Analysis & PDE最新文献

We develop a functional-analytic approach to the study of the Kramers and kinetic Fokker–Planck equations which parallels the classical $H^1$ theory of uniformly elliptic equations. In particular, we identify a function space analogous to $H^1$ and develop a well-posedness theory for weak solutions in this space. In the case of a conservative force, we identify the weak solution as the minimizer of a uniformly convex functional. We prove new functional inequalities of Poincaré- and Hörmander-type and combine them with basic energy estimates (analogous to the Caccioppoli inequality) in an iteration procedure to obtain the $C^{\infty }$ regularity of weak solutions. We also use the Poincaré-type inequality to give an elementary proof of the exponential convergence to equilibrium for solutions of the kinetic Fokker–Planck equation which mirrors the classic dissipative estimate for the heat equation. Finally, we prove enhanced dissipation in a weakly collisional limit.

We describe the asymptotic behavior of positive solutions $u_{𝜀}$ of the equation $�-\mathrm{\Delta }u�+�au�=3u^{5-𝜀}$ in $\mathrm{\Omega }�\subset {ℝ}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon, and the functions $u_{𝜀}$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brezis and Peletier (1989). Similar results are also obtained for solutions of the equation $�-\mathrm{\Delta }u�+�(a�+�𝜀V�)u�=3u^5$ in $\mathrm{\Omega }$.

Motivated by the strong cosmic censorship conjecture in the presence of matter, we study the Einstein equations coupled with a charged/massive scalar field with spherically symmetric characteristic data relaxing to a Reissner–Nordström event horizon. Contrary to the vacuum case, the relaxation rate is conjectured to be slow (nonintegrable), opening the possibility that the matter fields and the metric coefficients blow up in amplitude at the Cauchy horizon, not just in energy. We show that whether this blow-up in amplitude occurs or not depends on a novel oscillationcondition on the event horizon which determines whether or not a resonance is excited dynamically:

• If the oscillation condition is satisfied, then the resonance is not excited and we show boundedness and continuous extendibility of the matter fields and the metric across the Cauchy horizon.

• If the oscillation condition is violated, then by the combined effect of slowdecay and the resonance being excited, we show that the massive uncharged scalar field blows up in amplitude.

  In a companion paper, we will show that in that case a novel nullcontraction singularity forms at the Cauchy horizon, across which the metric is not continuously extendible in the usual sense.

Heuristic arguments in the physics literature indicate that the oscillation condition should be satisfied generically on the event horizon. If these heuristics are true, then our result falsifies the$C^0$-formulationof strong cosmic censorship by means of oscillation.

Inspired by recent work of Mourgoglou and the second author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincaré inequalities in open sets $\mathrm{\Omega }�\subset {ℝ}^{n+1}$, with codimension-$1$ Ahlfors–David regular boundaries. First, we prove that if $\mathrm{\Omega }$ satisfies both the local John condition and the exterior corkscrew condition, then $\mathrm{\Omega }$ also satisfies the Harnack chain condition (and hence is a chord-arc domain). Second, we show that if $\mathrm{\Omega }$ is a $2$-sided chord-arc domain, then the boundary $\partial \mathrm{\Omega }$ supports a Heinonen–Koskela-type weak $1$-Poincaré inequality. We also construct an example of a set $\mathrm{\Omega }�\subset {ℝ}^{n+1}$ such that the boundary $\partial \mathrm{\Omega }$ is Ahlfors–David regular and supports a weak boundary $1$-Poincaré inequality but $\mathrm{\Omega }$ is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincaré theories.

We consider the isoperimetric problem defined on the whole ${ℝ}^n$ by the Allen–Cahn energy functional. For nondegenerate double-well potentials, we prove sharp quantitative stability inequalities of quadratic type which are uniform in the length scale of the phase transitions. We also derive a rigidity theorem for critical points analogous to the classical Alexandrov theorem for constant mean curvature boundaries.

This paper deals with classical and semiclassical nonvanishing magnetic fields on a Riemannian manifold of arbitrary dimension. We assume that the magnetic field $B�=�dA$ has constant rank and admits a discrete well. On the classical part, we exhibit a harmonic oscillator for the Hamiltonian $H�=�|p�-�A(q){|}^2$ near the zero-energy surface: the cyclotron motion. On the semiclassical part, we describe the semiexcited spectrum of the magnetic Laplacian ${\mathcal{ℒ}}_{\hslash }�={(i\hslash d�+�A)}^{\ast }(i\hslash d�+�A)$. We construct a semiclassical Birkhoff normal form for ${\mathcal{ℒ}}_{\hslash }$ and deduce new asymptotic expansions of the smallest eigenvalues in powers of ${\hslash }^{1∕2}$ in the limit $\hslash �\to 0$. In particular we see the influence of the kernel of $B$ on the spectrum: it raises the energies at order ${\hslash }^{3∕2}$.

We prove the only blowup solutions to the focusing, quintic nonlinear Schrödinger equation with mass equal to the mass of the soliton are rescaled solitons or the pseudoconformal transformation of those solitons.

This paper is devoted to the study of noncommutative maximal operators with rough kernels. More precisely, we prove the weak-type $(1,1)$ boundedness for noncommutative maximal operators with rough kernels. The proof of the weak-type $(1,1)$ estimate is based on the noncommutative Calderón–Zygmund decomposition. To deal with the rough kernel, we use the microlocal decomposition in the proofs of both the bad and good functions.

We desingularise the union of three Grim paraboloids along Costa–Hoffman–Meeks surfaces in order to obtain complete embedded translating solitons of the mean curvature flow with three ends and arbitrary finite genus.

We propose a gradient flow perspective to the spatially homogeneous Landau equation for soft potentials. We construct a tailored metric on the space of probability measures based on the entropy dissipation of the Landau equation. Under this metric, the Landau equation can be characterized as the gradient flow of the Boltzmann entropy. In particular, we characterize the dynamics of the PDE through a functional inequality which is usually referred as the energy dissipation inequality (EDI). Furthermore, analogous to the optimal transportation setting, we show that this interpretation can be used in a minimizing movement scheme to construct solutions to a regularized Landau equation.