Showing papers in "Communications in Contemporary Mathematics in 2016"
TL;DR: In this paper, the authors studied the fractional Laplacian (−Δ)σ/2 on the n-dimensional torus and proved interior and boundary Harnack inequalities when 0 < σ < 2.
Abstract: We study the fractional Laplacian (−Δ)σ/2 on the n-dimensional torus 𝕋n, n ≥ 1. First, we present a general extension problem that describes any fractional power Lγ, γ > 0, where L is a general nonnegative self-adjoint operator defined in an L2-space. This generalizes to all γ > 0 and to a large class of operators the previous known results by Caffarelli and Silvestre. In particular, it applies to the fractional Laplacian on the torus. The extension problem is used to prove interior and boundary Harnack’s inequalities for (−Δ)σ/2, when 0 < σ < 2. We deduce regularity estimates on Holder, Lipschitz and Zygmund spaces. Finally, we obtain the pointwise integro-differential formula for the operator. Our method is based on the semigroup language approach.
94 citations
TL;DR: In this paper, a class of fractional elliptic problems of the form (−Δ)su = f(u) in the half-space ℝ+N := {x ∈ №ℝN:x 1 > 0} with the complementary Dirichlet condition u ≡ 0 in ΩN √ n √ N + n + n was studied and it was shown that bounded positive solutions are increasing in x 1.
Abstract: We study a class of fractional elliptic problems of the form (−Δ)su = f(u) in the half-space ℝ+N := {x ∈ ℝN:x 1 > 0} with the complementary Dirichlet condition u ≡ 0 in ℝN∖ℝ +N. Under mild assumptions on the nonlinearity f, we show that bounded positive solutions are increasing in x1. For the special case f(u) = uq, we deduce nonexistence of positive bounded solutions in the case where q > 1 and q < N−1+2s N−1−2s if N ≥ 1 + 2s. We do not require integrability assumptions on the solutions we study.
57 citations
TL;DR: For polynomials with Gaussian coefficients, the error term O(1) was shown to be 2 π log n + o(log n) in this paper.
Abstract: Roots of random polynomials have been studied intensively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdős–Offord, showed that the expectation of the number of real roots is 2 πlog n + o(log n). In this paper, we determine the true nature of the error term by showing that the expectation equals 2 πlog n + O(1). Prior to this paper, the error term O(1) has been known only for polynomials with Gaussian coefficients.
56 citations
TL;DR: In this article, a class of nonlinear nonautonomous scalar field equations with fractional diffusion, critical power nonlinearity and a subcritical term is investigated, where the involved potentials are allowed for vanishing behavior at infinity.
Abstract: We investigate a class of nonlinear nonautonomous scalar field equations with fractional diffusion, critical power nonlinearity and a subcritical term. The involved potentials are allowed for vanishing behavior at infinity. The problem is set on the whole space and compactness issues have to be tackled.
46 citations
TL;DR: In this paper, a variational Schrodinger equation in dimension 2 is studied for radial stationary states under the presence of a vortex at the origin, and the global behavior of that functional is studied.
Abstract: This paper is motivated by a gauged Schrodinger equation in dimension 2. We are concerned with radial stationary states under the presence of a vortex at the origin. Those states solve a nonlinear nonlocal PDE with a variational structure. We will study the global behavior of that functional, extending known results for the regular case.
42 citations
TL;DR: In this article, the decay of the semigroup generated by the damped wave equation in an unbounded domain was studied, and it was shown that under a natural geometric control condition, the exponential decay of a semigroup can be computed under the assumption that the initial data are smooth.
Abstract: We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilization and control.
37 citations
TL;DR: In this paper, the singular semilinear elliptic equation (SSE) was studied under zero Dirichlet boundary conditions, and it was shown that for β > 0 and f ∈ L1(Ω), the solution is unique.
Abstract: Given Ω a bounded open subset of ℝN, we consider non-negative solutions to the singular semilinear elliptic equation − Δu = f uβ in Hloc1(Ω), under zero Dirichlet boundary conditions. For β > 0 and f ∈ L1(Ω), we prove that the solution is unique.
36 citations
TL;DR: In this article, the authors proved the existence of a unique positive measure μ on ℝ, with respect to which certain normalized cotangent sums are equidistributed.
Abstract: Cotangent sums are associated to the zeros of the Estermann zeta function. They have also proven to be of importance in the Nyman–Beurling criterion for the Riemann Hypothesis. The main result of the paper is the proof of the existence of a unique positive measure μ on ℝ, with respect to which certain normalized cotangent sums are equidistributed. Improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are obtained. We also prove an asymptotic formula for a more general cotangent sum as well as asymptotic results for the moments of the cotangent sums under consideration. We also give an estimate for the rate of growth of the moments of order 2k, as a function of k.
32 citations
TL;DR: In this paper, the authors studied Dirac-harmonic maps from surfaces to manifolds with torsion, motivated from the superstring action considered in theoretical physics, and discussed analytic and geometric properties of such maps and outline an existence result for uncoupled solutions.
Abstract: We study Dirac-harmonic maps from surfaces to manifolds with torsion, which is motivated from the superstring action considered in theoretical physics. We discuss analytic and geometric properties of such maps and outline an existence result for uncoupled solutions.
31 citations
TL;DR: In this paper, the authors derived monotonicity formulae for solutions of the fractional Henon-Lane-Emden equation (−Δ)su = |x|a|u|p−1uin ℝn, when 0 0 and p > 1.
Abstract: We derive monotonicity formulae for solutions of the fractional Henon–Lane–Emden equation (−Δ)su = |x|a|u|p−1uin ℝn, when 0 0 and p > 1. Then, we apply these formulae to classify stable solutions of the above equation.
28 citations
TL;DR: In this article, a parabolic system in divergence form with measurable coefficients in a cylindrical space-time domain with nonsmooth base is considered and the associated nonhomogeneous term is assumed to belong to a suitable weighted Orlicz space.
Abstract: We consider a parabolic system in divergence form with measurable coefficients in a cylindrical space–time domain with nonsmooth base. The associated nonhomogeneous term is assumed to belong to a suitable weighted Orlicz space. Under possibly optimal assumptions on the coefficients and minimal geometric requirements on the boundary of the underlying domain, we generalize the Calderon–Zygmund theorem for such systems by essentially proving that the spatial gradient of the weak solution gains the same weighted Orlicz integrability as the nonhomogeneous term.
TL;DR: The concept of profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach-Alaoglu weak-star compactness t... as mentioned in this paper.
Abstract: The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach-Alaoglu weak-star compactness t ...
TL;DR: In this paper, it was shown that the Brezis-nirenberg problem has a solution with the shape of a tower of two bubbles with alternate signs, centered at the center of symmetry of the domain, for all ϵ > 0 sufficiently small.
Abstract: In this paper, we prove that the Brezis–Nirenberg problem: −Δu = |u|p−1u + ϵuinΩ,u = 0on∂Ω, where Ω is a symmetric bounded smooth domain in ℝN, N ≥ 7 and p = N+2 N−2, has a solution with the shape of a tower of two bubbles with alternate signs, centered at the center of symmetry of the domain, for all ϵ > 0 sufficiently small.
TL;DR: In this paper, the authors considered the Dirichlet boundary value problem for the biharmonic equation defined in a thin T-like shaped structure and constructed an asymptotic expansion of its solution.
Abstract: In the paper, we consider the Dirichlet boundary value problem for the biharmonic equation defined in a thin T-like shaped structure. Our goal is to construct an asymptotic expansion of its solution. We provide error estimates and also introduce and justify the asymptotic partial domain decomposition for this problem.
TL;DR: In this paper, the Morse index of some radial solutions to the problem is computed for the case f(λ,u) = λ, where λ is a smooth nonlinearity and α, λ are real numbers with α > 0.
Abstract: In this paper, we study the problem −Δu = 2+α 22|x|αf(λ,u),in B 1, u > 0, in B1, u = 0, on ∂B1, (𝒫) where B1 is the unit ball of ℝ2, f is a smooth nonlinearity and α, λ are real numbers with α > 0. From a careful study of the linearized operator, we compute the Morse index of some radial solutions to (𝒫). Moreover, using the bifurcation theory, we prove the existence of branches of nonradial solutions for suitable values of the positive parameter λ. The case f(λ,u) = λeu provides more detailed informations.
TL;DR: In this paper, the Frobenius-Lusztig kernel is associated to a different Lie algebra than the initial Lie algebra, and with the image of the universal enveloping algebra.
Abstract: For a finite-dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite-dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite-dimensional Hopf kernel and with the image of the universal enveloping algebra. In this article, we define and completely describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases, the Frobenius–Lusztig kernel is associated to a different Lie algebra than the initial Lie algebra. Moreover, the Frobenius homomorphism often switches short and long roots and/or maps to a braided category.
TL;DR: In this article, the authors studied the partial regularity of the suitable weak solutions to the fractional Navier-Stokes equations in ℝn for n = 2, 3.
Abstract: In this paper, we are concerned with the partial regularity of the suitable weak solutions to the fractional MHD equations in ℝn for n = 2, 3. In comparison with the work of the 3D fractional Navier–Stokes equations obtained by Tang and Yu in [Partial regularity of suitable weak solutions to the fractional Navier–Stokes equations, Comm. Math. Phys. 334 (2015) 1455–1482], our results include their endpoint case α = 3/4 and the external force belongs to a more general parabolic Morrey space. Moreover, we prove some interior regularity criteria just via the scaled mixed norm of the velocity for the suitable weak solutions to the fractional MHD equations.
TL;DR: In this paper, Yang, Su and Kong showed that more weighty curvature implies more powerful improvements in Rellich inequalities on Finsler-Hadamard manifolds.
Abstract: In this paper, we are dealing with quantitative Rellich inequalities on Finsler–Hadamard manifolds where the remainder terms are expressed by means of the flag curvature. By exploring various arguments from Finsler geometry and PDEs on manifolds, we show that more weighty curvature implies more powerful improvements in Rellich inequalities. The sharpness of the involved constants is also studied. Our results complement those of Yang, Su and Kong [Hardy inequalities on Riemannian manifolds with negative curvature, Commun. Contemp. Math. 16 (2014), Article ID: 1350043, 24 pp.].
TL;DR: In this paper, the authors studied a class of Trudinger-Moser inequality in the Sobolev space W1,2(ℝ2) and proved that extremal functions for l(α) can be found if 0 ≤ α < 1.
Abstract: In this paper, we study a class of Trudinger–Moser inequality in the Sobolev space W1,2(ℝ2). Setting l(α) :=supu∈W1,2(ℝ2):∥u∥1,2=1∫ℝ2(e4π(1+α∥u∥22)u2 − 1 − 4π(1 + α∥u∥22)u2)dx we prove: (1) l(α) 1, and (3) there exist extremal functions for l(α) if 0 ≤ α < 1. Blow-up analysis, elliptic estimates and a version of compactness result due to Lions are used to prove (1) and (3). The proof of (2) is based on computations of testing functions which are a combination of eigenfunctions with the Moser sequence.
TL;DR: In this paper, the existence of an exponential attractor for the wave equation with structural damping and supercritical nonlinearity was established by constructing a bounded absorbing set with higher global regularity and using weak quasi-stability estimates.
Abstract: The paper studies the existence of an exponential attractor for the wave equation with structural damping and supercritical nonlinearity utt − Δu + γ(−Δ)αu t + f(u) = g(x). By constructing a bounded absorbing set with higher global regularity (rather than the long-standing partial regularity) and by using the weak quasi-stability estimates (rather than the strong ones as usual), we establish the existence of an exponential attractor in the natural energy space.
TL;DR: In this article, it was shown that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli space.
Abstract: We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are shown to respect the evaluation morphisms in the sense that evaluation commutes with tropicalization. With this particular setting in mind, we prove a general correspondence theorem for enumerative problems which are defined via “evaluation maps” in both the algebraic and tropical world. Applying this to our motivational example, we show that the tropicalizations of the curves in a given toric variety which intersect the boundary divisors in their interior and with prescribed multiplicities, and pass through an appropriate number of generic points are precisely the tropical curves in the corresponding tropical toric variety satisfying the analogous condition. Moreover, the intersection-theoretically defined multiplicities of the tropical curves are equal to the numbers of algebraic curves tropicalizing to them.
TL;DR: In this paper, the global Holder estimates for solutions of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions a la Cabre were proved for the first time.
Abstract: We prove global Holder estimates for solutions of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions a la Cabre.
TL;DR: In this paper, Cazacu and Zuazua studied the optimization problem for the case when all the poles are located on the boundary, and showed that μ⋆(Ω) = N2/n2 if Ω is either a ball, the exterior of a ball or a half-space.
Abstract: In this paper, we study the optimization problem μ⋆(Ω) := inf u∈𝒟1,2(Ω)∫Ω|∇u|2dx ∫ΩV u2dx in a suitable functional space 𝒟1,2(Ω). Here, V is the multi-singular potential given by V :=∑1≤i μ⋆(ℝN) when n ≥ 3 and μ⋆(Ω) = μ⋆(ℝN) when n = 2 (it is known from [C. Cazacu and E. Zuazua, Improved multipolar hardy inequalities, in Studies in Phase Space Analysis with Application to PDEs, Progress in Nonlinear Differential Equations and Their Applications, Vol. 84 (Birkhauser, New York, 2013), pp. 37–52] that μ⋆(ℝN) = (N − 2)2/n2). In the situation when all the poles are located on the boundary, we show that μ⋆(Ω) = N2/n2 if Ω is either a ball, the exterior of a ball or a half-space. Our results do not depend on the distances between ...
TL;DR: In this paper, the decay properties of the Bresse-Cattaneo system in the whole space were investigated and it was shown that the L2-norm of the solution decays with the rate of (1 + t)−1/12.
Abstract: In this paper, we investigate the decay properties of the Bresse–Cattaneo system in the whole space. We show that the coupling of the Bresse system with the heat conduction of the Cattaneo theory leads to a loss of regularity of the solution and we prove that the decay rate of the solution is very slow. In fact, we show that the L2-norm of the solution decays with the rate of (1 + t)−1/12. The behavior of solutions depends on a certain number α (which is the same stability number for the Timoshenko–Cattaneo system [Damping by heat conduction in the Timoshenko system: Fourier and Cattaneo are the same, J. Differential Equations 255(4) (2013) 611–632; The stability number of the Timoshenko system with second sound, J. Differential Equations 253(9) (2012) 2715–2733]) which is a function of the parameters of the system. In addition, we show that we obtain the same decay rate as the one of the solution for the Bresse–Fourier model [The Bresse system in thermoelasticity, to appear in Math. Methods Appl. Sci.].
TL;DR: In this article, the authors define a notion of entropy for connections over ℝn which have shrinking Yang-Mills solitons as critical points, and prove a theorem characterizing entropy stability in terms of the spectrum of a certain linear operator associated to the soliton.
Abstract: Following [T. Colding and W. Minicozzi, II, Generic mean curvature flow I; generic singularities, Ann. of Math. 175(2) (2012) 755–833], we define a notion of entropy for connections over ℝn which has shrinking Yang–Mills solitons as critical points. As in [T. Colding and W. Minicozzi, II, Generic mean curvature flow I; generic singularities, Ann. of Math. 175(2) (2012) 755–833], this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stability in terms of the spectrum of a certain linear operator associated to the soliton. This leads furthermore to a gap theorem for solitons. These results point to a broader strategy of studying “generic singularities” of the Yang–Mills flow, and we discuss the differences in this strategy in dimension n = 4 versus n ≥ 5.
TL;DR: In this article, the fundamental groups of normal complex algebraic varieties share many properties of smooth varieties, such as the jump loci of rank one local systems on a normal variety.
Abstract: We show that the fundamental groups of normal complex algebraic varieties share many properties of the fundamental groups of smooth varieties. The jump loci of rank one local systems on a normal variety are related to the jump loci of a resolution and of a smoothing of this variety.
TL;DR: In this article, a weighted Hardy-Littlewood-Sobolev (HLS) inequality on the upper half space using weighted Hardy type inequality was established, and the existence of extremal functions based on symmetrization argument was discussed.
Abstract: In this paper, we establish a weighted Hardy–Littlewood–Sobolev (HLS) inequality on the upper half space using a weighted Hardy type inequality on the upper half space with boundary term, and discuss the existence of extremal functions based on symmetrization argument. As an application, we can show a weighted Sobolev–Hardy trace inequality with p-biharmonic operator.
TL;DR: In this paper, the authors studied the existence of subharmonic solutions of the prescribed curvature equation − (u′/1 + u′2)′=f(t,u).
Abstract: We study the existence of subharmonic solutions of the prescribed curvature equation − (u′/1 + u′2)′=f(t,u). According to the behavior at zero, or at infinity, of the prescribed curvature f, we prove the existence of arbitrarily small classical subharmonic solutions, or bounded variation subharmonic solutions with arbitrarily large oscillations.
TL;DR: In this article, it was shown that for all but finitely many complex α, the α-cosine transform is a composition of the (α + 2)-Cosine transform with an explicitly written (though complicated) O(n)-invariant differential operator.
Abstract: The goal of this paper is to describe the α-cosine transform on functions on real Grassmannian Gri(ℝn) in analytic terms as explicitly as possible. We show that for all but finitely many complex α the α-cosine transform is a composition of the (α + 2)-cosine transform with an explicitly written (though complicated) O(n)-invariant differential operator. For all exceptional values of α except one, we interpret the α-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the transform corresponding to the last remaining value α, which is − (min{i,n − i} + 1), is still an open problem.
TL;DR: In this paper, the modular invariance of pseudotraces of logarithmic intertwining operators was studied and a genus-one correlation function was defined for the space of solutions of these differential equations with regular singular points.
Abstract: This is the first of two papers in which we study the modular invariance of pseudotraces of logarithmic intertwining operators. We construct and study genus-one correlation functions for logarithmic intertwining operators among generalized modules over a positive-energy and C2-cofinite vertex operator algebra V. We consider grading-restricted generalized V-modules which admit a right action of some associative algebra P, and intertwining operators among such modules which commute with the action of P (P-intertwining operators). We obtain duality properties, i.e. suitable associativity and commutativity properties, for P-intertwining operators. Using pseudotraces introduced by Miyamoto and studied by Arike, we define formal q-traces of products of P-intertwining operators, and obtain certain identities for these formal series. This allows us to show that the formal q-traces satisfy a system of differential equations with regular singular points, and therefore are absolutely convergent in a suitable region and can be extended to yield multivalued analytic functions, called genus-one correlation functions. Furthermore, we show that the space of solutions of these differential equations is invariant under the action of the modular group.