The aim of this paper is to prove a qualitative property, namely the preservation of positivity, for Schrödinger-type operators acting on $L^p$ functions defined on (possibly incomplete) Riemannian manifolds. A key assumption is a control of the behaviour of the potential of the operator near the Cauchy boundary of the manifolds. As a by-product, we establish the essential self-adjointness of such operators, as well as its generalization to the case $p\neq 2$, i.e. the fact that smooth compactly supported functions are an operator core for the Schrödinger operator in $L^p$.

We consider the three-dimensional sloshing problem on a triangular prism whose angles with the sloshing surface are of the form ${\pi}/{2q}$ , where q is an integer. We are interested in finding a two-term asymptotic expansion of the eigenvalue counting function. When both angles are ${\pi}/{4}$ , we compute the exact value of the second term. As for the general case, we conjecture an asymptotic expansion by constructing quasimodes for the problem and computing the counting function of the related quasi-eigenvalues. These quasimodes come from solutions of the sloping beach problem and correspond to two kinds of waves, edge waves and surface waves. We show that the quasi-eigenvalues are exponentially close to real eigenvalues of the sloshing problem. The asymptotic expansion of their counting function is closely related to a lattice counting problem inside a perturbed ellipse where the perturbation is in a sense random. The contribution of the angles can then be detected through that perturbation.

We consider two natural gradings on the space of symmetric functions: by degree and by length. We introduce a differential operator T that leaves the components of this double grading invariant and exhibit a basis of bihomogeneous symmetric functions in which this operator is triangular. This allows us to compute the eigenvalues of T, which turn out to be nonnegative integers.

In this article, we study some Kramers–Fokker–Planck operators with a polynomial potential $V(q)$ of degree greater than two having quadratic limiting behaviour. This work provides an accurate global subelliptic estimate for Kramers–Fokker–Planck operators under some conditions imposed on the potential.

We consider an elliptic problem of the type

$$\left\{ {\matrix{ {-\Delta u = f(x,u)\quad } \hfill & {{\rm in}\,\Omega } \hfill \cr {u = 0} \hfill & {{\rm on}\,\Gamma _1} \hfill \cr {\displaystyle{{\partial u} \over {\partial \nu }} = g(x,u)} \hfill & {{\rm on}\,\Gamma _2} \hfill \cr } } \right.$$

$\partial \Omega = \overline {\Gamma _1} \cup \overline {\Gamma _2} $

Under a Morse index condition, we prove cylindrical symmetry results for solutions of the above problem.

As an intermediate step, we relate the Morse index of a solution of the nonlinear problem to the eigenvalues of the following linear eigenvalue problem

$$\left\{ {\matrix{ {-\Delta w_j + c(x)w_j = \lambda _jw_j} \hfill & {{\rm in }\Omega } \hfill \cr {w_j = 0} \hfill & {{\rm on }\Gamma _1} \hfill \cr {\displaystyle{{\partial w_j} \over {\partial \nu }} + d(x)w_j = \lambda _jw_j} \hfill & {{\rm on }\Gamma _2} \hfill \cr } } \right.$$

In this paper, we prove the existence and uniqueness of a positive solution for a nonlocal logistic equation arising from the birth-jump processes. For this, we establish a sub-super solution method for nonlocal elliptic equations, we perform a study of the eigenvalue problems associated with these equations and we apply these results to the nonlocal logistic equation.

This paper is devoted to dimensional reductions via the norm-resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its application on seemingly different partial differential equation problems from various areas of mathematical physics; all are analysed in a unified manner, known results are recovered and new ones established.

This paper systematically compares the numerical implementation and computational cost between the Fourier spectral iterative perturbation method (FSIPM) and the finite element method (FEM) in solving partial differential equilibrium equations with inhomogeneous material coefficients and eigen-fields (e.g., stress-free strain and spontaneous electric polarization) involved in phase-field models. Four benchmark numerical examples, including inhomogeneous elastic, electrostatic, and steady-state heat conduction problems demonstrate that (1) the FSIPM rigorously requires uniform hexahedral (3D) and quadrilateral (2D) mesh and periodic boundary conditions for numerical implementation while the FEM permits arbitrary mesh and boundary conditions; (2) the FSIPM solutions are comparable to their FEM counterparts, and both of them agree with the analytic solutions, (3) the FSIPM is much faster in solving equilibrium equations than the FEM to achieve the accurate solutions, thus exhibiting a greater potential for large-scale 3D computations.

We consider an infinite planar straight strip perforated by small holes along a curve. In such a domain, we consider a general second-order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm-resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm-resolvent convergence, we prove the convergence of the spectrum.

Let Ω be an open connected cone in ℝn with vertex at the origin. Assume that the Operator

is subcritical in Ω, where δΩ is the distance function to the boundary of Ω and μ ⩽ 1/4. We show that under some smoothness assumption on Ω the improved Hardy-type inequality

holds true, and the Hardy-weight λ(μ)|x|–2 is optimal in a certain definite sense. The constant λ(μ) > 0 is given explicitly.

Spectral and dynamical properties of some one-dimensional continuous Schrödinger and Dirac operators with a class of sparse potentials (which take non-zero values only at some sparse and suitably randomly distributed positions) are studied. By adapting and extending to the continuous setting some of the techniques developed for the corresponding discrete operator cases, the Hausdorff dimension of their spectral measures and lower dynamical bounds for transport exponents are determined. Furthermore, it is found that the condition for the spectral Hausdorff dimension to be positive is the same for the existence of a singular continuous spectrum.

Let $(M,g)$ be a manifold of bounded geometry with metric $g$. We consider a Schrödinger-type differential expression $H=\Delta_M+V$, where $\Delta_M$ is the scalar Laplacian on $M$ and $V$ is a non-negative locally integrable function on $M$. We give a sufficient condition for $H$ to have an $m$-accretive realization in the space $L^1(M)$.špace{-4pt}

Dans cet article on étudie la différence entre les deux premières valeurs propres, le splitting, d’un opérateur de Klein–Gordon semi-classique unidimensionnel, dans le cas d’un potentiel symétrique présentant un double puits. Dans le cas d’une petite barrière de potentiel, B. Helffer et B. Parisse ont obtenu des résultats analogues à ceux existant pour l’opérateur de Schrödinger. Dans le cas d’une grande barrière de potentiel, on obtient ici des estimations des tranformées de Fourier des fonctions propres qui conduisent à une conjecture du splitting. Des calculs numériques viennent appuyer cette conjecture.