Showing papers in "Revista Matematica Iberoamericana in 2020"

Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

Abstract: We study various partial data inverse boundary value problems for the semilinear elliptic equation Δu+a(x,u)=0 in a domain in Rn by using the higher order linearization technique introduced by Lassas–Liimatainen–Lin–Salo and Feizmohammadi–Oksanen. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of a(x,z) at z=0 under general assumptions on a(x,z). The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calderon problem by Ferreira–Kenig–Sjostrand–Uhlmann, and implies the solution of partial data problems for certain semilinear equations Δu+a(x,u)=0 also proved by Krupchyk–Uhlmann.

Structure of globally hyperbolic spacetimes-with-timelike-boundary

Abstract: Globally hyperbolic spacetimes-with-timelike-boundary (M¯¯¯¯¯=M∪∂M,g) are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if ∂M is obtained by means of a conformal embedding) can be posed. ∂M represents the naked singularities and can be identified with a part of the intrinsic causal boundary. Apart from general properties of ∂M, the splitting of any globally hyperbolic (M¯¯¯¯¯,g) as an orthogonal product R×Σ¯ with Cauchy slices-with-boundary {t}×Σ¯ is proved. This is obtained by constructing a Cauchy temporal function~τ with gradient ∇τ tangent to ∂M on the boundary. To construct such a~τ, results on stability of both global hyperbolicity and Cauchy temporal functions are obtained. Apart from having their own interest, these results allow us to circumvent technical difficulties introduced by ∂M. The techniques also show that M¯¯¯¯¯ is isometric to the closure of some open subset in a globally hyperbolic spacetime (without boundary). As a trivial consequence, the interior M both splits orthogonally and can be embedded isometrically in some LN, extending so properties of globally spacetimes without boundary to a class of causally continuous ones.

Lipschitz regularity for orthotropic functionals with nonstandard growth conditions

Abstract: We consider a model convex functional with orthotropic structure and super-quadratic nonstan-dard growth conditions. We prove that bounded local minimizers are locally Lipschitz, with no restrictions on the ratio between the highest and the lowest growth rate.

Supports and extreme points in Lipschitz-free spaces

Abstract: For a complete metric space M, we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space FM are precisely the elementary molecules (δ(p)−δ(q))/d(p,q) defined by pairs of points p,q in M such that the triangle inequality d(p,q)

Dynamical aspects of the generalized Schrödinger problem via Otto calculus – A heuristic point of view

Abstract: The defining equation $(\ast):\ \dot \omega_t=-F'(\omega_t),$ of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation $(\ast)$ into the family of slowed down gradient flow equations: $\dot \omega ^{ \varepsilon}_t=- \varepsilon F'( \omega ^{ \varepsilon}_t),$ where $\varepsilon>0$, and (ii) by considering the \emph{accelerations} $\ddot \omega ^{ \varepsilon}_t$. We shall focus on Wasserstein gradient flows. Our approach is mainly heuristic. It relies on Otto calculus.A special formulation of the Schrodinger problem consists in minimizing some action on the Wasserstein space of probability measures on a Riemannian manifold subject to fixed initial and final data. We extend this action minimization problem by replacing the usual entropy, underlying Schrodinger problem, with a general function of the Wasserstein space. The corresponding minimal cost approaches the squared Wasserstein distance when some fluctuation parameter tends to zero. We show heuristically that the solutions satisfy a Newton equation, extending a recent result of Conforti. The connection with Wasserstein gradient flows is established and various inequalities, including evolutional variational inequalities and contraction inequality under curvature-dimension condition, are derived with a heuristic point of view. As a rigorous result we prove a new and general contraction inequality for the Schrodinger problem under a Ricci lower bound on a smooth and compact Riemannian manifold.

Non-cutoff Boltzmann equation with polynomial decay perturbations

Abstract: The Boltzmann equation without the angular cutoff is considered when the initial data is a small perturbation of a global Maxwellian and decays algebraically in the velocity variable. We obtain a well-posedness theory in the perturbative framework for both mild and strong angular singularities. The three main ingredients in the proof are the moment propagation, the spectral gap of the linearized operator, and the regularizing effect of the linearized operator when the initial data is in a Sobolev space with a negative index. A carefully designed pseudo-differential operator plays a central role in capturing the regularizing effect. In addition, some intrinsic symmetry with respect to the collision operator and an intrinsic functional in the coercivity estimate are essentially used in the commutator estimates for the collision operator with velocity weights.

On compactness of commutators of multiplication and bilinear pseudodifferential operators and a new subspace of BMO

Abstract: It is known that the compactness of the commutators of point-wise multiplication with bilinear homogeneous Calderon–Zygmund operators acting on product of Lebesgue spaces is characterized by the multiplying function being in the space CMO. This space is the closure in BMO of its subspace of smooth functions with compact support. It is shown in this work that for bilinear Calderon–Zygmund operators arising from smooth (inhomogeneous) bilinear Fourier multipliers or bilinear pseudodifferential operators, one can actually consider multiplying functions in a new subspace of BMO larger than CMO.

Poincaré profiles of groups and spaces

Abstract: We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincare profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini-Schramm-Timar. In this paper we focus on properties of the Poincare profiles of groups with polynomial growth, and of hyperbolic spaces, where we deduce a connection between these profiles and conformal dimension. As applications, we use these invariants to show the non-existence of coarse embeddings in a variety of examples.

The Dirichlet problem in domains with lower dimensional boundaries

Abstract: Even in the context of the classical case d=n−1, (the analogues of) our results are new. The conditions on the coefficients are more relaxed than the previously known ones (most notably, we do not impose any restrictions whatsoever on the first n−1 rows of the matrix of coefficients) and the results are more general. We establish local rather than global estimates between the square function and the non-tangential maximal function and, perhaps even more importantly, we establish new Moser-type estimates at the boundary and improve the interior ones.

A sharp necessary condition for rectifiable curves in metric spaces

Abstract: In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones β -numbers, numbers measuring flatness in a given scale and location. This work was generalized to Rn by Okikiolu, to Hilbert space by the second author, and has many variants in a variety of metric settings. Notably, in 2005, Hahlomaa gave a sufficient condition for a subset of a metric space to be contained in a rectifiable curve. We prove the sharpest possible converse to Hahlomaa’s theorem for doubling curves, and then deduce some corollaries for subsets of metric and Banach spaces, as well as the Heisenberg group.

On the rate of convergence of semigroups of holomorphic functions at the Denjoy–Wolff point

Abstract: Let {ϕt} be a semigroup of holomorphic self-maps of the unit disc D with Denjoy–Wolff point τ∈∂D. We study the rate of convergence of the semigroup to τ, that is, given z∈D¯¯¯¯, we discuss the behavior of |ϕt(z)−τ| as t goes to +∞.

An abundance of simple left braces with abelian multiplicative Sylow subgroups

Abstract: Braces were introduced by Rump to study involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation. A constructive method for producing all such finite solutions from a description of all finite left braces has been recently discovered. It is thus a fundamental problem to construct and classify all simple left braces, as they can be considered as building blocks for the general theory. This program recently has been initiated by Bachiller and the authors. In this paper we study the simple finite left braces such that the Sylow subgroups of their multiplicative groups are abelian. We provide several new families of such simple left braces. In particular, they lead to the main, surprising result, that shows that there is an abundance of such simple left braces.

Newton-Okounkov bodies of exceptional curve valuations

Abstract: We prove that the Newton–Okounkov body associated to the flag E∙:={X=Xr⊃Er⊃{q}}, defined by the surface X and the exceptional divisor Er given by any divisorial valuation of the complex projective plane P2, with respect to the pull-back of the line-bundle OP2(1) is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton–Okounkov bodies which turn out to be triangular.

On Landis’ conjecture in the plane for some equations with sign-changing potentials

Abstract: In this article, we investigate the quantitative unique continuation properties of real-valued solutions to elliptic equations in the plane. Under a general set of assumptions on the operator, we establish quantitative forms of Landis’ conjecture. Of note, we prove a version of Landis’ conjecture for solutions to −Δu+Vu=0, where V is a bounded function whose negative part exhibits polynomial decay at infinity. The main mechanism behind the proofs is an order of vanishing estimate in combination with an iteration scheme. To prove the order of vanishing result, we present a new idea for constructing positive multipliers and use it reduce the equation to a Beltrami system. The resulting first-order equation is analyzed using the similarity principle and the Hadamard three-quasi-circle theorem.

Stein–Weiss inequalities with the fractional Poisson kernel

Abstract: In this paper, we establish the following Stein–Weiss inequality with the fractional Poisson kernel. Then we prove that there exist extremals for the Stein–Weiss inequality (⋆), and that the extremals must be radially decreasing about the origin. We also provide the regularity and asymptotic estimates of positive solutions to the integral systems which are the Euler–Lagrange equations of the extremals to the Stein–Weiss inequality (⋆) with the fractional Poisson kernel. Our result is inspired by the work of Hang, Wang and Yan, where the Hardy–Littlewood–Sobolev type inequality was first established when γ=2 and α=β=0. The proof of the Stein–Weiss inequality (⋆) with the fractional Poisson kernel in this paper uses recent work on the Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel by Chen, Lu and Tao, and the present paper is a further study in this direction.

Weighted mixed-norm $L_p$-estimates for elliptic and parabolic equations in non-divergence form with singular coefficients

Abstract: In this paper, we study non-divergence form elliptic and parabolic equations with singular coefficients. Weighted and mixed-norm Lp-estimates and solvability are established under suitable partially weighted BMO conditions on the coefficients. When the coefficients are constants, the operators are reduced to extensional operators which arise in the study of fractional heat equations and fractional Laplace equations. Our results are new even in this setting and in the unmixed norm case. For the proof, we explore and utilize the special structures of the equations to show both interior and boundary Lipschitz estimates for solutions and for higher-order derivatives of solutions to homogeneous equations. We then employ the perturbation method by using the Fefferman–Stein sharp function theorem, the Hardy–Littlewood maximal function theorem, as well as a weighted Hardy’s inequality.

On scattering for the cubic defocusing nonlinear Schrödinger equation on the waveguide $\mathbb R^2 \times \mathbb T$

Abstract: In this article, we will show the scattering of the cubic defocusing nonlinear Schrodinger equation on the waveguide R2×T in H1. We first establish the linear profile decomposition in H1(R2×T) motivated by the linear profile decomposition of the mass-critical Schrodinger equation in L2(R2). Then by using the solution of the cubic resonant nonlinear Schrodinger system to approximate the nonlinear profile, we can prove scattering in H1 by using the concentration-compactness/rigidity method.

On a class of nonlinear Schrödinger–Poisson systems involving a nonradial charge density

Abstract: In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schrodinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schrodinger–Poisson system {−Δu+u+ρ(x)ϕu=|u|p−1u,−Δϕ=ρ(x)u2,x∈R3,x∈R3, under different assumptions on ρ:R3→R+ at infinity. Our results cover the range p∈(2,3) where the lack of compactness phenomena may be due to the combined effect of the invariance by translations of a 'limiting problem' at infinity and of the possible unboundedness of the Palais–Smale sequences. Moreover, we find necessary conditions for concentration at points to occur for solutions to the singularly perturbed problem

Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

Abstract: We study regularity for solutions of quasilinear elliptic equations of the form divA(x,u,∇u)=divF in bounded domains in Rn. The vector field A is assumed to be continuous in u, and its growth in ∇u is like that of the p-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions u to the equation under a small BMO condition in x for A. As a consequence, we obtain that ∇u is in the classical Morrey space Mq,λ or weighted space Lqw whenever |F|1/(p−1) is respectively in Mq,λ or Lqw, where q is any number greater than p and w is any weight in the Muckenhoupt class Aq/p. In addition, our two-weight estimate allows the possibility to acquire the regularity for ∇u in a weighted Morrey space that is different from the functional space that the data |F|1/(p−1) belongs to.

Elliptic fibrations on K3 surfaces with a non-symplectic involution fixing rational curves and a curve of positive genus

Abstract: In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Neron–Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibrations on the K3 surfaces explicitly and to study their specializations.

A characterization of Krull monoids for which sets of lengths are (almost) arithmetical progressions

Abstract: Let H be a Krull monoid with finite class group G and suppose that every class contains a prime divisor. Then sets of lengths in H have a well-defined structure which depends only on the class group G. With methods from additive combinatorics we establish a characterization of those class groups G guaranteeing that all sets of lengths are (almost) arithmetical progressions.

On a Pólya functional for rhombi, isosceles triangles, and thinning convex sets

Abstract: Let Ω be an open convex set in Rm with finite width, and with boundary ∂Ω. Let vΩ be the torsion function for Ω, i.e., the solution of −Δv=1,v|∂Ω=0. An upper bound is obtained for the product of ∥vΩ∥L∞(Ω)λ(Ω), where λ(Ω) is the bottom of the spectrum of the Dirichlet Laplacian acting in L2(Ω). The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area 1, it is shown that ∥vΩ∥L1(Ω)λ(Ω)≥π2/24, and that this bound is sharp.

Ground state solutions for quasilinear Schrödinger equations with variable potential and superlinear reaction

Abstract: This paper is concerned with the following quasilinear Schrodinger equation: −Δu+V(x)u−12Δ(u2)u=g(u),x∈RN, where N≥3, V∈C(RN,[0,∞)) and g∈C(R,R) is superlinear at infinity. By using variational and some new analytic techniques, we prove the above problem admits ground state solutions under mild assumptions on V and g. Moreover, we establish a minimax characterization of the ground state energy. Especially, we impose some new conditions on V and more general assumptions on g. For this, some new tricks are introduced to overcome the competing effect between the quasilinear term and the superlinear reaction. Hence our results improve and extend recent theorems in several directions.

Sub-Riemannian structures do not satisfy Riemannian Brunn–Minkowski inequalities

Abstract: We prove that no Brunn–Minkowski inequality from the Riemannian theories of curvature-dimension and optimal transportation can be satisfied by a strictly sub-Riemannian structure. Our proof relies on the same method as for the Heisenberg group together with new investigations by Agrachev, Barilari and Rizzi on ample normal geodesics of sub-Riemannian structures and the geodesic dimension attached to them.

On measures that improve L-q dimension under convolution

Abstract: The Lq dimensions, quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the Lq dimension improve under convolution? This can be seen as a variant of the well-known Lp-improving property. Our main result asserts that uniformly perfect measures (which include Ahlfors-regular measures as a proper subset) have the property that convolving with them results in a strict increase of the Lq dimension. We also study the case q=∞, which corresponds to the supremum of the Frostman exponents of the measure. We obtain consequences for repeated convolutions and for the box dimension of sumsets. Our results are derived from an inverse theorem for the Lq norms of convolutions due to the second author.

Teichmüller space of circle diffeomorphisms with Hölder continuous derivatives

Abstract: Based on the quasiconformal theory of the universal Teichmuller space, we introduce the Teichmuller space of diffeomorphisms of the unit circle with α-Holder continuous derivatives as a subspace of the universal Teichmuller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the Bers embedding. Then, we provide a complex Banach manifold structure for it and prove that its topology coincides with the one induced by local C1+α-topology at the base point.

Generators for the $C^m$-closures of ideals

Abstract: Let R denote the ring of real polynomials on Rn. Fix m≥0, and let A1,…,AM∈R. The \emph{Cm-closure} of (A1,…,AM), denoted here by [A1,…,AM;Cm], is the ideal of all f∈R expressible in the form f=F1A1+⋯+FMAM with each Fi∈Cm(Rn). In this paper we exhibit an algorithm for computing generators for [A1,…,AM;Cm].

Regularity for fully nonlinear parabolic equations with oblique boundary data

Abstract: We obtain, up to a flat boundary, regularity results in parabolic Holder spaces for viscosity solutions of fully nonlinear parabolic equations with oblique boundary conditions.

Solutions to a System of Equations for $C^m$ Functions

Abstract: Fix m≥0, and let A=(Aij(x))1≤i≤N,1≤j≤M be a matrix of semialgebraic functions on Rn or on a compact subset E⊂Rn. Given f=(f1,…,fN)∈C∞(Rn,RN), we consider the following system of equations: ∑j=1MAij(x)Fj(x)=fi(x)for i=1,…,N. In this paper, we give algorithms for computing a finite list of linear partial differential operators such that AF=f admits a Cm(Rn,RM) solution F=(F1,…,FM) if and only if f=(f1,…,fN) is annihilated by the linear partial differential operators.

Young and rough differential inclusions

Abstract: We define in this work a notion of Young differential inclusion dz_t ∈ F(z_t) dx_t , for an α-Holder control x, with α > 1/2, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, γ-Holder continuous set-valued map on the interval [0,1] has a selection with finite p-variation, for p > 1/γ. We also give a notion of solution to the rough differential inclusion dz_t ∈ F(z_t) dt+G(z_t) dX_t , for an α-Holder rough path X with α ∈ (1/3 , 1/2] , a set-valued map F and a single-valued one form G. Then, we prove the existence of a solution to the inclusion when F is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values.