Related papers: An eigenvalue problem for a generalized polyharmon…

In this paper, we investigate the Dirchlet eigenvalue problems of poly-Laplacian with any order and quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first $k$ eigenvalues which…

We consider a class of nonlinear integro-differential operators and prove existence of two principal (half) eigenvalues in bounded smooth domains with exterior Dirichlet condition. We then establish simplicity of the principal…

Our studies are directed to the existence of weak solutions to a parabolic problem containing a multi-valued term. The problem is formulated in the language of maximal monotone graphs. We assume that the growth and coercivity conditions of…

We consider elliptic problems with nonclassical boundary conditions that contain additional unknown functions on the border of the domain of the elliptic equation and also contain boundary operators of higher orders with respect to the…

We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p-Laplacian operator and subcritical nonlinearities satisfying…

We prove the existence of p-harmonic functions under the form u(r, $\sigma$) = r --$\beta$ $\omega$($\sigma$) in any cone C S generated by a spherical domain S and vanishing on $\partial$C S. We prove the uniqueness of the exponent $\beta$…

We study a Dirichlet spectral problem for a second-order elliptic operator with locally periodic coefficients in a thin domain. The boundary of the domain is assumed to be locally periodic. When the thickness of the domain $\varepsilon$…

We present the first systematic work for deriving a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin (DG) formulation for solving eigenvalue problems associated with second…

We extend holomorphically polyharmonic functions on a real ball to a complex set being the union of rotated balls. We solve a Dirichlet type problem for complex polyharmonic functions with the boundary condition given on the union of…

We study second-order stochastic parabolic equations in a cylindrical domain with homogeneous Dirichlet boundary conditions. Under a natural compatibility condition on the gradient-type noise, we establish global Schauder estimates in…

In this paper, we generalize several results of the article "Analytic continuation of eigenvalues of a quartic oscillator" of A. Eremenko and A. Gabrielov. We consider a family of eigenvalue problems for a Schr\"odinger equation with even…

Given a Schr\"odinger differential expression on an exterior Lipschitz domain we prove strict inequalities between the eigenvalues of the corresponding selfadjoint operators subject to Dirichlet and Neumann or Dirichlet and mixed boundary…

We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic…

In this article, we consider eigenfunctions $u$ of the bi-harmonic operator, i.e., $\triangle^2u=\lambda^2u$ on $\Omega$ with some homogeneous linear boundary conditions. We assume that $\Omega\subseteq\mathbb{R}^n$ ($n\geq2$) is a…

This paper investigates the asymptotic behavior of the eigenvalues of the biharmonic operator on a thin set with Steklov boundary condition. The thin set is taken to be a tubular neighborhood of a planar smooth domain. We show that, as the…

In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or…

We consider the Dirichlet problem u_t &= \Delta u + f(x, u, \nabla u)+ h(x, t),& \qquad &(x, t) \in \Omega \times (0, \infty), u &= 0, & \qquad &(x, t) \in \partial\Omega \times (0, \infty), on a bounded domain $\Omega \subset…

We prove existence results for optimization problems for the $k$th Laplace eigenvalue on closed Riemannian manifolds of dimension $m \geq 3$, depending on the choice of normalization. One such normalization leads to eigenvalue optimization…

In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace…

We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation…