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We investigate the Hardy–Schrödinger operator on smooth domains whose boundaries contain the singularity . We prove a Hopf-type result and optimal regularity for variational solutions of corresponding linear and nonlinear Dirichlet boundary value problems, including the equation , where , and is the critical Hardy–Sobolev exponent. We also give a complete description of the profile of all positive solutions — variational or not — of the corresponding linear equation on the punctured domain. The value turns out to be a critical threshold for the operator . When , a notion of Hardysingular boundary mass associated to the operator can be assigned to any conformally bounded domain such that . As a byproduct, we give a complete answer to problems of existence of extremals for Hardy–Sobolev inequalities, and consequently for those of Caffarelli, Kohn and Nirenberg. These results extend previous contributions by the authors in the case , and by Chern and Lin for the case . More specifically, we show that extremals exist when if the mean curvature of at is negative. On the other hand, if , extremals then exist whenever the Hardy singular boundary mass of the domain is positive.
We give a technically simple approach to the maximal regularity problem in parabolic tent spaces for second-order, divergence-form, complex-valued elliptic operators. By using the associated Hardy space theory combined with certain - off-diagonal estimates, we reduce the tent space boundedness in the upper half-space to the reverse Riesz inequalities in the boundary space. This way, we also improve recent results obtained by P. Auscher et al.
We study the exponential stabilization problem for a nonlinear Korteweg-de Vries equation on a bounded interval in cases where the linearized control system is not controllable. The system has Dirichlet boundary conditions at the end-points of the interval and a Neumann nonhomogeneous boundary condition at the right end-point, which is the control. We build a class of time-varying feedback laws for which the solutions of the closed-loop systems with small initial data decay exponentially to . We present also results on the well-posedness of the closed-loop systems for general time-varying feedback laws.
Using suitable modified energies, we study higher-order Sobolev norms’ growth in time for the nonlinear Schrödinger equation (NLS) on a generic 2- or 3-dimensional compact manifold. In two dimensions, we extend earlier results that dealt only with cubic nonlinearities, and get polynomial-in-time bounds for any higher-order nonlinearities. In three dimensions, we prove that solutions to the cubic NLS grow at most exponentially, while for the subcubic NLS we get polynomial bounds on the growth of the norm.
We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schrödinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear Schrödinger equation (DNLS). For (DNLS), Wu (2015) proved that the solution with the initial data is global if by the sharp Gagliardo–Nirenberg inequality. The variational argument gives us another proof of the global existence for (DNLS). Moreover, by the variational argument, we can show that the solution to (DNLS) is global if the initial data satisfies and the momentum is negative.
We study the minimizers of an energy functional with a self-consistent magnetic field, which describes a quantum gas of almost-bosonic anyons in the average-field approximation. For the homogeneous gas we prove the existence of the thermodynamic limit of the energy at fixed effective statistics parameter, and the independence of such a limit from the shape of the domain. This result is then used in a local density approximation to derive an effective Thomas–Fermi-like model for the trapped anyon gas in the limit of a large effective statistics parameter (i.e., “less-bosonic” anyons).
We go back to the question of the regularity of the “velocity average” when and both belong to , and the variable lies in a discrete subset of . First of all, we provide a rate, depending on the number of velocities, for the defect of regularity which is reached when ranges over a continuous set. Second of all, we show that the regularity holds in expectation when the set of velocities is chosen randomly. We apply this statement to investigate the consistency with the diffusion asymptotics of a Monte Carlo-like discrete velocity model.
This paper is concerned with the existence of viscosity solutions of nonlocal fully nonlinear equations that are not translation-invariant. We construct a discontinuous viscosity solution of such a nonlocal equation by Perron’s method. If the equation is uniformly elliptic, we prove the discontinuous viscosity solution is Hölder continuous and thus it is a viscosity solution.
We prove that bilinear forms associated to the rough homogeneous singular integrals
where has vanishing average and , and to Bochner–Riesz means at the critical index in are dominated by sparse forms involving averages. This domination is stronger than the weak- estimates for and for Bochner–Riesz means, respectively due to Seeger and Christ. Furthermore, our domination theorems entail as a corollary new sharp quantitative -weighted estimates for Bochner–Riesz means and for homogeneous singular integrals with unbounded angular part, extending previous results of Hytönen, Roncal and Tapiola for . Our results follow from a new abstract sparse domination principle which does not rely on weak endpoint estimates for maximal truncations.
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