Related papers: The Second Dirichlet Eigenvalue is Simple on Every…

We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet…

We study small perturbations of the Dirichlet problems for second order elliptic equations that degenerate on the boundary. The limit of the solution, as the perturbation tends to zero, is calculated. The result is based on a certain…

We prove the $W^{1,2}_p$-estimate and solvability for the Dirichlet problem of second-order parabolic equations in simple convex polytopes with time irregular coefficients, when $p\in (1,2]$. We also consider the corresponding Neumann…

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices,…

We prove global second-order regularity for a class of quasilinear elliptic equations, both with homogeneous Dirichlet and Neumann boundary conditions. A condition on the integrability of the second fundamental form on the boundary of the…

In this paper we prove existence of (viscosity) solutions of Dirichlet problems concerning fully nonlinear elliptic operator, which are either degenerate or singular when the gradient of the solution is zero. For this class of operators it…

We prove the existence of classical solutions to the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on Riemannian manifolds. We also derive new second derivative boundary estimates which allows us to…

We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser-Jerison in a series of works on convex planar…

In this paper, using new correction to the Crouzeix-Raviart finite element eigenvalue approximations, we obtain lower eigenvalue bounds for the Steklov eigenvalue problem with variable coefficients on d-dimensional domains (d = 2,3). In…

The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $\Omega := \mathbb R^n \setminus \mathbb R^d$…

We put forward a conjecture about an universal asymptotical behaviour of the symbol of the Dirichlet-to-Neumann operator (considered as a pseudodifferential operator) in the 2D exterior problem for the Hemholtz equation. The conjecture is…

In this paper, we continue the study of eigenfunctions on triangles initiated by the first author in \cite{Chr-tri} and \cite{Chr-simp}. The Neumann data of Dirichlet eigenfunctions on triangles enjoys an equidistribution law, being…

In Part I of this paper, we introduced a 2D eigenvalue problem (2DEVP) and presented theoretical results of the 2DEVP and its intrinsic connetion with the eigenvalue optimizations. In this part, we devise a Rayleigh quotient iteration…

Let $L$ be an infinitely degenerate second-order linear operator defined on a bounded smooth Euclidean domain. Under weaker conditions than those of H\"ormander, we show that the Dirichlet problem associated with $L$ has a unique smooth…

We find conditions on the potential of the non-self-adjoint Mathieu-Hill operator such that the all eigenvalues of the periodic, antiperiodic, Dirichlet and Neumann boundary value problems are simple.

In this paper, we propose a new finite element approach, which is different than the classic Babuska-Osborn theory, to approximate Dirichlet eigenvalues. The Dirichlet eigenvalue problem is formulated as the eigenvalue problem of a…

The main difficulty in solving the Helmholtz equation within polygons is due to non-analytic vertices. By using a method nearly identical to that used by Fox, Henrici, and Moler in their 1967 paper; it is demonstrated that such eigenvalue…

A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The proof utilizes recently developed guaranteed computation methods for both eigenvalues and…

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of…

In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong…