We establish the strong unique continuation property for differences of solutions to the Navier–Stokes system with Gevrey forcing. For this purpose, we provide Carlemantype inequalities with the same singular weight for the Laplacian and the heat operator.

We relate the fractional powers of the discrete Laplacian with a standard time-fractional derivative in the sense of Liouville by encoding the iterative nature of the discrete operator through a time-fractional memory term.

We study here a class of quasilinear elliptic systems involving the p-Laplacian operator. Under some suitable assumptions on the nonlinearities, we show the existence result by using a fixed point theorem.

The use of Laplacian eigenfunctions is ubiquitous in a wide range of computer graphics and geometry processing applications. In particular, Laplacian eigenbases allow generalizing the classical Fourier analysis to manifolds. A key drawback of such bases is their inherently global nature, as the Laplacian eigenfunctions carry geometric and topological structure of the entire manifold. In this pa...

Given V a finite set, a self–adjoint operator on C(V ), K, is called elliptic if it is positive semi–definite and its lowest eigenvalue is simple. Therefore, there exists a unique, up to sign, unitary function ω ∈ C(V ) satisfying K(ω) = λω and then K is named (λ, ω)–elliptic. Clearly, a (λ, ω)–elliptic operator is singular iff λ = 0. Examples of elliptic operators are the so–called Schrödinger...

Let M be a compact Riemannian manifold with an action by isometries of a compact Lie group G. Suppose that this action could be lifted to an action by isometries on a Clifford bundle E over M. We use the method of the Witten deformation to compute the virtual representation-valued index of a transversally elliptic Dirac operator on E. We express the multiplicities of the associated representati...

In the paper, we derive a formula for computing the determinant of a Schrödinger operator on a compact metric graph. This formula becomes very explicit in the case of the Laplacian with the Neumann boundary conditions.

We consider the p–Laplacian operator on a domain equipped with a Finsler metric. We recall relevant properties of its first eigenfunction for finite p and investigate the limit problem as p → ∞.

The Dirichlet Laplacian in curved tubes of arbitrary constant cross-section rotating together with the Tang frame along a bounded curve in Euclidean spaces of arbitrary dimension is investigated in the limit when the volume of the cross-section diminishes. We show that spectral properties of the Laplacian are in this limit approximated well by those of the sum of the Dirichlet Laplacian in the ...

One of the fundamental problems is to study the eigenvalue problem for the differential operator in geometric analysis. In this article, we introduce the recent developments of the eigenvalue problem for the Finsler Laplacian. M.S.C. 2010: 53C60; 35P30; 35J60.