Showing papers in "Revista Matematica Iberoamericana in 2019"

On Kato–Ponce and fractional Leibniz

Abstract: We show that in the Kato–Ponce inequality ∥Js(fg)−fJsg∥p≲∥∂f∥∞∥Js−1g∥p+∥Jsf∥p∥g∥∞, the Jsf term on the right-hand side can be replaced by Js−1∂f. This solves a question raised in Kato–Ponce [14]. We propose a new fractional Leibniz rule for Ds=(−Δ)s/2 and similar operators, generalizing the Kenig–Ponce–Vega estimate [15] to all s>0. We also prove a family of generalized and refined Kato–Ponce type inequalities which include many commutator estimates as special cases. To showcase the sharpness of the estimates at various endpoint cases, we construct several counterexamples. In particular, we show that in the original Kato–Ponce inequality, the L∞-norm on the right-hand side cannot be replaced by the weaker BMO norm. Some divergence-free counterexamples are also included.

Representation and uniqueness for boundary value elliptic problems via first order systems

Abstract: Given any elliptic system with t-independent coefficients in the upper- half space, we obtain representation and trace for the conormal gradient of so- lutions in the natural classes for the boundary value problems of Dirichlet and Neumann types with area integral control or non-tangential maximal control. The trace spaces are obtained in a natural range of boundary spaces which is parametrized by properties of some Hardy spaces. This implies a complete picture of uniqueness vs solvability and well-posedness.

Characterization of the essential spectrum of the Neumann–Poincaré operator in 2D domains with corner via Weyl sequences

Abstract: The Neumann–Poincare (NP) operator naturally appears in the context of metamaterials as it may be used to represent the solutions of elliptic transmission problems via potentiel theory. In particular, its spectral properties are closely related to the well-posedness of these PDE’s, in the typical case where one considers a bounded inclusion of homogeneous plasmonic metamaterial embedded in a homogeneous background dielectric medium. In a recent work, M. Perfekt and M. Putinar have shown that the NP operator of a 2D curvilinear polygon has an essential spectrum, which depends only on the angles of the corners. Their proof is based on quasi-conformal mappings and techniques from complex-analysis. In this work, we characterise the spectrum of the NP operator for a 2D domain with corners in terms of elliptic corner singularity functions, which gives insight on the behaviour of generalized eigenmodes.

Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$

Abstract: We deal with the existence of positive solutions for the following fractional Schrodinger equation: e2s(−Δ)su+V(x)u=f(u)in RN, where e>0 is a parameter, s∈(0,1), N≥2, (−Δ)s is the fractional Laplacian operator, and V:RN→R is a positive continuous function. Under the assumptions that the nonlinearity f is either asymptotically linear or superlinear at infinity, we prove the existence of a family of positive solutions which concentrates at a local minimum of V as e tends to zero.

A weak type estimate for rough singular integrals

Abstract: We obtain a weak type (1, 1) estimate for a maximal operator associated with the classical rough homogeneous singular integrals TΩ. In particular, this provides a different approach to a sparse domination for TΩ obtained recently by Conde-Alonso, Culiuc, Di Plinio and Ou.

Overdetermined problems and constant mean curvature surfaces in cones

Abstract: We consider a partially overdetermined problem in a sector-like domain Ω in a cone Σ in RN, N≥2, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that Ω is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces Γ with boundary which satisfy a 'gluing' condition with respect to the cone Σ. We prove that if either the cone is convex or the surface is a radial graph then Γ must be a spherical cap. Finally we show that, under the condition that the relative boundary of the domain or the surface intersects orthogonally the cone, no other assumptions are needed.

The Flux Limited Keller-Segel System; Properties and Derivation from Kinetic Equations

Abstract: The flux limited Keller-Segel (FLKS) system is a macroscopic model describing bacteria motion by chemotaxis which takes into account saturation of the velocity. The hyper-bolic form and some special parabolic forms have been derived from kinetic equations describing the run and tumble process for bacterial motion. The FLKS model also has the advantage that traveling pulse solutions exist as observed experimentally. It has attracted the attention of many authors recently. We design and prove a general derivation of the FLKS departing from a kinetic model under stiffness assumption of the chemotactic response and rescaling the kinetic equation according to this stiffness parameter. Unlike the classical Keller-Segel system, solutions of the FLKS system do not blow-up in finite or infinite time. Then we investigate the existence of radially symmetric steady state and long time behaviour of this flux limited Keller-Segel system.

Endpoint Sobolev and BV continuity for maximal operators, II

Abstract: In this paper we study some questions about the continuity of classical and fractional maximal operators in the Sobolev space W1,1, in both the continuous and discrete setting, giving a positive answer to two questions posed recently, one of them regarding the continuity of the map f↦(M˜βf)′ from W1,1(R) to Lq(R), for q=1/(1−β). Here M˜β denotes the non-centered fractional maximal operator on R, with β∈(0,1). The second one is related to the continuity of the discrete centered maximal operator in the space of functions of bounded variation BV(Z), complementing some recent boundedness results.

Uniqueness of the solution to the 2D Vlasov-Navier-Stokes system

Abstract: We prove a uniqueness result for weak solutions to the Vlasov–Navier–Stokes system in two dimensions, both in the whole space and in the periodic case, under a mild decay condition on the initial distribution function. The main result is achieved by combining methods from optimal transportation (introduced in this context by G. Loeper) with the use of Hardy’s maximal function, in order to obtain some fine Wasserstein-like estimates for the difference of two solutions of the Vlasov equation.

Definition of fractional Laplacian for functions with polynomial growth

Abstract: We introduce a notion of fractional Laplacian for functions which grow more than linearly at infinity. In such case, the operator is not defined in the classical sense: nevertheless, we can give an ad-hoc definition, which (in addition to the various results that we prove here) can also be useful for applications in various fields, such as blowup and free boundary problems.

Semi-classical states for the nonlinear Choquard equations: Existence, multiplicity and concentration at a potential well

Abstract: We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation −e2Δv+V(x)v=1eα(Iα∗F(v))f(v)in RN, where N≥3, α∈(0,N), Iα(x)=Aα/|x|N−α is the Riesz potential, F∈C1(R,R), F′(s)=f(s) and e>0 is a small parameter. We develop a new variational approach and we show the existence of a family of solutions concentrating, as e→0, to a local minima of V(x) under general conditions on F(s). Our result is new also for f(s)=|s|p−2s and applicable for p∈(N+αN,N+αN−2). Especially, we can give the existence result for locally sublinear case p∈(N+αN,2), which gives a positive answer to an open problem arisen in recent works of Moroz and Van Schaftingen. We also study the multiplicity of positive single-peak solutions and we show the existence of at least cupl(K)+1 solutions concentrating around~K as e→0, where K⊂Ω is the set of minima of V(x) in a bounded potential well Ω, that is, m0≡infx∈ΩV(x)

On the energy variant of the sum-product conjecture

Abstract: We prove new exponents for the energy version of the Erdős–Szemeredi sum-product conjecture, raised by Balog and Wooley. They match the previously established milestone values for the standard formulation of the question, both for general fields and the special case of real or complex numbers, and appear to be the best ones attainable within the currently available technology. Further results are obtained about multiplicative energies of additive shifts and a strengthened energy version of the “few sums, many products” inequality of Elekes and Ruzsa. The latter inequality enables us to obtain a minor improvement of the state-of-the-art sum-product exponent over the reals due to Konyagin and the second author, up to 4/3 + 1/1509. An application of energy estimates to an instance of arithmetic growth in prime residue fields is presented.

Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian

Abstract: We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, min{(−Δ)su,u−φ}=0 in Rn, for general obstacles φ. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in those of Garofalo–Petrosyan to all s∈(0,1).

Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems

Abstract: We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk. In many instances, we find the minimum possible modulus of occurring zeros via a nonlinear extremal problem associated with norms of Jacobi matrices. We examine global properties of these zeros and prove Jentzsch-type theorems describing where they accumulate. As a consequence, we obtain detailed information regarding zeros of reproducing kernels in weighted spaces of analytic functions.

Domains for Dirac-Coulomb min-max levels

Abstract: We consider a Dirac operator in three space dimensions, with an electrostatic (i.e., real-valued) potential V(x), having a strong Coulomb-type singularity at the origin. This operator is not always essentially self-adjoint but admits a distinguished self-adjoint extension DV. In a first part we obtain new results on the domain of this extension, complementing previous works of Esteban and Loss. Then we prove the validity of min-max formulas for the eigenvalues in the spectral gap of DV, in a range of simple function spaces independent of V. Our results include the critical case lim infx→0|x|V(x)=−1, with units such that ℏ=mc2=1, and they are the first ones in this situation. We also give the corresponding results in two dimensions.

Decomposition of Jacobian varieties of curves with dihedral actions via equisymmetric stratification

Abstract: Given a compact Riemann surface X with an action of a finite group G, the group algebra Q[G] provides an isogenous decomposition of its Jacobian variety JX, known as the group algebra decomposition of JX. We consider the set of equisymmetric Riemann surfaces M(2n−1,D2n,θ) for all n≥2.

Sharp $L^p$ estimates for Schrödinger groups on spaces of homogeneous type

Abstract: We prove an Lp estimate ∥e−itLφ(L)f∥p≲(1+|t|)s∥f∥p,t∈R,s=n∣∣12−1p∣∣ for the Schrodinger group generated by a semibounded, self-adjoint operator L on a metric measure space X of homogeneous type (where n is the doubling dimension of X). The assumptions on L are a mild Lp0→Lp′0 smoothing estimate and a mild L2→L2 off-diagonal estimate for the corresponding heat kernel e−tL. The estimate is uniform for φ varying in bounded sets of S(R),or more generally of a suitable weighted Sobolev space.

A polynomial Carleson operator along the paraboloid

Abstract: In this work we extend consideration of the well-known polynomial Carleson operator to the setting of a Radon transform acting along the paraboloid in Rn+1 for n≥2. Inspired by work of Stein and Wainger on the original polynomial Carleson operator, we develop a method to treat polynomial Carleson operators along the paraboloid via van der Corput estimates. A key new step in the approach of this paper is to approximate a related maximal oscillatory integral operator along the paraboloid by a smoother operator, which we accomplish via a Littlewood–Paley decomposition and the use of a square function. The most technical aspect then arises in the derivation of bounds for oscillatory integrals involving integration over lower-dimensional sets. The final theorem applies to polynomial Carleson operators with phase belonging to a certain restricted class of polynomials with no linear terms and whose homogeneous quadratic part is not a constant multiple of the defining function |y|2 of the paraboloid in Rn+1.

Existence results on k-normal elements over finite fields

Abstract: An element α∈Fqn is normal over Fq if α and its conjugates α,αq,…,αqn−1 form a basis of Fqn over Fq. In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of k-normal elements, generalizing the normal elements. In the past few years, many questions concerning the existence and number of k-normal elements with specified properties have been proposed. In this paper, we discuss some of these questions and, in particular, we provide many general results on the existence of k-normal elements with additional properties like being primitive or having large multiplicative order. We also discuss the existence and construction of k-normal elements in finite fields, providing a connection between k-normal elements and the factorization of xn−1 over Fq

On critical $L^p$-differentiability of BD-maps

Abstract: We prove that functions of locally bounded deformation on Rn are Ln/(n−1)-differentiable Ln-almost everywhere. More generally, we show that this critical Lp-differentiability result holds for functions of locally bounded A-variation, provided that the first order, homogeneous differential operator A has finite dimensional null-space.

Quantitative invertibility and approximation for the truncated Hilbert and Riesz transforms

Abstract: In this article we derive quantitative uniqueness and approximation properties for (perturbations) of Riesz transforms. Seeking to provide robust arguments, we adopt a PDE point of view and realize our operators as harmonic extensions, which makes the problem accessible to PDE tools. In this context we then invoke quantitative propagation of smallness estimates in combination with qualitative Runge approximation results. These results can be viewed as quantifications of the approximation properties which have recently gained prominence in the context of nonlocal operators.

Asymptotics of fast rotating density-dependent incompressible fluids in two space dimensions

Abstract: In the present paper we study the fast rotation limit for viscous incompressible fluids with variable density, whose motion is influenced by the Coriolis force. We restrict our analysis to two dimensional flows. In the case when the initial density is a small perturbation of a constant state, we recover in the limit the convergence to the homogeneous incompressible Navier–Stokes equations (up to an additional term, due to density fluctuations). For general non-homogeneous fluids, the limit equations are instead linear, and the limit dynamics is described in terms of the vorticity and the density oscillation function: we lack enough regularity on the latter to prove convergence on the momentum equation itself. The proof of both results relies on a compensated compactness argument, which enables one to treat also the possible presence of vacuum.

Notions of Dirichlet problem for functions of least gradient in metric measure spaces

Abstract: We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a (1, 1)-Poincare inequality. Since one of the two notions is not amenable to the direct method of the calculus of variations, we construct, based on an approach of Juutinen and Mazon-Rossi–De Leon, solutions by considering the Dirichlet problem for p-harmonic functions, p>1, and letting p→1. Tools developed and used in this paper include the inner perimeter measure of a domain.

Dimension free bounds for the vector-valued Hardy–Littlewood maximal operator

Abstract: In this article, Fefferman-Stein inequalities in $L^p(\mathbb R^d;\ell^q)$ with bounds independent of the dimension $d$ are proved, for all $1 < p, q < + \infty.$ This result generalizes in a vector-valued setting the famous one by Stein for the standard Hardy-Littlewood maximal operator. We then extend our result by replacing $\ell^q$ with an arbitrary UMD Banach lattice. Finally, we prove similar dimensionless inequalities in the setting of the Grushin operators.

Towards a reversed Faber–Krahn inequality for the truncated Laplacian

Abstract: We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for the very degenerate elliptic operator P+1 mapping a function u to the maximum eigenvalue of its Hessian matrix. The aim is to show that, at least for square type domains having fixed volume, the symmetry of the domain maximizes the principal eigenvalue, contrary to what happens for the Laplacian.

On the failure of the Hörmander multiplier theorem in a limiting case

Abstract: We discuss the Hormander multiplier theorem for Lp boundedness of Fourier multipliers in which the multiplier belongs to a fractional Sobolev space with smoothness s. We show that this theorem does not hold in the limiting case |1/p−1/2|=s/n.

Nonlinear Calderón–Zygmund theory involving dual data

Abstract: We consider elliptic equations of the p-Laplace type with dual data. We obtain interior and boundary gradient estimates for their weak solutions under sharp conditions on the dual data, coefficients and boundaries of their domains.

Characterizations of a limiting class B ∞ of Békollé–Bonami weights

Abstract: We explore properties of the class of Bekolle–Bonami weights B∞ introduced by the authors in a previous work. Although Bekolle–Bonami weights are known to be ill-behaved because they do not satisfy a reverse Holder property, we prove that when restricting to a class of weights that are “nearly constant on top halves”, one recovers some of the classical properties of Muckenhoupt weights. We also provide an application of this result to the study of the spectra of certain integral operators.

Weighted fractional chain rule and nonlinear wave equations with minimal regularity

Abstract: We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data: It has been known that the problem is well-posed for s≥2 and ill-posed for s 3/2 and thus fill the gap which was left open for many years. For the purpose, we also obtain a weighted fractional chain rule, which is of independent interest. Our method here also works for a class of nonlinear wave equations with general power type nonlinearities which contain the space-time derivatives of the unknown functions. In particular, we prove the Glassey conjecture in the radial case, with minimal regularity assumption.

Whitney's extension problem in o-minimal structures

Abstract: In 1934, H. Whitney asked how one can determine whether a real-valued function on a closed subset of Rn is the restriction of a Cm-function on Rn. A complete answer to this question was found much later by C. Fefferman in the early 2000s. Here, we work in an o-minimal expansion of a real closed field and solve the C1-case of Whitney's extension problem in this context. Our main tool is a definable version of Michael's selection theorem, and we include other another application of this theorem, to solving linear equations in the ring of definable continuous functions.