The aim of this paper is to study the spectrum of the fourth order eigenvalue boundary value problem ⎧Δ²u = αu + βΔu in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω. where (α,β) ∈ ℝ². We prove the existence of a first nontrivial curve of this spectrum and we give its variational characterization. Moreover we prove some properties of this curve, e.g., continuity, convexity, and asymptotic behavior. As an application, we study the non-resonance of solutions below...

This paper reviews popular acceleration techniques to converge the non-linear self-consistent field equations appearing in quantum chemistry calculations with localized basis sets. The different methodologies, as well as their advantages and limitations are discussed within the same framework. Several illustrative examples of calculations are presented. This paper attempts to describe recent achievements and remaining challenges in this field.

For a bounded and sufficiently smooth domain $\mathrm{\Omega }$ in ${\mathbb{R}}^N$, $N\ge 2$, let ${\left({\lambda }_k\right)}_{k=1}^{\mathrm{\infty }}$ and ${\left({\phi }_k\right)}_{k=1}^{\mathrm{\infty }}$ be respectively the eigenvalues and the corresponding eigenfunctions of the problem (with Neumann boundary conditions) $$-\text{div}\left(a\left(x\right)\nabla {\phi }_k\right)+q\left(x\right){\phi }_k={\lambda }_k\varrho \left(x\right){\phi }_k\text{ in }\mathrm{\Omega },a\frac{\partial }{\partial \mathbf{n}}{\phi }_k=0\text{ su }\partial \mathrm{\Omega }.$$ We prove that knowledge of the Dirichlet boundary spectral data ${\left({\lambda }_k\right)}_{k=1}^{\mathrm{\infty }}$, ${\left({\phi }_{k|\partial \mathrm{\Omega }}\right)}_{k=1}^{\mathrm{\infty }}$ determines uniquely the Neumann-to-Dirichlet (or the Steklov- Poincaré) map $\gamma$ for a related elliptic problem. Under suitable hypothesis on the coefficients $a,q,\varrho$ their identifiability is then proved. We prove also analogous results for Dirichlet...

Dubrovin type equations for the N -gap solution of a completely integrable system associated with a polynomial pencil is constructed and then integrated to a system of functional equations. The approach used to derive those results is a generalization of the familiar process of finding the 1-soliton (1-gap) solution by integrating the ODE obtained from the soliton equation via the substitution u = u(x + λt).

We consider the linear eigenvalue problem -Δu = λV(x)u, $u\in D_0^{1,2}\left(\Omega \right)$, and its nonlinear generalization $-{\Delta }_{p}u=\lambda V\left(x\right){\left|u\right|}^{p-2}u$, $u\in D_0^{1,p}\left(\Omega \right)$. The set Ω need not be bounded, in particular, $\Omega ={ℝ}^N$ is admitted. The weight function V may change sign and may have singular points. We show that there exists a sequence of eigenvalues ${\lambda }_n\to \infty$.

We consider the nonlinear eigenvalue problem $$-{div\left(\right|\nabla u|}^{p-2}{\nabla u)=\lambda g\left(x\right)|u|}^{p-2}u$$ in ${𝐑}^N$ with $p>1$. A condition on indefinite weight function $g$ is given so that the problem has a sequence of eigenvalues tending to infinity with decaying eigenfunctions in $W^{1,p}\left({𝐑}^N\right)$. A nonexistence result is also given for the case $p\ge N$.

We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$-Laplacian and the Navier $p$-biharmonic operator on a ball of radius $R$ in ${ℝ}^N$ and its asymptotics for $p$ approaching $1$ and $\infty$. Let $p$ tend to $\infty$. There is a critical radius $R_C$ of the ball such that the principal eigenvalue goes to $\infty$ for $0<R\le R_C$ and to $0$ for $R>R_C$. The critical radius is $R_C=1$ for any $N\in \mathbb{N}$ for the $p$-Laplacian and $R_C=\sqrt{2N}$ in the case of the $p$-biharmonic operator. When $p$ approaches $1$, the principal eigenvalue of the Dirichlet...