In this note, we study the maximal perimeter of a convex set in ℝn with respect to various classes of measures. Firstly, we show that for a probability measure μ on ℝn, satisfying very mild assumptions, there exists a convex set of μ-perimeter at least CnVar|X|4𝔼|X|. This implies, in particular, that for any isotropic log-concave measure μ, one may find a convex set of μ-perimeter of order n18. Secondly

In this work, we study the following problem (𝒫)−Δu=λg(x)u+12∗f(x)Hu(u,v),inℝN,−Δv=μh(x)v+12∗f(x)Hv(u,v),inℝN, where N≥4, 2∗:=2N/(N−2), Hu and Hv are the partial derivatives of the homogeneous function H∈C1(ℝ+2,ℝ), where ℝ+2=[0,∞)×[0,∞), g+≢0, h+≢0, f+≢0, g,h,f∈L∞(ℝN), g+,h+∈LN/2(ℝN) and λ,μ>0.

In this paper, we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form κ(𝒰)≤K1log1δK2. This is done in a rather general setting, i.e. for the δ-complement of a polynomial zero-level hypersurface Y0 and for the regular level hypersurfaces Yc themselves with no assumptions

In this paper, we study the asymptotic behavior as 𝜀→0+ of solutions u𝜀 to the nonlocal stationary Fisher-KPP type equation 1𝜀m∫ℝNJ𝜀(x−y)(u𝜀(y)−u𝜀(x))dy+u𝜀(x)(a(x)−u𝜀(x))=0 in ℝN, where 𝜀>0 and 0≤m<2. Under rather mild assumptions and using very little technology, we prove that there exists one and only one positive solution u𝜀 and that u𝜀→a+ as 𝜀→0+ where a+=max{0,a}. This generalizes

This paper analyzes the superlinear indefinite prescribed mean curvature problem −div∇u/1+|∇u|2=λa(x)h(u)in Ω,u=0on ∂Ω, where Ω is a bounded domain in ℝN with a regular boundary ∂Ω, h∈C0(ℝ) satisfies h(s)∼sp, as s→0+, p>1 being an exponent with p0 represents a parameter, and a∈C0(Ω¯) is a sign-changing function. The main result establishes the existence of positive regular solutions when λ is sufficiently

In this work, we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms of the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the

For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions if some blowup points coincide with the singularities of the Dirac data. If the strength of the Dirac mass at each blowup point is not a multiple of 4π, we prove that bubbling solutions are unique. This paper extends previous results of Lin-Yan [C. S. Lin and S. S. Yan, On the mean

The Chow rings of hyperKähler varieties are conjectured to have a particularly rich structure. In this paper, we focus on the locally complete family of double EPW sextics and establish some properties of their Chow rings. First, we prove a Beauville–Voisin type theorem for zero-cycles on double EPW sextics; precisely, we show that the codimension-4 part of the subring of the Chow ring of a double

This paper is a work in progress on Bloch’s conjecture asserting the vanishing of the Pontryagin product of a p codimensional cycle on an abelian variety by p+1 zero cycles of degree zero. We prove an infinitesimal version of the conjecture and we discuss in particular, the case of 3-dimensional cycles.

We investigate how the motive of hyper-Kähler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the

In this paper, by Morse theory we will study the Kirchhoff type equation with an additional critical nonlinear term, and the main results are to compute the critical groups including the cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. As an application, the multiplicity of nontrivial solutions for this equation with the parameter across the first eigenvalue is

The Kubo–Martin–Schwinger (KMS) condition is a widely studied fundamental property in quantum statistical mechanics which characterizes the thermal equilibrium states of quantum systems. In the seventies, Gallavotti and Verboven, proposed an analogue to the KMS condition for infinite classical mechanical systems and highlighted its relationship with the Kirkwood–Salzburg equations and with the Gibbs

The Brunn–Minkowski and Prékopa–Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions so it seems that convexity has no special signficance. On the other hand, it was recently shown that the Brunn–Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance

We study some qualitative properties of ancient solutions of superlinear heat equations on a Riemannian manifold, with particular interest in positivity and constancy in space.

In this paper, we introduce the notion of pseudoheaviness of closed subsets of closed symplectic manifolds and prove the existence of pseudoheavy fibers of moment maps. In particular, we generalize Entov and Polterovich’s theorem, which ensures the existence of non-displaceable fibers. As its application, we provide a partial answer to a problem posed by them, which asks the existence of heavy fibers

In this paper, we study totally geodesic subvarieties Y⊂Ag of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for g≥4. We prove that if Y is generically contained in the Torelli locus, then dimY≤(7g−2)/3.

A key basis for seeking solutions of the Camassa–Holm equation is to understand the associated spectral problem y″=14y+λm(t)y. We will study in this paper the optimal lower bound of the smallest eigenvalue for the Camassa–Holm equation with the Neumann boundary condition when the L1 norm of potentials is given. First, we will study the optimal lower bound for the smallest eigenvalue in the measure

This paper shows that the non-expansive transport map constructed by Y.-H. Kim and E. Milman using heat flow interpolation is in general different from the optimal transport Brenier map.

We prove local boundedness of local minimizers of scalar integral functionals ∫Ωf(x,∇u(x))dx, Ω⊂ℝn where the integrand satisfies (p,q)-growth of the form |z|p≲f(x,z)≲|z|q+1 under the optimal relation 1p−1q≤1n−1.

We consider the buckling eigenvalue problem for a clamped plate in the annulus. We identify the first eigenvalue in dependence of the inner radius, and study the number of nodal domains of the corresponding eigenfunctions. Moreover, in order to investigate the asymptotic behavior of eigenvalues and eigenfunctions as the inner radius approaches the outer one, we provide an analytical study of the buckling

From a finite set in a lattice, we can define a toric variety embedded in a projective space. In this paper, we give a combinatorial description of the dual defect of the toric variety using the structure of the finite set as a Cayley sum with suitable conditions. We also interpret the description geometrically.

Hecke–Kiselman monoids HKΘ and their algebras K[HKΘ], over a field K, associated to finite oriented graphs Θ are studied. In the case Θ is a cycle of length n≥3, a hierarchy of certain unexpected structures of matrix type is discovered within the monoid Cn=HKΘ and this hierarchy is used to describe the structure and the properties of the algebra K[Cn]. In particular, it is shown that K[Cn] is a right

In this paper, we study various forms of the Hardy inequality for Dunkl operators, including the classical inequality, Lp inequalities, an improved Hardy inequality, as well as the Rellich inequality and a special case of the Caffarelli–Kohn–Nirenberg inequality. As a consequence, one-dimensional many-particle Hardy inequalities for generalized root systems are proved, which in the particular case

The focus of the paper is the behavior under iterations of the filtered and local Floer homology of a Hamiltonian on a symplectically aspherical manifold. The Floer homology of an iterated Hamiltonian comes with a natural cyclic group action. In the filtered case, we show that the supertrace of a generator of this action is equal to the Euler characteristic of the homology of the un-iterated Hamiltonian

For d≥4, the Noether–Lefschetz locus NLd parametrizes smooth, degree d surfaces in ℙ3 with Picard number at least 2. A conjecture of Harris states that there are only finitely many irreducible components of the Noether–Lefschetz locus of non-maximal codimension. Voisin showed that the conjecture is false for sufficiently large d, but is true for d≤5. She also showed that for d=6,7, there are finitely

We prove the existence of ground state solution for the nonlocal problem m∫ℝ2(|∇u|2+b(x)u2)dx(−Δu+b(x)u)=A(x)f(u)in ℝ2, where m is a Kirchhoff type function, b may be negative and noncoercive, A is locally bounded and the function f has critical exponential growth. We also obtain new results for the classical Schrödinger equation, namely the local case m≡1. In the proofs, we apply Variational Methods

Frank and Lieb gave a new, rearrangement-free, proof of the sharp Hardy–Littlewood–Sobolev inequalities by exploiting their conformal covariance. Using this they gave new proofs of sharp Sobolev inequalities for the embeddings Wk,2(ℝn)↪L2nn−2k(ℝn). We show that their argument gives a direct proof of the latter inequalities without passing through Hardy–Littlewood–Sobolev inequalities, and, moreover

This paper is to prove global regularity estimates for solutions to the second-order elliptic equation in non-divergence form with BMO coefficients in a C1,1 domain on weighted variable exponent Lebesgue spaces. Our approach is based on the representations for the solutions to the non-divergence elliptic equations and the domination technique by sparse operators in harmonic analysis.

Let (M,g) be a Riemannian manifold with finite volume and U be a linear topological space. We consider the strongly indefinite superlinear problem Au=|u|p−2u+∂∂uψ(x,u),u∈H, where A:L2(M,U)→L2(M,U) is a self-adjoint linear operator, H is a real Hilbert space with the compact embedding H↪Lp(M,U) if p∈[2,q) for some q>2, and (∂∂uψ(x,u),u)≤C(|u|2+|u|p). We obtain the existence of two solutions provided

The Wills functional 𝒲(K) of a convex body K, defined as the sum of its intrinsic volumes Vi(K), turns out to have many interesting applications and properties. In this paper, we make profit of the fact that it can be represented as the integral of a log-concave function, which, furthermore, is the Asplund product of other two log-concave functions, and obtain new properties of the Wills functional

We consider the inverse problem of determining the coupling coefficients in a two-state Schrödinger system. We prove a Lipschitz stability inequality for the zeroth- and first-order coupling terms by finitely many partial lateral measurements of the solution to the coupled Schrödinger equations.

Many earlier works were devoted to the study of the breakdown of superconductivity in type-II superconducting bounded planar domains, submitted to smooth magnetic fields. In the present contribution, we consider a new situation where the applied magnetic field is piecewise-constant, and the discontinuity jump occurs along a smooth curve meeting the boundary transversely. To handle this situation, we

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: (−Δ)α2u−γu|x|α=λf(x)|u|q−2u+g(x)|u|p−2u|x|sin Ω,u=0in ℝn∖Ω, where Ω⊂ℝn is a smooth bounded domain in ℝn containing 0 in its interior, and f,g∈C(Ω¯) with f+,g+≢0 which may change sign in Ω¯. We use the variational methods and the Nehari

We construct a colored operad whose category of algebras is the category of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped with an additional structure that we call an orthogonality relation. This allows us to describe different types of quantum field theories, including theories on a fixed

This paper studies the Cauchy problem of radial inhomogeneous Schrödinger maps (ISM) which arises from the integrable model of the inhomogeneous spherically symmetric Heisenberg ferromagnetic spin system. Through a complex transformation the radial ISM is equivalent to an integro-differential Schrödinger equation. A new weighted Sobolev space 𝒲β1,l(ℝ+) is introduced and the well-posedness of integro-differential

This paper is a fundamental study of the Real 2-representation theory of 2-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a 2-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real 2-representations as a Real variant of the Burnside ring of the fundamental group of the 2-group

In this paper, we are concerned with the existence of multi-bump solutions for the following class of p(x)-Laplacian equations: −div(|∇u|p(x)−2∇u)+(λV(x)+Z(x))|u|p(x)−2u=αf(x,u)+uq(x)−1,in ℝN,u∈W1,p(x)(ℝN),u>0, where α>0 and λ≥1 are two real parameters, the nonlinearity f:ℝN×ℝ→ℝ is a continuous function with subcritical growth, N>p+=supx∈ℝNp(x), the exponent q(x) can be equal to the critical exponent

We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the Laplace equation with the same dynamical boundary condition.

In this paper, we study the horizontal Newton transformations, which are nonlinear operators related to the natural splitting of the second fundamental form for hypersurfaces in a complex space form. These operators allow to prove the classical Minkowski formulas in the case of real space forms: unlike the real case, the horizontal ones are not divergence-free. Here, we consider the highest order of

Two observations in support of the thesis that trusses are inherent in ring theory are made. First, it is shown that every equivalence class of a congruence relation on a ring or, equivalently, any element of the quotient of a ring R by an ideal I is a paragon in the truss T(R) associated to R. Second, an extension of a truss by a one-sided module is described. Even if the extended truss is associated

We consider a parametric double phase Dirichlet problem. Using variational tools together with suitable truncation and comparison techniques, we show that for all parametric values λ>λ∗ the problem has at least three nontrivial solutions, two of which have constant sign. Also, we identify the critical parameter λ∗ precisely in terms of the spectrum of the q-Laplacian.

In this paper, we give a detailed proof of a result due to Torsten Ekedahl, describing complex tori admitting a rigid group action and showing explicitly their projectivity and their structure in terms of CM-fields. In the appendix, joint with Claudon, we show, using a method of Green-Voisin, that all group actions on complex tori deform to projective ones.

We explore ways in which the covariance ellipsoid ℬ={v∈ℝd:𝔼〈X,v〉2≤1} of a centered random vector X in ℝd can be approximated by a simple set. The data one is given for constructing the approximating set is X1,…,XN that are independent and distributed as X. We present a general method that can be used to construct such approximations and implement it for two types of approximating sets. We first construct

In this paper, we study several aspects related with solutions of nonlocal problems whose prototype is ut=∫ℝNJ(x−y)(u(y,t)−u(x,t))𝒢(u(y,t)−u(x,t))dyin Ω×(0,T),u(x,0)=u0(x)in Ω, where we take, as the most important instance, 𝒢(s)∼1+μ2s1+μ2s2 with μ∈ℝ as well as u0∈L1(Ω), J is a smooth symmetric function with compact support and Ω is either a bounded smooth subset of ℝN, with nonlocal Dirichlet boundary

We use techniques of proof mining to extract a uniform rate of metastability (in the sense of Tao) for the strong convergence of approximants to fixed points of uniformly continuous pseudocontractive mappings in Banach spaces which are uniformly convex and uniformly smooth, i.e. a slightly restricted form of the classical result of Reich. This is made possible by the existence of a modulus of uniqueness

In recent years, the study of the bienergy functional has attracted the attention of a large community of researchers, but there are not many examples where the second variation of this functional has been thoroughly studied. We shall focus on this problem and, in particular, we shall compute the exact index and nullity of some known examples of proper biharmonic maps. Moreover, we shall analyze a

This paper deals with an analog of the Mahler volume product related to the 𝒥 transform acting in the class of geometric convex functions Cvx0(ℝn). We provide asymptotically sharp bounds for the quantity sJ(f)=∫e−𝒥f∫e−f and characterize all the extremal functions.

We present a necessary and sufficient condition for a random product of maps on a compact metric space to be (strongly) synchronizing on average.

We consider the family of convex bodies obtained from an origin symmetric convex body K by multiplication with diagonal matrices, by forming Minkowski sums of the transformed sets, and by taking limits in the Hausdorff metric. Support functions of these convex bodies arise by an integral transform of measures on the family of diagonal matrices, equivalently, on Euclidean space, which we call K-transform

In this paper, we will consider 4-dimensional manifolds with nonnegative scalar curvature and constant mean curvature (CMC) boundary. For compact manifolds with boundary, the influence of the nonnegativity of the region scalar curvature to the geometry of the boundary is considered. Some inequalities are established for manifolds with inner boundary and outer boundary. Even for compact manifolds without

In this paper, we obtain gradient continuity estimates for viscosity solutions of ΔpNu=f in terms of the scaling critical L(n,1)-norm of f, where ΔpN is the normalized p-Laplacian operator. Our main result corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potential 𝕀̃qf. Moreover, for f∈Lm with m>n, we also obtain C1,α estimates. This improves one of the regularity

We consider the following quasilinear Schrödinger equation −𝜀2Δw+V(x)w−𝜀2Δ(w2)w=h(w),w>0,x∈ℝN, where N≥4, liminf|x|→∞V(x)>infx∈ℝNV(x)=0, and h satisfies a weaker growth condition than the Ambrosetti–Rabinowitz type condition in Byeon and Wang [Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal.165(4) (2002) 295–316; Standing waves with a critical

We consider the eigenvalue problem for the restricted fractional Laplacian in a bounded domain with homogeneous Dirichlet boundary conditions. We introduce the notion of fractional capacity for compact subsets, with the property that the eigenvalues are not affected by the removal of zero fractional capacity sets. Given a simple eigenvalue, we remove from the domain a family of compact sets which are

In this paper, we consider the Cauchy–Ventcel problem in an inhomogeneous medium with dynamic boundary conditions subject to a nonlinear damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. Uniform decay rates of the associated energy are established and, in addition, the exact internal controllability for the linear problem is also proved. For this

In this paper, we prove three fundamental types of limit theorems for the q-norm of random vectors chosen at random in an ℓpn-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guédon, Mendelson, and Naor. It includes the normalized

We consider a partially hinged rectangular plate and its normal modes. The dynamical properties of the plate are influenced by the spectrum of the associated eigenvalue problem. In order to improve the stability of the plate, we place a certain amount of denser material in appropriate regions. If we look at the partial differential equation appearing in the model, this corresponds to insert a suitable

The aim of this paper is to investigate the existence of one or more critical points of a family of functionals which generalizes the model problem J̄(u)=1p∫ΩĀ(x,u)|∇u|pdx−∫ΩG(x,u)dx in the Banach space X=W01,p(Ω)∩L∞(Ω), where Ω⊂ℝN is an open bounded domain, 1

We show that automorphic Lie algebras which contain a Cartan subalgebra with a constant-spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianization of the

We study the Γ-convergence of a family of non-local, non-convex functionals in Lp(I) for p≥1, where I is an open interval. We show that the limit is a multiple of the W1,p(I) semi-norm to the power p when p>1 (respectively, the BV(I) semi-norm when p=1). In dimension one, this extends earlier results which required a monotonicity condition.

We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3[n]-type and of generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.