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For distributions, we build a theory of higher-order pointwise differentiability comprising, for order zero, Łojasiewicz’s notion of point value. Results include Borel regularity of differentials, higher-order rectifiability of the associated jets, a Rademacher–Stepanov-type differentiability theorem, and a Lusin-type approximation. A substantial part of this development is new also for zeroth order. Moreover, we establish a Poincaré inequality involving the natural norms of negative order of differentiability. As a corollary, we characterise pointwise differentiability in terms of point values of distributional partial derivatives.
We are concerned with the short-time observability constant of the heat equation from a subdomain of a bounded domain . The constant is of the form , where depends only on the geometry of and . Luc Miller (J. Differential Equations 204:1 (2004), 202–226) conjectured that is (universally) proportional to the square of the maximal distance from to a point of . We show in particular geometries that may blow up like when is a ball of radius , hence disproving the conjecture. We then prove in the general case the associated upper bound on this blowup. We also show that the conjecture is true for positive solutions of the heat equation.
The proofs rely on the study of the maximal vanishing rate of (sums of)eigenfunctions. They also yield lower and upper bounds for other geometric constants appearing as tunneling constants or approximate control costs.
As an intermediate step in the proofs, we provide a uniform Carleman estimate for Lipschitz metrics. The latter also implies uniform spectral inequalities and observability estimates for the heat equation in a bounded class of Lipschitz metrics, which are of independent interest.
We prove the global stability of the Minkowski space viewed as the trivial solution of the Einstein–Vlasov system. To estimate the Vlasov field, we use the vector field and modified vector field techniques we previously developed in 2017. In particular, the initial support in the velocity variable does not need to be compact. To control the effect of the large velocities, we identify and exploit several structural properties of the Vlasov equation to prove that the worst nonlinear terms in the Vlasov equation either enjoy a form of the null condition or can be controlled using the wave coordinate gauge. The basic propagation estimates for the Vlasov field are then obtained using only weak interior decay for the metric components. Since some of the error terms are not time-integrable, several hierarchies in the commuted equations are exploited to close the top-order estimates. For the Einstein equations, we use wave coordinates and the main new difficulty arises from the commutation of the energy-momentum tensor, which needs to be rewritten using the modified vector fields.
We consider the following perturbed critical Dirichlet problem involving the Hardy–Schrödinger operator:
when is small, , and where , , is a smooth bounded domain with . We show that there exists a sequence in with such that, if for any and , then the above equation has for small, a positive — in general nonminimizing — solution that develops a bubble at the origin. If moreover , then for any integer , the equation has for small enough a sign-changing solution that develops into a superposition of bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition is not necessary. Indeed, it is known that, if and is a ball , then there is no radial positive solution for small. We complete the picture here by showing that, if , then the above problem has no radial sign-changing solutions for small. These results recover and improve what is already known in the nonsingular case, i.e., when .
We derive local estimates for complete noncompact translating solitons of the Gauss curvature flow in which are graphs over a convex domain . This is closely is related to deriving local estimates for the degenerate Monge–Ampère equation. As a result, given a weakly convex bounded domain , we establish the existence of a translating soliton. In particular, when the boundary has line segments, we show the existence of flat sides of the translator from a local a priori nondegeneracy estimate near the free boundary.
We consider the dispersive logarithmic Schrödinger equation in a semiclassical scaling. We extend the results of Carles and Gallagher (Duke Math. J. 167:9 (2018), 1761–1801) about the large-time behavior of the solution (dispersion faster than usual with an additional logarithmic factor and convergence of the rescaled modulus of the solution to a universal Gaussian profile) to the case with semiclassical constant. We also provide a sharp convergence rate to the Gaussian profile in the Kantorovich–Rubinstein metric through a detailed analysis of the Fokker–Planck equation satisfied by this modulus. Moreover, we perform the semiclassical limit of this equation thanks to the Wigner transform in order to get a (Wigner) measure. We show that those two features are compatible and the density of a Wigner measure has the same large-time behavior as the modulus of the solution of the logarithmic Schrödinger equation. Lastly, we discuss about the related kinetic equation (which is the kinetic isothermal Euler system) and its formal properties, enlightened by the previous results and a new class of explicit solutions.
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