Showing papers in "Communications in Contemporary Mathematics in 2015"
TL;DR: In this article, the authors consider nonlinear Choquard equation where N ≥ 3, V ∈ L∞(ℝN) is an external potential and Iα(x) is the Riesz potential of order α ∈ (0, N).
Abstract: We consider nonlinear Choquard equation where N ≥ 3, V ∈ L∞(ℝN) is an external potential and Iα(x) is the Riesz potential of order α ∈ (0, N). The power in the nonlocal part of the equation is critical with respect to the Hardy–Littlewood–Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if then the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis–Lieb lemma.
161 citations
TL;DR: In this paper, the authors complete the study made in [The elliptic Kirchhoff equation in ℝN perturbed by a local nonlinearity, Differential Integral Equations 25 (2012) 543-554] on a kirchhoff type equation with a Berestycki-Lions non-linearity.
Abstract: In this note, we complete the study made in [The elliptic Kirchhoff equation in ℝN perturbed by a local nonlinearity, Differential Integral Equations 25 (2012) 543–554] on a Kirchhoff type equation with a Berestycki–Lions nonlinearity. We also correct Theorem 0.6 inside.
72 citations
TL;DR: In this article, the logarithmic version of K-stability was introduced by generalizing Donaldson's algebraic formulation of Kstability, which was later generalized to the algebraic version of kstability.
Abstract: We introduce the logarithmic version of K-stability by generalizing Donaldson's algebraic formulation of K-stability.
42 citations
TL;DR: Cao et al. as mentioned in this paper considered a perturbation of the Ricci solitons equation proposed in [J.-P. Bourguignon, Ricci curvature and Einstein metrics, in Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, Vol. 838 (Springer, Berlin, 1981), pp. 42-63] and studied in [H.-D Cao, Geometry of Ricci Solitons, Chinese Ann. Math. Ser. B 27(2) (2006) 121-142] and classified noncompact gradient
Abstract: In this paper, we consider a perturbation of the Ricci solitons equation proposed in [J.-P. Bourguignon, Ricci curvature and Einstein metrics, in Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, Vol. 838 (Springer, Berlin, 1981), pp. 42–63] and studied in [H.-D. Cao, Geometry of Ricci solitons, Chinese Ann. Math. Ser. B 27(2) (2006) 121–142] and we classify noncompact gradient shrinkers with bounded non-negative sectional curvature.
42 citations
TL;DR: In this paper, it was shown that the minimizer of the Thomas-Fermi energy per particle in Bravais lattices with fixed density is a triangular lattice composed of equilateral triangles.
Abstract: We prove in this paper that the minimizer of Lennard–Jones energy per particle among Bravais lattices is a triangular lattice, i.e. composed of equilateral triangles, in ℝ2 for large density of points, while it is false for sufficiently small density. We show some characterization results for the global minimizer of this energy and finally we also prove that the minimizer of the Thomas–Fermi energy per particle in ℝ2 among Bravais lattices with fixed density is triangular.
36 citations
TL;DR: In this paper, the authors try to understand complete types over a somewhat saturated model of a complete first-order theory which is dependent (previously called NIP), by "decomposition theorems for such types".
Abstract: We try to understand complete types over a somewhat saturated model of a complete first-order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory is the combination of the one for stable theories and the one for the theory of dense linear order or trees (and first, we should try to understand the quite saturated case). As a measure of our progress, we give several applications considering some test questions; in particular, we try to prove the generic pair conjecture and do it for measurable cardinals.
35 citations
TL;DR: In this paper, the authors studied the nonexistence of positive solutions for the following elliptic equation, where Lαu = Δxu + (α + 1)2|x|2αΔyu, α > 0, (x, y) ∈ ℝm × k.
Abstract: In this paper, we study the nonexistence of positive solutions for the following elliptic equation where Lαu = Δxu + (α + 1)2|x|2αΔyu, α > 0, (x, y) ∈ ℝm × ℝk. We will prove that this problem possesses no positive solutions under some assumptions on the nonlinear term f. The main technique we use is the moving plane method in an integral form.
34 citations
TL;DR: In this article, a non-local Neumann problem driven by a nonhomogeneous elliptic differential operator is studied, where the reaction term is a nonlinearity function that exhibits psuperlinear growth but need not satisfy the Ambrosetti-Rabinowitz condition.
Abstract: We study a nonlocal Neumann problem driven by a nonhomogeneous elliptic differential operator. The reaction term is a nonlinearity function that exhibits p-superlinear growth but need not satisfy the Ambrosetti–Rabinowitz condition. By using an abstract linking theorem for smooth functionals, we prove a multiplicity result on the existence of weak solutions for such problems. An explicit example illustrates the main abstract result of this paper.
29 citations
TL;DR: Wu et al. as mentioned in this paper studied concave-convex semilinear elliptic problems with sign-changing weights and obtained three positive solutions under the predefined conditions of fλ(x) and gμ(x).
Abstract: In this paper, we study the following concave–convex elliptic problems: where N ≥ 3, 1 0 and μ < 0 are two parameters. By using several variational methods and a perturbation argument, we obtain three positive solutions to this problem under the predefined conditions of fλ(x) and gμ(x), which simultaneously extends the result of [T. Hsu, Multiple positive solutions for a class of concave–convex semilinear elliptic equations in unbounded domains with sign-changing weights, Bound. Value Probl. 2010 (2010), Article ID 856932, 18pp.; T. Wu, Multiple positive solutions for a class of concave–convex elliptic problems in ℝN involving sign-changing weight, J. Funct. Anal. 258 (2010) 99–131]. We also study the concentration behavior of these three solutions both for λ → 0 and μ → -∞.
23 citations
TL;DR: In this paper, it was shown that the local existence and singularity structures of low regularity solution to the semilinear generalized Tricomi equation with typical discontinuous initial data (u, x), ∂tu(0, x)) = (0, φ(x)), where m ∈ ℕ, x = (x1,…,xn), n ≥ 2, and f(t, x, u) is C∞ smooth on its arguments.
Abstract: In this paper, we are concerned with the local existence and singularity structures of low regularity solution to the semilinear generalized Tricomi equation with typical discontinuous initial data (u(0, x), ∂tu(0, x)) = (0, φ(x)), where m ∈ ℕ, x = (x1,…,xn), n ≥ 2, and f(t, x, u) is C∞ smooth on its arguments. When the initial data φ(x) is homogeneous of degree zero or piecewise smooth along the hyperplane {t = x1 = 0}, it is shown that the local solution u(t, x) ∈ L∞([0, T] × ℝn) exists and is C∞ away from the forward cuspidal conic surface or the cuspidal wedge-shaped surfaces respectively. On the other hand, for n = 2 and piecewise smooth initial data φ(x) along the two straight lines {t = x1 = 0} and {t = x2 = 0}, we establish the local existence of a solution and further show that in general due to the degenerate character of the equation under study, where . This is an essential difference to the well-known result for solution to the two-dimensional semilinear wave equation with (v(0, x), ∂tv(0, x)) = (0, φ(x)), where Σ0 = {t = |x|}, and .
22 citations
TL;DR: In this article, the WP-class Teichmuller space of Riemann surfaces with biholomorphisms was defined, and it was shown that the rigged moduli space of such surfaces is a complex Hilbert manifold.
Abstract: We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disk removed. We define a refined Teichmuller space of such Riemann surfaces (which we refer to as the WP-class Teichmuller space) and demonstrate that in the case that 2g + 2 - n > 0, this refined Teichmuller space is a Hilbert manifold. The inclusion map from the refined Teichmuller space into the usual Teichmuller space (which is a Banach manifold) is holomorphic. We also show that the rigged moduli space of Riemann surfaces with non-overlapping holomorphic maps, appearing in conformal field theory, is a complex Hilbert manifold. This result requires an analytic reformulation of the moduli space, by enlarging the set of non-overlapping mappings to a class of maps intermediate between analytically extendible maps and quasiconformally extendible maps. Finally, we show that the rigged moduli space is the quotient of the refined Teichmuller space by a properly discontinuous group of biholomorphisms.
TL;DR: In this article, the authors considered a nonlinear logistic type equation with constant sign and showed that the problem admits nontrivial solutions of constant sign for all big values of the parameter, and in fact established the existence of extremal constant sign solutions.
Abstract: We consider a nonlinear logistic type equation. For all big values of the parameter, we show that the problem admits nontrivial solutions of constant sign and in fact we establish the existence of extremal constant sign solutions. Using these extremal solutions, we produce a nodal (sign-changing) solution. We also investigate the uniqueness and continuous dependence on the parameter of positive solutions. Finally, we study the degenerate p-logistic equation.
TL;DR: In this article, the authors considered the delay differential equation u′(t) ∈ Au (t) + f(t, ut), t ∈ ℝ+, where A is the infinitesimal generator of a nonlinear semigroup of contractions in a Banach space X and f is continuous, subjected to a general mixed nonlocal + local initial condition.
Abstract: We consider the delay differential equation u′(t) ∈ Au(t) + f(t, ut), t ∈ ℝ+, where A is the infinitesimal generator of a nonlinear semigroup of contractions in a Banach space X and f is continuous, subjected to a general mixed nonlocal + local initial condition of the form u(t) = g(u)(t) + ψ(t), t ∈ [-τ, 0]. We prove that under natural conditions on A, f, g and ψ the problem above has at least one C0-solution. Applications to T-periodic and to T-anti-periodic problems, as well as an illustrative example referring to the nonlinear diffusion equation are also included.
TL;DR: In this paper, the authors studied the nondegeneracy properties of positive finite energy solutions of the equation Δ𝔹u - λu = |u|p-1u in the hyperbolic space and showed that the degeneracy occurs only in an N-dimensional subspace.
Abstract: In this article, we will study the nondegeneracy properties of positive finite energy solutions of the equation -Δ𝔹u - λu = |u|p-1u in the hyperbolic space. We will show that the degeneracy occurs only in an N-dimensional subspace. We will prove that the positive solutions are nondegenerate in the case of geodesic balls.
TL;DR: In this paper, the authors generalize the results of Chang and Wang and prove sharp inequalities between the volume and the integral of the kth mean curvature for (k + 1)-convex domains in the Euclidean space for all k.
Abstract: In this paper, we prove sharp inequalities between the volume and the integral of the kth mean curvature for (k + 1)-convex domains in the Euclidean space for all k. We generalize the recent results of Chang and Wang [Some higher order isoperimetric inequalities via the method of optimal transport, preprint (2013); arXiv:1305.3004], where they prove the case k = 1, 2. The idea is the same as [S.-Y. A. Chang and Y. Wang, Some higher order isoperimetric inequalities via the method of optimal transport, preprint (2013); arXiv:1305.3004], but calculations are involved.
TL;DR: In this paper, it was shown that the set of the initial data for which the solution is global is not star-shaped with respect to the origin, and that the solution with initial value ϑup blows up in finite time if |ϑ - 1| > 0 is sufficiently small and if p is sufficiently close to the critical exponent.
Abstract: Consider the nonlinear heat equation vt - Δv = |v|p-1v in a bounded smooth domain Ω ⊂ ℝn with n > 2 and Dirichlet boundary condition. Given up a sign-changing stationary classical solution fulfilling suitable assumptions, we prove that the solution with initial value ϑup blows up in finite time if |ϑ - 1| > 0 is sufficiently small and if p is sufficiently close to the critical exponent . Since for ϑ = 1 the solution is global, this shows that, in general, the set of the initial data for which the solution is global is not star-shaped with respect to the origin. This phenomenon had been previously observed in the case when the domain is a ball and the stationary solution is radially symmetric.
TL;DR: In this paper, the L2-Alexander invariant of a knot complement is computed using any presentation of default 1 of the knot group, and a method for computing the invariant is given.
Abstract: This paper deals with the study of a new family of knot invariants: the L2-Alexander invariant. A main result is to give a method of computation of the L2-Alexander invariant of a knot complement using any presentation of default 1 of the knot group.
TL;DR: In this article, a sufficient condition for the completeness of a time-dependent vector field in ℝN is provided, generalizing the well-known left-invariance condition on Lie groups.
Abstract: We provide a sufficient condition for the completeness of a time-dependent vector field in ℝN, generalizing the well-known left-invariance condition on Lie groups. This result can be applied to the construction of Lie groups associated to suitable families X of Hormander vector fields, without the need to use the Third Fundamental Theorem of Lie. Further applications are given to the control-theoretic distance related to X, and to the existence of the relevant geodesics.
TL;DR: In this article, it was shown that the Chevalley-Eilenberg complex associated to this representation of C possesses the structure 𝕏 of a strong homotopy Lie-Rinehart algebra.
Abstract: An involutive distribution C on a smooth manifold M is a Lie-algebroid acting on sections of the normal bundle TM/C. It is known that the Chevalley–Eilenberg complex associated to this representation of C possesses the structure 𝕏 of a strong homotopy Lie–Rinehart algebra. It is natural to interpret 𝕏 as the (derived) Lie–Rinehart algebra of vector fields on the space P of integral manifolds of C. In this paper, we show that 𝕏 is embedded in an A∞-algebra 𝔻 of (normal) differential operators. It is natural to interpret 𝔻 as the (derived) associative algebra of differential operators on P. Finally, we speculate about the interpretation of 𝔻 as the universal enveloping strong homotopy algebra of 𝕏.
TL;DR: In this article, the authors studied the symmetry properties of solutions of elliptic systems of the type where x ∈ ℝm with 1 ≤ m < N, X = (x, y) ∈ ∄°m × �°N-m, and F 1,…,Fn are the derivatives with respect to ξ 1, ξn of some F = F(x,ξ1, etc, n) such that for any i = 1, ǫ, n and any fixed ξi → F (x
Abstract: We study the symmetry properties for solutions of elliptic systems of the type where x ∈ ℝm with 1 ≤ m < N, X = (x, y) ∈ ℝm × ℝN-m, and F1,…,Fn are the derivatives with respect to ξ1,…,ξn of some F = F(x,ξ1,…,ξn) such that for any i = 1,…,n and any fixed (x,ξ1,…,ξi-1,ξi+1,…,ξn) ∈ ℝm × ℝn-1 the map ξi → F(x,ξ1,…,ξi,…,ξn) belongs to C2(ℝ). We obtain a Poincare-type formula for the solutions of the system and we use it to prove a symmetry result both for stable and for monotone solutions.
TL;DR: In this article, sufficient conditions for convergence of continued fractions are provided for convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre-Fenchel and Artstein-Avidan-Milman transforms.
Abstract: In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalization of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.
TL;DR: In this paper, the existence of meromorphic first integrals for Pfaff systems of arbitrary codimension on complex manifolds has been studied and conditions for their existence have been established.
Abstract: We present results expressing conditions for the existence of meromorphic first integrals for Pfaff systems of arbitrary codimension on complex manifolds. Some of the results presented improve previous ones due to Jouanolou and Ghys. We also present an enumerative result counting the number of hypersurfaces invariant by a projective holomorphic foliation with split tangent sheaf.
TL;DR: In this paper, the existence and nonexistence of radial solutions in the first open case was shown to be open for 0 < α < 2 and, for 2 < α > N and, and for N ≤ α > 2N - 2 and.
Abstract: Several existence and nonexistence results are known for positive solutions u ∈ D1,2(ℝN) ∩ L2(ℝN, ∣x∣-αdx) ∩ Lp(ℝN) to the equation resting upon compatibility conditions between α and p. Letting 2α := 2N/(N - α) and , the problem is still open for 0 < α < 2 and , for 2 < α < N and , and for N ≤ α < 2N - 2 and . Here we give a negative answer to the problem of the existence of radial solutions in the first open case.
TL;DR: In this paper, the validity of the circular law for random matrices with non-i.i.d. entries was investigated and it was shown that as the dimension n goes to infinity, the empirical spectral distribution of M tends to the uniform law on the unit disc of the complex plane.
Abstract: We explore the validity of the circular law for random matrices with non-i.i.d. entries. Let M be an n × n random real matrix obeying, as a real random vector, a log-concave isotropic (up to normalization) unconditional law, with mean squared norm equal to n. The entries are uncorrelated and obey a symmetric law of zero mean and variance 1/n. This model allows some dependence and non-equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries, known as the real Ginibre Ensemble. Our main result states that as the dimension n goes to infinity, the empirical spectral distribution of M tends to the uniform law on the unit disc of the complex plane.
TL;DR: In this paper, the existence and regularity of the solution to the multivalued equation -ΔΦu ∈ ∂j(u) + λh in Ω, where Ω ⊂ RN is a bounded smooth domain, Φ is an N-function, ΔΦ is the corresponding Φ-Laplacian, λ > 0 is a parameter, h is a measurable function, and j is a continuous function with critical growth.
Abstract: We study the existence and regularity of the solution to the multivalued equation -ΔΦu ∈ ∂j(u) + λh in Ω, where Ω ⊂ RN is a bounded smooth domain, Φ is an N-function, ΔΦ is the corresponding Φ-Laplacian, λ > 0 is a parameter, h is a measurable function, and j is a continuous function with critical growth where ∂j(u) denotes its subdifferential. We apply the Ekeland Variational Principle to an associated locally Lipschitz energy functional. A major point in our study is that in order to deal with the obtained Ekeland sequence we developed a generalized version for the framework of Orlicz–Sobolev spaces of a well-known Brezis–Lieb lemma which was employed together with a variant of the Lions concentration-compactness theory to get a solution of the equation.
TL;DR: In this paper, the inequality for Henneaux-Teitelboim's total energy momentum for asymptotically anti-de Sitter initial data sets was established.
Abstract: We establish the inequality for Henneaux–Teitelboim's total energy–momentum for asymptotically anti-de Sitter initial data sets which are asymptotic to arbitrary t-slice in anti-de Sitter spacetime...
TL;DR: In this article, the Ricci curvature bound of the boundary of an n-dimensional Riemannian manifold with boundary ∂M is shown to be bounded from below by (n − 1)k for k ∈ ℝ.
Abstract: Let Mn be an n-dimensional Riemannian manifold with boundary ∂M. Assuming that Ricci curvature is bounded from below by (n - 1)k, for k ∈ ℝ, we give a sharp estimate of the upper bound of ρ(x) = d(x, ∂M), in terms of the mean curvature bound of the boundary. When ∂M is compact, the upper bound is achieved if and only if M is isometric to a disk in space form. A Kahler version of estimation is also proved. Moreover, we prove a Laplacian comparison theorem for distance function to the boundary of Kahler manifold and also estimate the first eigenvalue of the real Laplacian.
TL;DR: In this article, the existence and uniqueness of the solutions for linear and nonlinear parabolic equations with time-dependent coefficients was studied in the class of bounded solutions satisfying appropriate conditions at infinity.
Abstract: We are concerned with existence and uniqueness of the solutions for linear and nonlinear parabolic equations with time-dependent coefficients, in the class of bounded solutions satisfying appropriate conditions at infinity.
TL;DR: In this paper, the global phase portraits in the Poincare disc of planar quadratic polynomial systems which admit invariant straight lines with total multiplicity two and Darboux invariants are presented.
Abstract: In this paper, we present the global phase portraits in the Poincare disc of the planar quadratic polynomial systems which admit invariant straight lines with total multiplicity two and Darboux invariants.
TL;DR: In this paper, the authors studied a class of timelike weakly extremal surfaces in flat Minkowski space ℝ1+n, characterized by the fact that they admit a C 1 parametrization (in general not an immersion) of a specific form.
Abstract: We study a class of timelike weakly extremal surfaces in flat Minkowski space ℝ1+n, characterized by the fact that they admit a C1 parametrization (in general not an immersion) of a specific form. We prove that if the distinguished parametrization is of class Ck, then the surface is regularly immersed away from a closed singular set of Euclidean Hausdorff dimension at most 1 + 1/k, and that this bound is sharp. We also show that, generically with respect to a natural topology, the singular set of a timelike weakly extremal cylinder in ℝ1+n is one-dimensional if n = 2, and it is empty if n ≥ 4. For n = 3, timelike weakly extremal surfaces exhibit an intermediate behavior.