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Eigenvalue problems for semidefinite operators with infinite dimensional kernels appear for instance in electromagnetics. Variational discretizations with edge elements have long been analyzed in terms of a discrete compactness property. As…
We prove a generalized Birman-Schwinger principle in the non-self-adjoint context. In particular, we provide a detailed discussion of geometric and algebraic multiplicities of eigenvalues of the basic operator of interest (e.g., a…
In the paper, we derive a sensitivity result for a nonlinear fractional ordinary elliptic system on a bounded interval with Dirichlet boundary conditions. More precisely, using a global implicit function theorem, we show that, for any…
Generalized eigenfunctions may be regarded as vectors of a basis in a particular direct integral of Hilbert spaces or as elements of the antidual space $\Phi^\times$ in a convenient Gelfand triplet…
In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is $L(\lambda)=-\triangle +(P(x)-\lambda)^2$ in $L^2(\R^d)$ where $P$ is a positive elliptic…
This paper classifies the set of supersolutions of a general class of periodic-parabolic problems in the presence of a positive supersolution. From this result we characterize the positivity of the underlying resolvent operator through the…
This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…
The spectral properties of a class of non-selfadjoint second order elliptic operators with indefinite weight functions on unbounded domains $\Omega$ are investigated. It is shown that under an abstract regularity assumption the nonreal…
The aim of this text is to extend the theory of generalized ordinary differential equations to the setting of metric spaces. We present existence and uniqueness theorems that significantly improve previous results even when restricted back…
We consider positive semidefinite kernels which have values given by bounded linear operators on certain bundles of Hilbert spaces and which are invariant under actions of $*$-semigroupoids. For these kernels, we prove that there exist…
We consider the weighted eigenvalue problem for a general non-local pseudo-differential operator, depending on a bounded weight function. For such problem, we prove that strict (decreasing) monotonicity of the eigenvalues with respect to…
The paper is concerned with a sequence of constants which appear in several problems. These problems include the minimal eigenvalue of certain positive definite Toeplitz matrices, the minimal eigenvalue of some higher-order ordinary…
We study the eigenvalue problem involving the mixed local-nonlocal operator $ L:= -\Delta +(-\Delta)^{s}+q\cdot\nabla$~ in a bounded domain $\Omega\subset\R^N,$ where a Dirichlet condition is posed on $\R^N\setminus\Omega.$ The field $q$…
This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted Sobolev classes. We establish…
The necessary and sufficient conditions for existence of a generalized representer theorem are presented for learning Hilbert space-valued functions. Representer theorems involving explicit basis functions and Reproducing Kernels are a…
This paper investigates the mathematical properties and numerical approximation of a class of nonlocal elliptic partial differential equations of the form \begin{equation*} -\Delta u + \lambda \, G(u) = f, \end{equation*} where $\Delta$…
The paper deals with homogenization problem for a non-local linear operator with a kernel of convolution type in a medium with a periodic structure. We consider the natural diffusive scaling of this operator and study the limit behaviour of…
The paper addresses the doubly elliptic eigenvalue problem $$\begin{cases} -\Delta u=\lambda u \qquad &\text{in $\Omega$,}\\ u=0 &\text{on $\Gamma_0$,}\\ -\Delta_\Gamma u +\partial_\nu u =\lambda u\qquad &\text{on $\Gamma_1$,} \end{cases}…
In this paper, we concern the kernel of linear operator for a class of Grushin equation. First, we study the kernel space of linear operator for a general Grushin equation. Then, we provide an exact expression for the kernel space of linear…