相关论文: The Sobolev Inequalities on Real Hyperbolic Spaces…

In this article we establish an Adam's Inequality in the Hyperbolic space. As an application we will also prove the asymptotic behaviour of the best constants in the Sobolev inequality and also discuss the solvability of Q curvature type…

Let $\Omega$ be a bounded open set in the Poincar\'e hyperbolic disk, $\mathbb{D}$. In this article, we consider the hyperbolic logarithmic potential operator $\mathcal{L}_h : L^2(\Omega) \to L^2(\Omega)$, defined by \begin{equation*}…

On a complete weighted Riemannian manifold $(M^n,g,\mu)$ satisfying the doubling condition and the Poincar{\'e} inequalities, we characterize the class of function $V$ such that the Schr{\"o}dinger operator $\Delta-V$ maps the homogeneous…

We study the eigenvalues of the magnetic Schroedinger operator associated with a magnetic potential A and a scalar potential q, on a compact Riemannian manifold M, with Neumann boundary conditions if the boundary is not empty. We obtain…

In this paper, we consider a generalized polyharmonic eigenvalue problem of the form $A(u)= \lambda h(u)$ in a bounded smooth domain with Dirichlet boundary conditions in the setting of higher-order Orlicz-Sobolev spaces. Here, $A$ is a…

We determine a fundamental solution for the differential operator (Delta - lambda_z)^n on the Riemannian symmetric space G/K, where G is any complex semi-simple Lie group, and K is a maximal compact subgroup. We develop a global zonal…

In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell,…

Morrey--Sobolev inequalities are established for functions in weighted Sobolev spaces on the $n$-dimensional half-space, where the weight is a power of the distance to the boundary, as well as for Sobolev spaces on the $n$-dimensional…

When an eigenvector of a semi-bounded operator is positive, we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue. This theorem is a substantial generalization of Barta-like…

We construct homotopy formulae $f=\overline\partial \mathcal H_q f+\mathcal H_{q+1}\overline\partial f$ on a bounded domain which is either $C^2$ strongly pseudoconvex or $C^{1,1}$ strongly $\mathbb C$-linearly convex. Such operators…

The paper studies the uniqueness problem for the one-dimensional Schr\"{o}dinger operator associated with the formal differential expression \begin{equation*} l[u] =-u''+qu + i[(ru)'+ru'], \end{equation*} in the complex Hilbert space…

Let $\Omega$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in…

The aim of this work is to study the continuity and compactness of the operators $W^{1, q}(\Omega ; \mathtt {V}_0, \mathtt {V}_1 ) \rightarrow L^{q_0} (\Omega ; \mathtt {V}_2)$ and $W^{1, q} (\Omega ; \mathtt {V}_0, \mathtt {V}_1 )…

We obtain semiclassical resolvent estimates for the Schr{\"o}dinger operator (ih$\nabla$ + b)^2 + V in R^d , d $\ge$ 3, where h is a semiclassical parameter, V and b are real-valued electric and magnetic potentials independent of h. Under…

In this article we establish a global subelliptic estimate for Kramers-Fokker-Planck operators with homogeneous potentials $V(q)$ under some conditions, involving in particular the control of the eigenvalues of the Hessian matrix of the…

This article deals with the inverse problem of determining the unbounded real-valued electric potential of the Robin Laplacian on a bounded domain of dimension 3 or greater, by incomplete knowledge of its boundary spectral data. Namely, the…

We obtain a new bound on the location of eigenvalues for a non-self-adjoint Schr\"odinger operator with complex-valued potentials by obtaining a weighted $L^2$ estimate for the resolvent of the Laplacian.

In our companion paper (S.N. Chandler Wilde, D.P. Hewett, A. Moiola, Sobolev spaces on non-Lipschitz subsets of $\mathbb{R}^n$ with application to boundary integral equations on fractal screens, 2016) we studied a number of different…

We consider the eigenvalue problem for certain classes of elliptic operators, namely inhomogeneous membrane operators $ L = \tfrac{1}{ \rho } ( -\Delta + V ) $ and divergence form operators $ L = -\operatorname{div} A \nabla $, on bounded…

For a large class of semiclassical pseudodifferential operators, including Schr\"odinger operators, $ P (h) = -h^2 \Delta_g + V (x) $, on compact Riemannian manifolds, we give logarithmic lower bounds on the mass of eigenfunctions outside…