We consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan–Moore–Gibson–Thompson (JMGT) equation arising in acoustics as an alternative model to the well-known Kuznetsov equation. We show a local existence result in appropriate function spaces, and, using the energy method together with a bootstrap argument, we prove a global existence result for small data, without

In this paper, by means of minimax techniques involving Cerami sequences, we prove the existence of at least one pair of positive solutions for a Hamiltonian system of Schrödinger equations in ℝN with potentials vanishing at infinity and subcritical nonlinearities which are superlinear at the origin and at infinity. We establish new estimates to prove the boundedness of a Cerami sequence.

Let X⊆ℙN be a non-degenerate normal projective variety of codimension e and degree d with isolated ℚ-Gorenstein singularities. We prove that the Castelnuovo–Mumford regularity reg(𝒪X)≤d−e, as predicted by the Eisenbud–Goto regularity conjecture. Such a bound fails for general projective varieties by a recent result of McCullough–Peeva. The main techniques are Noma’s classification of non-degenerate

In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form −div(𝒜(x,∇u))=μin Ω,u=0on ∂Ω, where μ is a finite signed Radon measure in Ω, Ω⊂ℝn is a bounded domain such that its complement ℝn∖Ω is uniformly p-thick and 𝒜 is a Carathéodory vector-valued function satisfying growth and monotonicity

We prove that the singularities of the R-matrix R(k) of the minimal quantization of the adjoint representation of the Yangian Y(𝔤) of a finite dimensional simple Lie algebra 𝔤 are the opposite of the roots of the monic polynomial p(k) entering in the OPE expansions of quantum fields of conformal weight 3/2 of the universal minimal affine W-algebra at level k attached to 𝔤.

The pointwise gradient estimate for weak solution pairs to the stationary Stokes system with Dini-BMO coefficients is established via the Havin–Maz’ya–Wolff type nonlinear potential of the nonhomogeneous term. In addition, we present a pointwise bound for the weak solutions under no extra regularity assumption on the coefficients.

The classical Aleksandrov–Bakel’man–Pucci estimate (ABP estimate) for a second-order elliptic operator in nondivergence form is one of the fundamental tools for the bounds of subsolutions. Cabre improved the ABP estimate by replacing a constant factor, the diameter of a given domain, with a geometric character, which can be defined and finite for some unbounded domains. In the proof, Cabre used the

We construct Ionel–Parker’s proposed refinement of the standard relative Gromov–Witten invariants in terms of abelian covers of the symplectic divisor and discuss in what sense it gives rise to invariants. We use it to obtain some vanishing results for the standard relative Gromov–Witten invariants. In a separate paper, we describe to what extent this refinement sharpens the usual symplectic sum formula

We prove that every 2-Segal space is unital.

In this paper, by utilizing a newly established variational principle on convex sets, we provide an existence and multiplicity result for a class of semilinear elliptic problems defined on the whole ℝN with nonlinearities involving linear and superlinear terms. We shall impose no growth restriction on the nonlinear term, and consequently, our problem can be supercritical in the sense of the Sobolev

In this paper, we study solitary wave solutions of the Cauchy problem for Half-wave-Schrödinger equation in the plane. First, we show the existence and the orbital stability of the ground states. Second, we prove that given any speed v, traveling wave solutions exist and converge to the zero wave as the velocity tends to 1. Finally, we solve the Cauchy problem for initial data in Lx2Hys(ℝ2), with s>12

In his PhD thesis [G. Goosen, Oriented 123-tqfts via string-nets and state-sums, PhD thesis, Stellenbosch University, Stellenbosch (2018)], Goosen combined the string-net and the generators-and-relations formalisms for arbitrary once-extended 3-dimensional topological quantum field theories (TQFTs). In this paper, we work this out in detail for the simplest nontrivial example, where the underlying

In this paper, we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators (∂t−ℒ)s, 0

Let G be a stratified homogeneous group with homogeneous dimension Q and whose Lie algebra is generated by the left-invariant vector fields X1,…,Xd1. Let 10. We prove that for any function f∈Ḟqα,p(G) there exists a function F∈L∞(G)∩Ḟqα,p(G) such that ∑i=1k∥Xi(f−F)∥Ḟqα−1,p(G)≤δ∥f∥Ḟqα,p(G),∥F∥L∞(G)+∥F∥Ḟqα,p(G)≤Cδ∥f∥Ḟqα,p(G) where k is the largest integer smaller than min(p,d1) and Cδ is a positive constant

We prove the analyticity of smooth critical points for O’Hara’s knot energies ℰα,p, with p=1 and 2<α<3, subject to a fixed length constraint. This implies, together with the already established regularity results for O’Hara’s knot energies, that bounded energy critical points of ℰα,1 subject to a fixed length constraint are not only C∞ but also analytic. Our approach is based on Cauchy’s method of

This work concerns with polynomial families of real planar vector fields having a monodromic nilpotent singularity. The families considered are those for which the centers are characterized by the existence of a formal inverse integrating factor vanishing at the singularity with a leading term of minimum (1,n)-quasihomogeneous weighted degree, being n the Andreev number of the singularity. These families

We improve known upper bounds for the minimal dispersion of a point set in the unit cube and its inverse in both the periodic and non-periodic settings. Some of our bounds are sharp up to logarithmic factors.

We study spectral problems for integro-differential equations arising in the theory of Gaussian processes similar to the fractional Brownian motion. We generalize the method of Chigansky–Kleptsyna and obtain the two-term eigenvalue asymptotics for such equations. Application to the small ball probabilities in L2-norm is given.

We obtain a pair of nontrivial solutions for a class of concave-linear-convex type elliptic problems that are either critical or subcritical. The solutions we find are neither local minimizers nor of mountain pass type in general. They are higher critical points in the sense that they each have a higher critical group that is nontrivial. This fact is crucial for showing that our solutions are nontrivial

We discuss a version of the Chevalley–Eilenberg cohomology in characteristic 2, where the alternating cochains are replaced by symmetric ones.

In this paper, we investigate the multiplicity of positive solutions for a class of Schrödinger–Poisson systems with concave and convex nonlinearities as follows: −Δu+λV(x)u+μϕu=a(x)|u|p−2u+b(x)|u|q−2uin ℝ3,−Δϕ=u2in ℝ3, where λ,μ>0 are two parameters, 1

In this paper, we consider the Hénon problem in the ball with Dirichlet boundary conditions. We study the asymptotic profile of radial solutions and then deduce the exact computation of their Morse index when the exponent p is close to 1. Next we focus on the planar case and describe the asymptotic profile of some solutions which minimize the energy among functions which are invariant for reflection

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space H0m(Ω), where Ω is any bounded, smooth, open subset of ℝ2m, m≥1. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques

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

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.

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

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

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

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

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.

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