In 2006, Dafermos and Holzegel formulated the so-called AdS instability conjecture, stating that there exist arbitrarily small perturbations to AdS initial data which, under evolution by the Einstein vacuum equations for Λ < 0 with reflecting boundary conditions on conformal infinity ℐ, lead to the formation of black holes. The numerical study of this conjecture in the simpler setting of the spherically

We study uniqueness of Dirichlet problems of second-order divergence-form elliptic systems with transversally independent coefficients on the upper half-space in the absence of regularity of solutions. To this end, we develop a substitute for the fundamental solution used to invert elliptic operators on the whole space by means of a representation via abstract single-layer potentials. We also show

We study eigenvalues of non-self-adjoint Schrödinger operators on nontrapping asymptotically conic manifolds of dimension n ≥ 3. Specifically, we are concerned with the following two types of estimates. The first one deals with Keller-type bounds on individual eigenvalues of the Schrödinger operator with a complex potential in terms of the Lp-norm of the potential, while the second one is a Lieb–Thirring-type

We solve the Dirichlet problem for the quaternionic Monge–Ampère equation with a continuous boundary data and the right-hand side in Lp for p > 2. This is the optimal bound on p. We prove also that the local integrability exponent of quaternionic plurisubharmonic functions is 2, which turns out to be less than an integrability exponent of the fundamental solution.

Columnar vortices are stationary solutions of the three-dimensional Euler equations with axial symmetry, where the velocity field only depends on the distance to the axis and has no component in the axial direction. Stability of such flows was first investigated by Lord Kelvin in 1880, but despite a long history the only analytical results available so far provide necessary conditions for instability

Some exotic compact objects, including supersymmetric microstate geometries and certain boson stars, possess evanescent ergosurfaces: time-like submanifolds on which a Killing vector field, which is time-like everywhere else, becomes null. We show that any manifold possessing an evanescent ergosurface but no event horizon exhibits a linear instability of a peculiar kind: either there are solutions

The theory of second-order complex-coefficient operators of the form ℒ = divA(x)∇ has recently been developed under the assumption of p-ellipticity. In particular, if the matrix A is p-elliptic, the solutions u to ℒu = 0 will satisfy a higher integrability, even though they may not be continuous in the interior. Moreover, these solutions have the property that |u|p∕2−1u ∈ Wloc1,2. These properties

We construct an example showing that the upper bound [w]A1 log(e+[w]A1) for the L1(w) → L1,∞(w) norm of the Hilbert transform cannot be improved in general.

We prove an 𝜖-regularity theorem for vector-valued p-harmonic maps, which are critical with respect to a partially free boundary condition, namely that they map the boundary into a round sphere. This does not seem to follow from the reflection method that Scheven used for harmonic maps with free boundary (i.e., the case p = 2): the reflected equation can be interpreted as a p-harmonic map equation

We consider a wide class of functionals with the property of changing their growth and ellipticity properties according to the modulating coefficients in the framework of Musielak–Orlicz spaces. In particular, we provide an optimal condition on the modulating coefficient to establish the Hölder regularity and Harnack inequality for quasiminimizers of the generalized double phase functional with (G

We prove uniform Sobolev estimates for the resolvent of Schrödinger operators with large scaling-critical potentials without any repulsive condition. As applications, global-in-time Strichartz estimates including some nonadmissible retarded estimates, a Hörmander-type spectral multiplier theorem, and Keller-type eigenvalue bounds with complex-valued potentials are also obtained.

We are interested in several questions raised mainly by Mancini and Martinazzi (2017) (see also work of McLeod and Peletier (1989) and Pruss (1996)). We consider the perturbed Moser–Trudinger inequality Iαg(Ω) at the critical level α = 4π, where g, satisfying g(t) → 0 as t → +∞, can be seen as a perturbation with respect to the original case g ≡ 0. Under some additional assumptions, ensuring basically

We address the local well-posedness of the hydrostatic Navier–Stokes equations. These equations, sometimes called reduced Navier–Stokes/Prandtl, appear as a formal limit of the Navier–Stokes system in thin domains, under certain constraints on the aspect ratio and the Reynolds number. It is known that without any structural assumption on the initial data, real-analyticity is both necessary and sufficient

We prove variation-norm estimates for certain oscillatory integrals related to Carleson’s theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. Such variation-norm estimates have applications to discrete analogues and ergodic theory. The proof relies on square function estimates for Schrödinger-like

Federer’s characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension 1. In complete metric spaces that are equipped with a doubling measure and support a Poincaré inequality, the “only if” direction was shown by Ambrosio (2002). By applying fine potential

We extend the results of our paper “Attractors for two-dimensional waves with homogeneous Hamiltonians of degree 0,” written with Laure Saint-Raymond, to the case of forced linear wave equations in any dimension. We prove that, in dimension 2, if the foliation on the boundary at infinity of the energy shell is Morse–Smale, we can apply Mourre’s theory and hence get the asymptotics of the forced solution

We show that the derivative nonlinear Schrödinger (DNLS) equation is globally well-posed in the weighted Sobolev space H2,2(ℝ). Our result exploits the complete integrability of the DNLS equation and removes certain spectral conditions on the initial data required by our previous work, thanks to Zhou’s analysis (Comm. Pure Appl. Math. 42:7 (1989), 895–938) on spectral singularities in the context of

We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension m ≥ 3 starting from any smooth, conformally hyperbolic initial metric. We do not require initial completeness or curvature bounds. With the same methods, we show rigidity of hyperbolic space under the Yamabe flow.

Motivated by the inequality ∥f + g∥22 ≤∥f∥22 + 2∥fg∥1 + ∥g∥22, Carbery (2009) raised the question of what is the “right” analogue of this estimate in Lp for p≠2. Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an Lp version of this inequality by providing upper bounds for ∥f + g∥pp in terms of the quantities ∥f∥pp, ∥g∥pp and ∥fg∥p∕2p∕2 when p ∈ (0,1] ∪ [2,∞), and lower bounds when p ∈ (−∞