Related papers: An extremal problem related to negative refraction
The aim of this note is to give a geometric insight into the classical second order optimality conditions for equality-constrained minimization problem. We show that the Hessian's positivity of the Lagrangian function associated to the…
We establish the optimal regularity of solutions to the Neumann problem for the fractional Laplacian, $(-\Delta)^s u=h$ in $\Omega$, with the external condition $\mathcal N^s u=0$ in $\Omega^c$. For this, a key point is to establish a 1D…
In this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on $\mathbb{R}^N$. These extremal functions minimize the $L^1(\mathbb{R}^N, |x|^{2\nu + 2 - N}dx)$-distance to the original…
Let $\mathbb{H}^n$ be the $n$-dimensional real hyperbolic space, $\Delta$ its nonnegative Laplace--Beltrami operator whose bottom of the spectrum we denote by $\lambda_{0}$, and $\sigma \in (0,1)$. The aim of this paper is twofold. On the…
We analyze Bergman spaces A p f (D) of generalized analytic functions of solutions to the Vekua equation $\partial$w = ($\partial$f /f)w in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions f and 1…
We discuss the solvability of a fairly general class of systems of perturbed Hammerstein integral equations with functional terms that depend on several parameters. The nonlinearities and the functionals are allowed to depend on the…
The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order…
In the paper "Bellman function for extremal problems in $\mathrm{BMO}$", the authors built the Bellman function for integral functionals on the $\mathrm{BMO}$ space. The present paper provides a development of the subject. We abandon the…
We construct effective Hamiltonians which despite their apparently nonrelativistic form incorporate relativistic effects by involving parameters which depend on the relevant momentum. For some potentials the corresponding energy eigenvalues…
A common problem in the sciences is that a signal of interest is observed only indirectly, through smooth functionals of the signal whose values are then obscured by noise. In such inverse problems, the functionals dampen or entirely…
In this paper, we study weakly nonlinear boundary value problems on infinite intervals. For such problems, we provide criteria for the existence of solutions as well as a qualitative description of the behavior of solutions depending on a…
We develop an approach to use nanostructured plasmonic materials as a non-magnetic negative-refractive index system at optical and near-infrared frequencies. In contrast to conventional negative refraction materials, our design does not…
We review the finite element approximation of the classical obstacle problem in energy and max-norms and derive error estimates for both the solution and the free boundary. On the basis of recent regularity results we present an optimal…
We establish sharp regularity and Fredholm theorems for the \bar{\partial}_b-Neumann problem on domains satisfying some non-generic geometric conditions. We use these domains to construct explicit examples of bad behaviour of the Kohn…
We introduce fractional Brownian motion processes (fBm) as an alternative model for the turbulent index of refraction. These processes allow to reconstruct most of the refractive index properties, but they are not differentiable. We…
We prove necessary optimality conditions, in the class of continuous functions, for variational problems defined with Jumarie's modified Riemann-Liouville derivative. The fractional basic problem of the calculus of variations with free…
We introduce variational problems on Riemannian manifolds with constrained acceleration and derive necessary conditions for normal extremals in the constrained variational problem. The problem consists on minimizing a higher-order energy…
Finite temperature density functional theory requires representations for the internal energy, entropy, and free energy as functionals of the local density field. A central formal difficulty for an orbital-free representation is…
A robust wedge setup is proposed to unambiguously demonstrate negative refraction for negative index metamaterials. We applied our setup to several optical metamaterials from the literature and distinctly observed the phenomena of negative…
The purpose of this paper is to study the weak solutions of the fractional elliptic problem \begin{equation}\label{000} \begin{array}{lll} (-\Delta)^\alpha u+\epsilon g(u)=k\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}\quad &{\rm…