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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 that such layer potentials are uniquely determined.
We study eigenvalues of non-self-adjoint Schrödinger operators on nontrapping asymptotically conic manifolds of dimension . 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 -norm of the potential, while the second one is a Lieb–Thirring-type bound controlling sums of powers of eigenvalues in terms of the -norm of the potential. We extend the results of Frank (2011), Frank and Sabin (2017), and Frank and Simon (2017) on the Keller- and Lieb–Thirring-type bounds from the case of Euclidean spaces to that of nontrapping asymptotically conic manifolds. In particular, our results are valid for the operator on with being a nontrapping compactly supported (or suitably short-range) perturbation of the Euclidean metric and complex-valued.
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 with reflecting boundary conditions on conformal infinity , lead to the formation of black holes. The numerical study of this conjecture in the simpler setting of the spherically symmetric Einstein-scalar field system was initiated by Bizón and Rostworowski (Phys. Rev. Lett. 107:3 (2011), art. id. 031102), followed by a vast number of numerical and heuristic works by several authors.
In this paper, we provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the spherically symmetric Einstein-massless Vlasov system, in the case when the Vlasov field is moreover supported only on radial geodesics. This system is equivalent to the Einstein-null dust system, allowing for both ingoing and outgoing dust. In order to overcome the breakdown of this system occurring once the null dust reaches the center , we place an inner mirror at and study the evolution of this system on the exterior domain . The structure of the maximal development and the Cauchy stability properties of general initial data in this setting are studied in our companion paper (2017, arXiv: 1704.08685).
The statement of the main theorem is as follows: We construct a family of mirror radii and initial data , , converging, as , to the AdS initial data in a suitable norm, such that, for any , the maximal development of contains a black hole region. Our proof is based on purely physical space arguments and involves the arrangement of the null dust into a large number of beams which are successively reflected off and , in a configuration that forces the energy of a certain beam to increase after each successive pair of reflections. As , the number of reflections before a black hole is formed necessarily goes to . We expect that this instability mechanism can be applied to the case of more general matter fields.
We solve the Dirichlet problem for the quaternionic Monge–Ampère equation with a continuous boundary data and the right-hand side in for . This is the optimal bound on . 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 under either planar or axisymmetric perturbations. The purpose of this paper is to show that columnar vortices are spectrally stable with respect to three-dimensional perturbations with no particular symmetry. Our result applies to a large family of velocity profiles, including the most common models in atmospheric flows and engineering applications. The proof is based on a homotopy argument which allows us, when analyzing the spectrum of the linearized operator, to concentrate on a small neighborhood of the imaginary axis, where unstable eigenvalues can be excluded using integral identities and a careful study of the so-called critical layers.
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 to the linear wave equation which concentrate a finite amount of energy into an arbitrarily small spatial region, or the energy of waves measured by a stationary family of observers can be amplified by an arbitrarily large amount. In certain circumstances we can rule out the first type of instability. We also provide a generalisation to asymptotically Kaluza–Klein manifolds. This instability bears some similarity with the “ergoregion instability” of Friedman (Comm. Math.Phys. 63:3 (1978), 243–255), and we use many of the results from the recent proof of this instability by Moschidis (Comm. Math. Phys. 358:2 (2018), 437–520).
The theory of second-order complex-coefficient operators of the form has recently been developed under the assumption of -ellipticity. In particular, if the matrix is -elliptic, the solutions to will satisfy a higher integrability, even though they may not be continuous in the interior. Moreover, these solutions have the property that . These properties of solutions were used by Dindoš and Pipher to solve the Dirichlet problem for -elliptic operators whose coefficients satisfy a further regularity condition, a Carleson measure condition that has often appeared in the literature in the study of real, elliptic divergence form operators. This paper contains two main results. First, we establish solvability of the regularity boundary value problem for this class of operators, in the same range as that of the Dirichlet problem. The regularity problem, even in the real elliptic setting, is more delicate than the Dirichlet problem because it requires estimates on derivatives of solutions. Second, the regularity results allow us to extend the previously established range of solvability of the Dirichlet problem using a theorem due to Z. Shen for general bounded sublinear operators.