Related papers: Analysis of degenerate elliptic operators of Gru\v…

We prove some Hardy type inequalities related to quasilinear second order degenerate elliptic differential operators L_p(u):=-\nabla_L^*(\abs{\nabla_L u}^{p-2}\nabla_L u). If \phi is a positive weight such that -L_p\phi>= 0, then the Hardy…

Let $S=\{S_t\}_{t\geq0}$ be the submarkovian semigroup on $L_2(\Ri^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients $c_{ij}$. Further let $\Omega$ be an open subset…

We consider a non-uniformly elliptic second-order differential operator with periodic coefficients that models composite media consisting of highly anisotropic cylindrical fibres periodically distributed in an isotropic background. The…

We consider the Kato problem and extensions for degenerate elliptic operators of arbitrary order $2m$ ($m\geq 1$), whose coefficients are measurable, complex-valued and satisfy the G$\mathring{a}$rding inequality with respect to a…

We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is…

We consider the Hodge Laplacian $\Delta$ on the Heisenberg group $H_n$, endowed with a left-invariant and U(n)-invariant Riemannian metric. For $0\le k\le 2n+1$, let $\Delta_k$ denote the Hodge Laplacian restricted to $k$-forms. Our first…

This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a…

We study the solvability of the regularity problem for degenerate elliptic operators in the block case for data in weighted spaces. More precisely, let $L_w$ be a degenerate elliptic operator with degeneracy given by a fixed weight $w\in…

Let $\Omega$ be an open subset of $\Ri^d$ with $0\in \Omega$. Further let $H_\Omega=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j$ be a second-order partial differential operator with domain $C_c^\infty(\Omega)$ where the coefficients…

We consider the problem of homogenization for non-self-adjoint second-order elliptic differential operators~$\mathcal{A}^{\varepsilon}$ of divergence form on $L_{2}(\mathbb{R}^{d_{1}}\times\mathbb{T}^{d_{2}})$, where $d_{1}$ is positive…

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms, whose coefficients lie in critical…

Let $\Omega\subset{\mathbb R}^n$ be bounded with a smooth boundary $\Gamma$ and let $S$ be the symmetric operator in $L^2(\Omega)$ given by the minimal realization of a second order elliptic differential operator. We give a complete…

Motivated by applications to stochastic differential equations, an extension of H\"{o}rmander's hypoellipticity theorem is proved for second-order degenerate elliptic operators with non-smooth coefficients. The main results are established…

Let $H:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is a radially symmetric inverse square potential. Let $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$ be the operator norm…

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…

In this paper we extend classical criteria for determining lower bounds for the least point of the essential spectrum of second-order elliptic differential operators on domains $\Omega\subset\R^n$ allowing for degeneracy of the coefficients…

We study a class of second-order degenerate linear parabolic equations in divergence form in $(-\infty, T) \times \mathbb R^d_+$ with homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial \mathbb R^d_+$, where $\mathbb…

In this article, we will consider second order uniformly elliptic operators of divergence form defined on R^n with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially…

Assume that $(X,d,\mu)$ is a metric space endowed with a non-negative Borel measure $\mu$ satisfying the doubling condition and the additional condition that $\mu(B(x,r))\gtrsim r^n$ for any $x\in X, \,r>0$ and some $n\geq1$. Let $L$ be a…

The paper deals with homogenization problem for a non-local linear operator with a kernel of convolution type in a medium with a periodic structure. We consider the natural diffusive scaling of this operator and study the limit behaviour of…