Related papers: A change of variable for Dahlberg-Kenig-Pipher ope…

We consider a slowly decaying oscillatory potential such that the corresponding 1D Schr\"odinger operator has a positive eigenvalue embedded into the absolutely continuous spectrum. This potential does not fall into a known class of initial…

We make a spectral analysis of discrete Schroedinger operators on the half-line, subject to complex Robin-type boundary couplings and complex-valued potentials. First, optimal spectral enclosures are obtained for summable potentials.…

We establish existence and uniqueness results for nonlinear elliptic Dirichlet boundary value problems on n-dimensional time scale domains. Time scales provide a unified framework that encompasses continuous, discrete, and hybrid settings.…

This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data,…

The goal of the paper is to develop a method that will combine the use of variational techniques with regularization methods in order to study existence and multiplicity results for the periodic and the Dirichlet problem associated to the…

A delicate problem is to obtain existence of solutions to the boundary blow-up elliptic equation% \begin{equation*} \sigma _{k}^{1/k}\left( \lambda \left( D^{2}u\right) \right) =g\left( u\right) \text{ in }\Omega \text{,…

We improve our previous result [L. Molinet and T. Tanaka, Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations, J. Funct. Anal. 283 (2022), 109490] on the Cauchy problem for one dimensional…

The aim of this article is the explicit construction of some barrier functions ("fundamental solutions") for the Pucci-Heisenberg operators. Using these functions we obtain the continuity property, up to the boundary, for the viscosity…

Divided difference operators are degree-reducing operators on the cohomology of flag varieties that are used to compute algebraic invariants of the ring (for instance, structure constants). We identify divided difference operators on the…

In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems,…

In this paper, we design Dirichlet-Neumann boundary feedback controllers for the Korteweg-de Vries (KdV) equation that act at the right endpoint of the domain. The length of the domain is allowed to be critical. Constructing backstepping…

We introduce and study the class of unbounded DunfordPettis operators. As consequences, we give basic properties and derive interesting results about the duality, domination problem and relationship with other known classes of operators.

We prove a Lieb--Thirring inequality for Schr\"odinger operators $-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+V$ on the semi-axis with Robin boundary condition at the origin. The result improves on a bound obtained by P.~Exner, A.~Laptev and…

We prove a Lieb-Thirring type inequality for a complex perturbation of a d-dimensional massive Dirac operator $D_m, m\geq 0$ whose spectrum is $]-\infty , -m]\cup[m , +\infty[$. The difficulty of the study is that the unperturbed operator…

In this paper, we survey recent results on index defects of elliptic operators on manifolds with boundary. Index defects are similar to the Hirzebruch signature defects in topology, where the defects appear as the correction terms to the…

Persistent Laplacians are matrix operators that track how the shape and structure of data transform across scales and are popularly adopted in biology, physics, and machine learning. Their eigenvalues are concise descriptors of geometric…

Let $\Omega =\omega\times\mathbb R$ where $\omega\subset \mathbb R^2$ be a bounded domain, and $V : \Omega \to\mathbb R$ a bounded potential which is $2\pi$-periodic in the variable $x_{3}\in \mathbb R$. We study the inverse problem…

We suggest a modification of the operator exponential method for the numerical solving the difference linear initial boundary value problems. The scheme is based on the representation of the difference operator for given boundary conditions…

For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $\lambda _1(\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ (with…

We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of…