Related papers: On an eigenvalue problem associated with mixed ope…

We consider the linear eigenvalue problem \tag{1} -u" = \lambda u, \quad \text{on $(-1,1)$}, where $\lambda \in \mathbb{R}$, together with the general multi-point boundary conditions \tag{2} \alpha_0^\pm u(\pm 1) + \beta_0^\pm u'(\pm 1) =…

The paper treats boundary value problems for the fractional Laplacian $(-\Delta )^a$, $a>0$, and more generally for classical pseudodifferential operators ($\psi $do's) $P$ of order $2a$ with even symbol, applied to functions on a smooth…

We consider eigenvalue problems for elliptic operators of arbitrary order $2m$ subject to Neumann boundary conditions on bounded domains of the Euclidean $N$-dimensional space. We study the dependence of the eigenvalues upon variations of…

This article is an expanded version of the plenary talk given by Evans Harrell at QMath98, a meeting in Prague, June 1998. We consider Laplace operators and Schr\"odinger operators with potentials containing curvature on certain regions of…

Let $s\in(0,1),$ $1<p<\frac{N}{s}$ and $\Omega\subset\mathbb{R}^N$ be an open bounded set. In this work we study the existence of solutions to problems ($E_\pm$) $Lu\pm g(u)=\mu$ and $u=0$ a.e. in $\mathbb{R}^N\setminus\Omega,$ where $g\in…

We study a fractional analogue of a plasma problem arising from physics. Specifically, for a fixed bounded domain $\Omega$ we study solutions to the eigenfunction equation \[ (- \Delta)^s u = \lambda(u- \gamma)_+ \] with $u \equiv 0$ on…

We study the eigenvalue problem for the $g-$Laplacian operator in fractional order Orlicz-Sobolev spaces, where $g=G'$ and neither $G$ nor its conjugated function satisfy the $\Delta_2$ condition. Our main result is the existence of a…

We study master equations of the form $$(\partial_t+L)^su=f\quad\hbox{in}~\mathbb{R}\times\Omega$$ where $L$ is a divergence form elliptic operator and $\Omega\subseteq\mathbb{R}^n$. These are nonlocal equations of order $2s$ in space and…

We apply the method of inverse iteration to the Laplace eigenvalue problem with Robin and mixed Dirichlet-Neumann boundary conditions, respectively. For each problem, we prove convergence of the iterates to a non-trivial principal…

We reformulate all general real coupled self-adjoint boundary value problems as integral operators and show that they are all finite rank perturbations of the free space Green's function on the real line. This free space Green's function…

The aim of the present work is to derive a error estimates for the Laplace eigenvalue problem in mixed form, by means of a virtual element method. With the aid of the theory for non-compact operators, we prove that the proposed method is…

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential…

In this chapter we are examining several iterative methods for solving nonlinear eigenvalue problems. These arise in variational image-processing, graph partition and classification, nonlinear physics and more. The canonical eigenproblem we…

The term interlacing refers to systematic inequalities between the sequences of eigenvalues of two operators defined on objects related by a specific oper- ation. In particular, knowledge of the spectrum of one of the objects then implies…

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 paper investigates the initial-boundary value problem for a nonlinear parabolic equation involving the $p$-Laplacian operator, nonlocal source terms, gradient absorption, and various nonlinearities: \[ \frac{\partial u}{\partial t} -…

In this work we study a general shape optimization problem where the state equation is given in terms of a nonlocal operator. Examples of the problems considered are monotone combinations of fractional eigenvalues. Moreover, we also analyze…

In this paper, we study existence, uniqueness and asymptotic behavior of the Laplace equation with dynamical boundary conditions on regular non-cylindrical domains. We write the problem as a non-autonomous Dirichlet-to-Neumann operator and…

In this paper, we investigate the eigenvalue problem for a non-local dispersal operator defined on a bounded spatial domain with Neumann-type boundary conditions. Unlike the classical Laplacian, the non-local operator lacks compactness,…

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of boundary mass concentration. We discuss the asymptotic behavior of the Neumann eigenvalues in a ball and we deduce that the Steklov…