We prove the existence of time quasiperiodic solutions with Liouvillean frequencies for the forced nonlinear Schrödinger equation (NLS). The tool utilized here is a new infinite-dimensional KAM theorem that applies to Liouvillean frequencies.

We study the following problem: given initial data on a compact Cauchy horizon, does there exist a unique solution to wave equations on the globally hyperbolic region? Our main results apply to any spacetime satisfying the null energy condition and containing a compact Cauchy horizon with surface gravity that can be normalised to a nonzero constant. Examples include the Misner spacetime and the Taub-NUT spacetime. We prove an energy estimate close to the Cauchy horizon for wave equations acting on sections of vector bundles. Using this estimate we prove that if a linear wave equation can be solved up to any order at the Cauchy horizon, then there exists a unique solution on the globally hyperbolic region. As a consequence, we prove several existence and uniqueness results for linear and nonlinear wave equations without assuming analyticity or symmetry of the spacetime and without assuming that the generators close. We overcome in particular the essential remaining difficulty in proving that vacuum spacetimes with a compact Cauchy horizon, with constant nonzero surface gravity, necessarily admit a Killing vector field. This work is therefore related to the strong cosmic censorship conjecture.

For domains of first kind we describe the qualitative behavior of the global bifurcation diagram of the unbounded branch of solutions of the Gelfand problem crossing the origin. At least to our knowledge this is the first result about the exact monotonicity of the branch of nonminimal solutions which is not just concerned with radial solutions and/or with symmetric domains. Toward our goal we parametrize the branch not by the $L^{\infty }\left(\mathrm{\Omega }\right)$-norm of the solutions but by the energy of the associated mean field problem. The proof relies on a refined spectral analysis of mean-field-type equations and some surprising properties of the quantities triggering the monotonicity of the Gelfand parameter.

The geometry of solutions to the higher-dimensional Einstein vacuum equations presents aspects that are absent in four dimensions, one of the most remarkable being the existence of stably trapped null geodesics in the exterior of asymptotically flat black holes. This paper investigates the stable trapping phenomenon for two families of higher-dimensional black holes, namely black strings and black rings, and how this trapping structure is responsible for the slow decay of linear waves on their exterior. More precisely, we study decay properties for the energy of solutions to the scalar, linear wave equation ${\square }_{g_{\text{ring}}}\mathrm{\Psi }=0$, where $g_{\text{ring}}$ is the metric of a fixed black ring solution to the five-dimensional Einstein vacuum equations. For a class $\mathfrak{𝔤}$ of black ring metrics, we prove a logarithmic lower bound for the uniform energy decay rate on the black ring exterior $\left(\mathcal{𝒟},g_{\text{ring}}\right)$, with $g_{\text{ring}}\in \mathfrak{𝔤}$. The proof generalizes the perturbation argument and quasimode construction of Holzegel and Smulevici (Anal. PDE 7:5 (2014), 1057–1090) to the case of a nonseparable wave equation and crucially relies on the presence of stably trapped null geodesics on $\mathcal{𝒟}$. As a by-product, the same logarithmic lower bound can be established for any five-dimensional black string.

Our result is the first mathematically rigorous statement supporting the expectation that black rings are dynamically unstable to generic perturbations. In particular, we conjecture a new nonlinear instability for five-dimensional black strings and thin black rings which is already present at the level of scalar perturbations and clearly differs from the mechanism driven by the well-known Gregory–Laflamme instability.

This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations. It deduces optimal estimates of the homogenization error from optimal growth estimates of the (extended) corrector. In line with the heuristics, there are transitions at dimension $d=2$, and for a correlation-decay exponent $\beta =2$ we capture the correct power of logarithms coming from these two sources of criticality.

The decay of correlations is sharply encoded in terms of a multiscale logarithmic Sobolev inequality (LSI) for the ensemble under consideration—the results would fail if correlation decay were encoded in terms of an $\alpha$-mixing condition. Among other ensembles popular in modeling of random media, this class includes coefficient fields that are local transformations of stationary Gaussian fields.

The optimal growth of the corrector $\varphi$ is derived from bounding the size of spatial averages $F=\int g\cdot \nabla \varphi$ of its gradient. This in turn is done by a (deterministic) sensitivity estimate of $F$, that is, by estimating the functional derivative $\frac{\partial F}{\partial a}$ of $F$ with respect to the coefficient field $a$. Appealing to the LSI in form of concentration of measure yields a stochastic estimate on $F$. The sensitivity argument relies on a large-scale Schauder theory for the heterogeneous elliptic operator $-\nabla \cdot a\nabla$. The treatment allows for nonsymmetric $a$ and for systems like linear elasticity.

The aim of this note is to prove sharp regularity estimates for solutions of the continuity equation, associated to $W^{1,p}$ vector fields for . The regularity is of “logarithmic order” and is measured by means of suitable seminorms.

We prove global existence and stability of the solution to the mass-critical stochastic nonlinear Schrödinger equation in $d=1$ with $L_{\omega }^{\infty }{L}_x^2$ initial data. Our construction starts with the existence of a solution to the truncated subcritical problem. With the presence of truncation, we construct the solution to the critical equation as the limit of subcritical solutions. We then obtain uniform bounds on the solutions to the truncated critical problems that allow us to remove truncation in the limit.

We give sharp asymptotic equivalents in the limit $h\to 0$ of the small eigenvalues of the Witten Laplacian, that is, the operator associated with the quadratic form

$$\psi \in H_0^1\left(\mathrm{\Omega }\right)\mapsto h^2{\int }_{\mathrm{\Omega }}|\nabla \left(e^{\frac{1}{h}f}\psi \right){|}^2{e}^{-\frac{2}{h}f},$$

where $\overline{\mathrm{\Omega }}=\mathrm{\Omega }\cup \partial \mathrm{\Omega }$ is an oriented $C^{\infty }$ compact and connected Riemannian manifold with nonempty boundary $\partial \mathrm{\Omega }$ and $f:\overline{\mathrm{\Omega }}\to ℝ$ is a $C^{\infty }$ Morse function. The function $f$ is allowed to admit critical points on $\partial \mathrm{\Omega }$, which is the main novelty of this work in comparison with the existing literature.