We consider Kolmorogov operator $-\Delta +b \cdot \nabla $ with drift b in the class of form-bounded vector fields (containing vector fields having critical-order singularities). We characterize quantitative dependence of the Sobolev and Hölder regularity of solutions to the corresponding elliptic equation on the value of the form-bound of b.

We prove sufficient and necessary conditions for compactness of the Sobolev embeddings of Besov and Triebel–Lizorkin spaces defined on bounded and unbounded uniformly E-porous domains. The asymptotic behaviour of the corresponding entropy numbers is calculated. Some applications to the spectral properties of elliptic operators are described.

Let L be a linear, closed, densely defined in a Hilbert space operator,not necessarily selfadjoint. Consider the corresponding wave equations

\begin{eqnarray} &(1) \quad \ddot{w}+ Lw=0, \quadw(0)=0,\quad \dot{w}(0)=f, \quad \dot{w}=\frac{dw}{dt}, \quad f \in H. \\ &(2)\quad \ddot{u}+Lu=f e^{-ikt}, \quad u(0)=0, \quad \dot{u}(0)=0,\end{eqnarray}$\begin{array}{ccc}& \mathrm{\left(}\mathrm{1}\mathrm{\right)}\hspace{1em}\begin{array}{}\mathrm{¨}\\ \mathit{w}\end{array}\mathrm{+}\mathit{Lw}\mathrm{=}\mathrm{0}\mathit{,}\hspace{1em}\mathit{w}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}\hspace{1em}\mathit{ẇ}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathit{f,}\hspace{1em}\mathit{ẇ}\mathrm{=}\frac{\mathit{dw}}{\mathit{dt}}\mathit{,}\hspace{1em}\mathit{f}\mathrm{\in }\mathit{H}\mathit{.}& \\ & \mathrm{\left(}\mathrm{2}\mathrm{\right)}\hspace{1em}\begin{array}{}\mathrm{¨}\\ \mathit{u}\end{array}\mathrm{+}\mathit{Lu}\mathrm{=}\mathit{f}{\mathit{e}}^{\mathrm{-}\mathit{ikt}}\mathit{,}\hspace{1em}\mathit{u}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}\hspace{1em}\mathit{u̇}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}& \end{array}$

where k > 0 is a constant. Necessary and sufficient conditions aregiven for the operator L not to have eigenvalues in the half-planeRez < 0 and not to have a positive eigenvalue at a given point kd2 > 0. These conditions are given in terms of the large-timebehavior of the solutions to problem (1) for generic f.

Sufficient conditions are given for the validity of a version of the limiting amplitudeprinciple for the operator L.

A relation between the limiting amplitude principle and the limiting absorption principleis established.

In this article, we begin by recalling the inversion formula for the convolution with the box spline. The equivariant cohomology and the equivariant $K$-theory with respect to a compact torus $G$ of various spaces associated to a linear action of $G$ in a vector space $M$ can both be described using some vector spaces of distributions, on the dual of the group $G$ or on the dual of its Lie algebra $\mathfrak{g}$. The morphism from $K$-theory to cohomology is analyzed, and multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semi-discrete convolution with a box spline. Finally, the multiplicities of the index of a $G$-transversally elliptic operator on $M$ are determined using the infinitesimal index of the symbol.

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation with the weight function. Firstly, we introduce local Carleman estimates for elliptic operators and deduce unique continuation properties as well as interpolation inequalities. These latter inequalities yield a remarkable spectral inequality and the null controllability of the heat equation. Secondly, we prove Carleman estimates for parabolic operators. We state them locally in space at first, and patch them together to obtain a global estimate. This second approach also yields the null controllability of the heat equation.

In this paper, the Babuška's theory of Lagrange multipliers is extended to higher order elliptic Dirichlet problems. The resulting variational formulation provides an efficient numerical squeme in meshless methods for the approximation of elliptic problems with essential boundary conditions.

In this paper we study the problem of $L^p$-independence of the spectrum of second-order elliptic operators with measurable coefficients. A new technique is developed to treat the problem in cases when the kernel of the corresponding semigroup does not satisfy upper bounds of Gaussian type and the semigroup itself exists only in a certain interval of the $L^p$-scale. This allows the authors to treat second-order elliptic operators of divergence type with singular coefficients in the main part and singular lower-order terms. A criterion of $L^p$-independence obtained in the paper applies also to certain classes of higher-order elliptic operators. The problem of stability of the essential spectrum is also studied. It is shown, under some mild conditions on the coefficients of the elliptic operators, that its essential spectrum is the same as the one of the Laplacian. 1991 Mathematics Subject Classification: 35P99, 35B25, 35J15.