounded regular domain $\Omega$ of $\R^N (N\ge2)$, subject to the condition (E2) $u=0$ in $\Omega^c$, where…

The delta set of a numerical semigroup $S$, denoted $\Delta(S)$, is a factorization invariant that measures the complexity of the sets of lengths of elements in $S$. We study the following problem: Which finite sets occur as the delta set…

In this note, we establish $L^p$-bounds for the semigroup $e^{-tq^w(x,D)}$, $t \ge 0$, generated by a quadratic differential operator $q^w(x,D)$ on $\mathbb{R}^n$ that is the Weyl quantization of a complex-valued quadratic form $q$ defined…

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we study a selfadjoint strongly elliptic operator $A_\varepsilon$ of order $2p$ given by the expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon) b({\mathbf D})$, $\varepsilon >0$. Here $g({\mathbf…

We give a uniform description of resolvents and complex powers of elliptic semiclassical cone differential operators as the semiclassical parameter $h$ tends to $0$. An example of such an operator is the shifted semiclassical Laplacian…

We provide a novel approach to the numerical solution of the family of nonlocal elliptic equations $(-\Delta)^su=f$ in $\Omega$, subject to some homogeneous boundary conditions $\mathcal{B}(u)=0$ on $\partial \Omega$, where $s\in(0,1)$,…

For a class of non-selfadjoint $h$--pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin.…

We explicitly construct a finite number of discrete components in the restriction of complementary series representations of rank one semisimple groups $G$ to rank one subgroups $G_1$. For this we use the realizations of complementary…

In $\mathbb R^d$, $d \geq 3$, consider the divergence and the non-divergence form operators \begin{equation} \tag{$i$} - \nabla \cdot a \cdot \nabla + b \cdot \nabla, \end{equation} \begin{equation} \tag{$ii$} - a \cdot \nabla^2 + b \cdot…

We analyse $C_0$-semigroups of contractive operators on real-valued $L^p$-spaces for $p \not= 2$ and on other classes of non-Hilbert spaces. We show that, under some regularity assumptions on the semigroup, the geometry of the unit ball of…

We establish two global subellipticity properties of positive symmetric second-order partial differential operators on $L_2(\Ri^d)$. First, if $m \in \Ni$ then we consider operators $H_0$ with coefficients in $W^{m+1,\infty}(\Ri^d)$ and…

Given an elliptic differential operator L of second order with smooth coefficients in a bounded domain with smooth boundary. We show that if the coefficients are H\"older-continuous up to the boundary and the boundary is…

The existence of Feller semigroups arising in the theory of multidimensional diffusion processes is studied. An elliptic operator of second order is considered on a plane bounded region $G$. Its domain of definition consists of continuous…

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-\Delta)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $\Omega$},\\ u \geq 0 \quad \mbox{in}…