Related papers: Eigenvalues for Maxwell's equations with dissipati…

This paper studies eigenvalues of some Steklov problems. Among other things, we show the following sharp estimtes. Let $\Omega$ be a bounded smooth domain in an $n(\geq 2)$-dimensional Hadamard manifold an let $0=\lambda_0 < \lambda_1\leq…

Let $G_n$ be an $n \times n$ matrix with real i.i.d. $N(0,1/n)$ entries, let $A$ be a real $n \times n$ matrix with $\Vert A \Vert \le 1$, and let $\gamma \in (0,1)$. We show that with probability $0.99$, $A + \gamma G_n$ has all of its…

In this paper we study the following eigenvalue boundary value problem for Monge-Amp\`{e}re equations: {equation} \{{array}{l} \det(D^2u)=\lambda^N f(-u)\,\, \text{in}\,\, \Omega, u=0,\,\text{on}\,\, \partial \Omega. {array}. {equation} We…

A plasmon of a bounded domain $\Omega\subset\mathbb{R}^n$ is a non-trivial bounded harmonic function on $\mathbb{R}^n\setminus\partial\Omega$ which is continuous at $\partial\Omega$ and whose exterior and interior normal derivatives at…

We find solutions $E:\Omega\to\mathbb{R}^3$ of the problem \[ \left\{\begin{aligned} &\nabla\times(\nabla\times E) + \lambda E = \partial_E F(x,E) &&\quad \text{in}\Omega\\ &\nu\times E = 0 &&\quad \text{on}\partial\Omega \end{aligned}…

Let $\Omega\subset \RR^2$ be a domain having a compact boundary $\Sigma$ which is Lipschitz and piecewise $C^4$ smooth, and let $\nu$ denote the inward unit normal vector on $\Sigma$. We study the principal eigenvalue $E(\beta)$ of the…

Let $A$ be an irreducible (entrywise) nonnegative $n\times n$ matrix with eigenvalues $$\rho, b+ic,b-ic, \lambda_4,\cdots,\lambda_n,$$ where $\rho$ is the Perron eigenvalue. It is shown that for any $t \in [0, \infty)$ there is a…

Let $\Omega$ be a curvilinear polygon and $Q^\gamma_{\Omega}$ be the Laplacian in $L^2(\Omega)$, $Q^\gamma_{\Omega}\psi=-\Delta \psi$, with the Robin boundary condition $\partial_\nu \psi=\gamma \psi$, where $\partial_\nu$ is the outer…

This paper is devoted to Maxwell modes in three-dimensional bounded electromagnetic cavities that have the form of a product of lower dimensional domains in some system of coordinates. The boundary conditions are those of the perfectly…

If a single particle obeys non-relativistic QM in R^d and has the Hamiltonian H = - Delta + f(r), where f(r)=sum_{i = 1}^{k}a_ir^{q_i}, 2\leq q_i < q_{i+1}, a_i \geq 0$, then the eigenvalues E = E_{n\ell}^{(d)}(\lambda) are given…

It is proved that for class $A_\gamma=\{q\in L_1[0,1]: q\geq 0, \int_0^1 q^\gamma\,dx=1\}$, where $\gamma\in (0,1)$, there exists a potential $q_*\in A_\gamma$ such that minimal eigenvalue $\lambda_1(q_*)$ of boundary problem $$…

It is constructively proved that for class $A_{r,\gamma}=\{q\in L_{1,loc}(0,1): q\leq 0, \int_0^1 rq^\gamma\,dx\leqslant 1\}$, where $r\in C[0,1]$ is uniformly positive weight and $\gamma>1$, there exists a unique potential $\hat q\in…

Let $\Omega\subset \mathbb R^2$ be a bounded planar domain, with piecewise smooth boundary $\partial \Omega$. For $\sigma>0$, we consider the Robin boundary value problem \[ -\Delta f =\lambda f, \qquad \frac{\partial f}{\partial n} +…

Given a star-shaped bounded Lipschitz domain $\Omega\subset{\mathbb R}^d$, we consider the Schr\"odinger operator $L_{\mathcal G}=-\Delta+V$ on $\Omega$ and its restrictions $L^{\Omega_t}_{\mathcal G}$ on the subdomains $\Omega_t$,…

Let $\Omega\subset \mathbb{R}^2$ be the exterior of a convex polygon whose side lengths are $\ell_1,...,\ell_M$. For $\alpha>0$, let $H^\Omega_\alpha$ denote the Laplacian in $\Omega$, $u\mapsto -\Delta u$, with the Robin boundary…

The eigenvalue bounds obtained earlier [J. Phys. A: Math. Gen. 31 (1998) 963] for smooth transformations of the form V(x) = g(x^2) + f(1/x^2) are extended to N-dimensions. In particular a simple formula is derived which bounds the…

This work deals with the Landau equation in a bounded domain with the Maxwell reflection condition on the boundary for any (possibly smoothly position dependent) accommodation coefficient and for the full range of interaction potentials,…

We consider the quasi-linear eigenvalue problem $-\Delta_p u = \lambda g(u)$ subject to Dirichlet boundary conditions on a bounded open set $\Omega$, where $g$ is a locally Lipschitz continuous functions. Imposing no further conditions on…

We prove explicit and sharp eigenvalue estimates for Neumann $p$-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if $\gamma$ denotes a non-closed curve in $\mathbb{R}^2$ symmetric with…

The paper is devoted to constructing the global solutions around global Maxwellians to the initial-boundary value problem on the Boltzmann equation in general bounded domains with isothermal diffuse reflection boundaries. We allow a class…