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We develop a functional model for operators arising in the study of boundary-value problems of materials science and mathematical physics. We then provide explicit formulae for the resolvents of the associated extensions of symmetric…
On locally conformally flat manifolds we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the…
We study the hemisphere threshold for the conformally covariant Escobar functional on compact Riemannian manifolds $(M^n,g)$ with boundary. The near-threshold landscape is organized by boundary invariants: the first-order coefficient…
On a bounded Lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to Robin boundary conditions, where the coefficients in the boundary conditions are functions.…
INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…
We investigate Sobolev inequalities for several rough operators. We prove that several operators satisfy a pointwise bound by the Riesz potential applied to the gradient. From this inequality, we derive several new Sobolev-type inequalities…
We give explicit formulas for conformally invariant operators with leading term an $m$-th power of Laplacian on the product of spheres with the natural pseudo-Riemannian product metric for all $m$.
We emphasize some properties of coherent state groups, i.e. groups whose quotient with the stationary groups, are manifolds which admit a holomorphic embedding in a projective Hilbert space. We determine the differential action of the…
We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the spectrum of the (differential) Laplacian on…
Over n-dimensional manifolds, I classify ternary differential operators acting on the spaces of weighted densities and invariant with respect to the Lie algebra of vector fields. For n=1, some of these operators can be expressed in terms of…
We propose probabilistic representations for inverse Stein operators (i.e. solutions to Stein equations) under general conditions; in particular we deduce new simple expressions for the Stein kernel. These representations allow to deduce…
The main objects of this paper include some degenerate and nonlocal elliptic operators which naturally arise in the conformal invariant theory of Poincar\'e-Einstein manifolds. These operators generally reflect the correspondence between…
The purpose of this note is to describe a unified approach to the fundamental results in the spectral theory of boundary value problems, restricted to the case of Dirac type operators. Even though many facts are known and well presented in…
We study spectral properties of second order elliptic operators with periodic coefficients in dimension two. These operators act in periodic simply-connected waveguides, with either Dirichlet, or Neumann, or the third boundary condition.…
A compact Riemannian manifold is associated with geometric data given by the eigenvalues of various Laplacian operators on the manifold and the triple overlap integrals of the corresponding eigenmodes. This geometric data must satisfy…
We describe the eigenvalues and eigenvectors of real-analytic, non-self-adjoint Berezin--Toeplitz operators, up to exponentially small error, on complex one-dimensional compact manifolds, under the hypothesis of regularity of the energy…
It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In…
We introduce a fourth order CR invariant operator on pluriharmonic functions on a three-dimensional CR manifold, generalizing to the abstract setting the operator discovered by Branson, Fontana and Morpurgo. For a distinguished class of…
We introduce a simplified (coarse) version of pseudo-differential calculus for operators of order zero on complete Riemannian manifolds. This calculus works for the usual Hormander (1,0) class of operators, as well as for…
Gaussian random fields over infinite-dimensional Hilbert spaces require the definition of appropriate covariance operators. The use of elliptic PDE operators to construct covariance operators allows to build on fast PDE solvers for…