ISSN 1439-8516 CN 11-2039/O1
Let Λ ⊂ Rn be a uniformly discrete set and let CΛ be the vector space consisting of all mean periodic functions whose spectrum is simple and contained in Λ. If Λ is a gentle set then for every f ∈ CΛ we have f(x)=O(ωΛ(x)) as|x|→ ∞ and ωΛ(x) can be estimated (Theorem 4.1). This line of research was proposed by Jean-Pierre Kahane in 1957.
We introduce a conjecture that we call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an approach we propose to the Hots Spots Conjecture of J. Rauch using deformation and Lipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincaré inequalities, Harnack inequalities, and NTA (non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible.
We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result:If L0=divA0(x)▽ +B0(x) · ▽ is a p-elliptic operator satisfying the assumptions of Theorem 1.1 then the Lp Dirichlet problem for the operator L0 is solvable in the upper half-space R+n. In this paper we prove that the Lp solvability is stable under small perturbations of L0. That is if L1 is another divergence form elliptic operator with complex coefficients and the coefficients of the operators L0 and L1 are sufficiently close in the sense of Carleson measures, then the Lp Dirichlet problem for the operator L1 is solvable for the same value of p. As a corollary we obtain a new result on Lp solvability of the Dirichlet problem for operators of the form L=divA(x)▽ + B(x) · ▽ where the matrix A satisfies weaker Carleson condition (expressed in term of oscillation of coefficients). In particular the coefficients of A need no longer be differentiable and instead satisfy a Carleson condition that controls the oscillation of the matrix A over Whitney boxes. This result in the real case has been established by Dindoš, Petermichl and Pipher.
A set E ⊂ Rd whose indicator function 1E has maximal Gowers norm, among all sets of equal measure, is an ellipsoid up to Lebesgue null sets. If 1E has nearly maximal Gowers norm then E nearly coincides with some ellipsoid, in the sense that their symmetric difference has small Lebesgue measure.
The Riesz-Sobolev inequality provides a sharp upper bound for a trilinear expression involving convolution of indicator functions of sets. Equality is known to hold only for indicator functions of appropriately situated intervals. We characterize ordered triples of subsets of R1 that nearly realize equality, with quantitative bounds of power law form with the optimal exponent.
This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {Lε} in divergence form with rapidly oscillating and periodic coefficients. We show that the (d -1)-dimensional Hausdorff measures of the nodal sets of solutions to Lε(uε)=0 in a ball in Rd are bounded uniformly in ε > 0. The proof relies on a uniform doubling condition and approximation of uε by solutions of the homogenized equation.
For the general second order linear differential operator
with complex-valued distributional coefficients ajk, bj, and c in an open set Ω ⊆ Rn (n ≥ 1), we present conditions which ensure that -L0 is accretive, i.e., Re<-L0φ, φ> ≥ 0 for all φ ∈ C0∞(Ω).
The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel on the upper half space
where f ∈ Lp(∂R+n), g ∈ Lq' (R+n) and p, q'∈ (1, +∞), 2 ≤ α <n satisfying
. Second, we utilize the technique combining the rearrangement inequality and Lorentz interpolation to show the attainability of best constant Cn,α,p,q'. Third, we apply the regularity lifting method to obtain the smoothness of extremal functions of the above inequality under weaker assumptions. Furthermore, in light of the Pohozaev identity, we establish the sufficient and necessary condition for the existence of positive solutions to the integral system of the Euler-Lagrange equations associated with the extremals of the fractional Poisson kernel. Finally, by using the method of moving plane in integral forms, we prove that extremals of the Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel must be radially symmetric and decreasing about some point ξ0 ∈ ∂R+n. Our results proved in this paper play a crucial role in establishing the Stein-Weiss inequalities with the Poisson kernel in our subsequent paper.
The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of n -1 dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new PDE approach to sets with lower dimensional boundaries. We define a suitable class of degenerate elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one.
In this paper we survey some results on existence, and when possible also uniqueness, of solutions to certain evolution equations obtained by injecting randomness either on the set of initial data or as a perturbative term.
In this paper we consider a coupled Wave-Klein-Gordon system in 3D, and prove global regularity and modified scattering for small and smooth initial data with suitable decay at infinity. This system was derived by Wang and LeFloch-Ma as a simplified model for the global nonlinear stability of the Minkowski space-time for self-gravitating massive fields.
We introduce the concept of transmission eigenvalues in scattering theory for automorphic forms on fundamental domains generated by discrete groups acting on the hyperbolic upper half complex plane. In particular, we consider Fuchsian groups of type Ⅰ. Transmission eigenvalues are related to those eigen-parameters for which one can send an incident wave that produces no scattering. The notion of transmission eigenvalues, or non-scattering energies, is well studied in the Euclidean geometry, where in some cases these eigenvalues appear as zeros of the scattering matrix. As opposed to scattering poles, in hyperbolic geometry such a connection between zeros of the scattering matrix and non-scattering energies is not studied, and the goal of this paper is to do just this for particular arithmetic groups. For such groups, using existing deep results from analytic number theory, we reveal that the zeros of the scattering matrix, consequently non-scattering energies, are directly expressed in terms of the zeros of the Riemann zeta function. Weyl's asymptotic laws are provided for the eigenvalues in those cases along with estimates on their location in the complex plane.
It is a well-known folklore result that quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the boundary of an open set Ω ⊂ Rn+1 with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω, with data in Lp(∂Ω) for some p < ∞. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with continuous boundary data, can be solved, one may seek to characterize the open sets for which Lp solvability holds, thus allowing for singular boundary data.
It has been known for some time that absolute continuity of harmonic measure is closely tied to rectifiability properties of ∂Ω, but also that rectifiability alone is not sufficient to guarantee absolute continuity. In this note, we survey recent progress in this area, culminating in a geometric characterization of the weak-A∞ property, and hence of solvability of the Lp Dirichlet problem for some finite p. This characterization, obtained under rather optimal background hypotheses, follows from a combination of the present author's joint work with Martell, and the work of Azzam, Mourgoglou and Tolsa.
We consider the Cauchy problem for the energy critical heat equation
in dimension n=5. More precisely we find that for given points q1, q2,..., qk and any sufficiently small T > 0 there is an initial condition u0 such that the solution u(x, t) of (0.1) blows-up at exactly those k points with rates type Ⅱ, namely with absolute size ～ (T -t)-α for
. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles.
In this note we show that on any compact subdomain of a Kähler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of Kähler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be treated by standard methods for the Calderón problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends earlier results from the case of Riemann surfaces to higher dimensional complex manifolds.
In this work we shall consider the initial value problem associated to the generalized derivative Schrödinger (gDNLS) equations
and
Following the argument introduced by Cazenave and Naumkin we shall establish the local well-posedness for a class of small data in an appropriate weighted Sobolev space. The other main tools in the proof include the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schrödinger equation established by Kenig-Ponce-Vega.
Let L be a second-order linear elliptic operator with complex coefficients. It is shown that if the Lp Dirichlet problem for the elliptic system L(u)=0 in a fixed Lipschitz domain Ω in Rd is solvable for some
, then it is solvable for all p satisfying
The proof is based on a real-variable argument. It only requires that local solutions of L(u)=0 satisfy a boundary Cacciopoli inequality.
We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary by measuring for every point and direction of entrance of a geodesic the point of exit and direction of exit and its length. The linearization of these two problems is tensor tomography. The question is whether one can determine a symmetric two-tensor from its integrals along geodesics. We emphasize recent results on boundary and lens rigidity and in tensor tomography in the partial data case, with further applications.
