Existence of cylindrically symmetric ground states to a nonlinear curl-curl equation with non-constant coefficients

We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U=f(x,|U|^2)U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations with Kerr-type nonlinear material laws. We prove the existence of a symmetric ground-state type solution for a bounded, cylindrically symmetric coefficient $V$ and subcritical cylindrically symmetric nonlinearity $f$. The new existence result extends the class of problems for which ground-state type solutions are known. It is based on compactness properties of symmetric functions due to Lions, new rearrangement type inequalities from Brock and the recent extension of the Nehari-manifold technique by Szulkin and Weth.Comment: 13 page

Real-valued, time-periodicweak solutions for a semilinear wave equation with periodic δ-potential

We consider the semilinear wave equation $V(x)$$u$$_{u}$ − $u$$_{xx}$ = $±|u|$$^{p-1}$$u$ with p ∈ (1, $\frac{5}{3}$) and a periodically extended delta potential $V(x)$ = $α + βδper(x)$. Both the “+” and the “-” case can be treated. We prove the existence of time-periodic real-valued solutions that are localized in the space direction. Our result builds upon a Fourier-Floquet-Bloch expansion of the solution and a detailed analysis of the spectrum of the wave operator. In fact, it turns out that by a careful choice of the parameters α, β and the spatial and temporal periods, the spectrum of the wave operator $V(x)$∂$\frac{2}{t}$ - ∂$\frac{2}{x}$ (considered on suitable space of time-periodic functions) is bounded away from 0. This allows to find weak solutions as critical points of a functional on a suitable Hilbert space and to apply tools for indefinite variational problems

Quenched invariance principle for random walks on dynamically averaging random conductances

We prove a quenched invariance principle for continuous-time random walks in a dynamically averaging environment on Z. In the beginning, the conductances may fluctuate substantially, but we assume that as time proceeds, the fluctuations decrease according to a typical diffusive scaling and eventually approach constant unit conductances. The proof relies on a coupling with the standard continuous time simple random walk.publishedVersio