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Abstract: Abstract Given a probability measure space \((X,\Sigma ,\mu )\) , it is well known that the Riesz space \(L^0(\mu )\) of equivalence classes of measurable functions \(f: X \rightarrow \mathbf {R}\) is universally complete and the constant function \(\varvec{1}\) is a weak order unit. Moreover, the linear functional \(L^\infty (\mu )\rightarrow \mathbf {R}\) defined by \(f \mapsto \int f\,\mathrm {d}\mu \) is strictly positive and order continuous. Here we show, in particular, that the converse holds true, i.e., any universally complete Riesz space E with a weak order unit \(e>0\) which admits a strictly positive order continuous linear functional on the principal ideal generated by e is lattice isomorphic onto \(L^0(\mu )\) , for some probability measure space \((X,\Sigma ,\mu )\) . PubDate: 2022-05-21

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Abstract: Abstract This paper concerns semilinear elliptic equations involving sign-changing weight function and a nonlinearity of subcritical nature understood in a generalized sense. Using an Orlicz–Sobolev space setting, we consider superlinear nonlinearities which do not have a polynomial growth, and state sufficient conditions guaranteeing the Palais–Smale condition. We study the existence of a bifurcated branch of classical positive solutions, containing a turning point, and providing multiplicity of solutions. PubDate: 2022-04-25

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Abstract: Abstract In this paper we introduce some variation functions associated to the rank of the Infinitesimal Variations of Hodge Structure for a family of smooth projective complex curves. We give some bounds and inequalities and, in particular, we prove that if X is a smooth plane curve, then, there exists a first order deformation \(\xi \in H^1(T_X)\) which deforms X as plane curve and such that \(\xi \cdot :H^0(\omega _X)\rightarrow H^1({\mathcal O}_{X})\) is an isomorphism. We also generalize the notions of variation functions to higher dimensional case and we analyze the link between IVHS and the weak and strong Lefschetz properties of the Jacobian ring of a smooth hypersurface. PubDate: 2022-04-25

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Abstract: Abstract We present and discuss connections between the problem of trend to equilibrium for one-dimensional Fokker–Planck equations modeling socio-economic problems, and one-dimensional functional inequalities of the type of Poincaré, Wirtinger and logarithmic Sobolev, with weight, for probability densities with polynomial tails. As main examples, we consider inequalities satisfied by inverse Gamma densities, taking values on \(\mathbb R_+\) , and Cauchy-type densities, taking values on \(\mathbb R\) . PubDate: 2022-03-28

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Abstract: Abstract Inspired by the modelization of 2D materials systems, we characterize arrangements of identical nonflat squares in 3D. We prove that the fine geometry of such arrangements is completely characterized in terms of patterns of mutual orientations of the squares and that these patterns are periodic and one-dimensional. In contrast to the flat case, the nonflatness of the tiles gives rise to nontrivial geometries, with configurations bending, wrinkling, or even rolling up in one direction. PubDate: 2022-03-24

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Abstract: Abstract The aim of this paper is to study the following time-space fractional diffusion problem $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \partial _t^\beta u+(-\Delta )^\alpha u+(-\Delta )^\alpha \partial _t^\beta u=\lambda f(x,u) +g(x,t) &{}\text{ in } \Omega \times {\mathbb {R}}^{+},\\ u(x,t)=0\ \ &{}\text{ in } ({\mathbb {R}}^N{\setminus }\Omega )\times {\mathbb {R}}^+,\\ u(x,0)=u_0(x)\ &{}\text{ in } \Omega ,\\ \end{array}\right. } \end{aligned}$$ where \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain with Lipschitz boundary, \((-\Delta )^{\alpha }\) is the fractional Laplace operator with \(0<\alpha <1\) , \(\partial _t^{\beta }\) is the Riemann-Liouville time fractional derivative with \(0<\beta <1\) , \(\lambda \) is a positive parameter, \(f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function, and \(g\in L^2(0,\infty ;L^2(\Omega ))\) . Under natural assumptions, the global and local existence of solutions are obtained by applying the Galerkin method. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions. Moreover, the blow-up property of solutions is also investigated. PubDate: 2022-03-22

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Abstract: Abstract We will consider the multidimensional truncated \(p \times p\) Hermitian matrix-valued moment problem. We will prove a characterisation of truncated \(p \times p\) Hermitian matrix-valued multisequence with a minimal positive semidefinite matrix-valued representing measure via the existence of a flat extension, i.e., a rank preserving extension of a multivariate Hankel matrix (built from the given truncated matrix-valued multisequence). Moreover, the support of the representing measure can be computed via the intersecting zeros of the determinants of matrix-valued polynomials which describe the flat extension. We will also use a matricial generalisation of Tchakaloff’s theorem due to the first author together with the above result to prove a characterisation of truncated matrix-valued multisequences which have a representing measure. When \(p = 1\) , our result recovers the celebrated flat extension theorem of Curto and Fialkow. The bivariate quadratic matrix-valued problem and the bivariate cubic matrix-valued problem are explored in detail. PubDate: 2022-03-19

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Abstract: Abstract We review the theory of Lagrangian fibrations of hyperkähler manifolds as initiated by Matsushita. We also discuss more recent work of Shen–Yin and Harder–Li–Shen–Yin. Occasionally, we give alternative arguments and complement the discussion by additional observations. PubDate: 2022-03-19

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Abstract: Abstract For nonautonomous differential equations depending on a parameter, we show that the normal form inherits the regularity on the parameter of the original equation, provided that the nonresonances allow a certain spectral gap. PubDate: 2022-01-16 DOI: 10.1007/s00032-021-00347-6

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Abstract: Abstract In this paper we deal with the existence of solutions for the following class of magnetic semilinear Schrödinger equation $$\begin{aligned} (P) \qquad \qquad \left\{ \begin{aligned}&(-i\nabla + A(x))^2u +u = u ^{p-2}u,\;\;\text{ in }\;\;\Omega ,\\&u=0\;\;\text{ on }\;\;\partial \Omega , \end{aligned} \right. \end{aligned}$$ where \(N \ge 3\) , \(\Omega \subset {\mathbb {R}}^N\) is an exterior domain, \(p\in (2, 2^*)\) with \(2^*=\frac{2N}{N-2}\) , and \(A: {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N\) is a continuous vector potential verifying \(A(x) \rightarrow 0\;\;\text{ as }\;\; x \rightarrow \infty .\) PubDate: 2021-12-07 DOI: 10.1007/s00032-021-00340-z

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Abstract: Abstract We study the behaviour of the solutions to a dynamic evolution problem for a viscoelastic model with long memory, when the rate of change of the data tends to zero. We prove that a suitably rescaled version of the solutions converges to the solution of the corresponding stationary problem. PubDate: 2021-11-30 DOI: 10.1007/s00032-021-00343-w

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Abstract: Abstract We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of those differential equations. This brings to the center of the analysis several classical results from algebraic geometry, including the Cayley-Bacharach theorem and some of its variants as Serret’s theorem, and the Brill-Noether Restsatz theorem. PubDate: 2021-11-22 DOI: 10.1007/s00032-021-00336-9

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Abstract: Abstract The class of Divergence-free symmetric tensors is ubiquitous in Continuum Mechanics. We show its invariance under projective transformations of the independent variables. This action, which preserves the positiveness, extends Sophus Lie’s group analysis of Newtonian dynamics. When applied to models of gas dynamics—such as Euler system or Boltzmann equation,—in combination with Compensated Integrability, this yields new dispersive estimates. The most accurate one is obtained for mono-atomic gases. Then the space-time integral of \(t\rho ^\frac{1}{d} p\) is bounded in terms of the total mass and moment of inertia alone. PubDate: 2021-11-20 DOI: 10.1007/s00032-021-00342-x

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Abstract: Abstract We consider linear elliptic systems whose prototype is 0.1 $$\begin{aligned} div \, \Lambda \left[ \,\exp (- x ) - \log x \,\right] I \, Du = div \, F + g \text { in}\, B. \end{aligned}$$ Here B denotes the unit ball of \(\mathbb {R}^n\) , for \(n > 2\) , centered in the origin, I is the identity matrix, F is a matrix in \(W^{1, 2}(B, \mathbb {R}^{n \times n})\) , g is a vector in \(L^2(B, \mathbb {R}^n)\) and \(\Lambda \) is a positive constant. Our result reads that the gradient of the solution \(u \in W_0^{1, 2}(B, \mathbb {R}^n)\) to Dirichlet problem for system (0.1) is weakly differentiable provided the constant \(\Lambda \) is not large enough. PubDate: 2021-11-16 DOI: 10.1007/s00032-021-00345-8

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Abstract: Abstract We give a new construction of sheaves on a relative site associated to a product \(X \times S\) where S plays the role of a parameter space, expanding the previous construction by the same authors, where the subanalytic structure on S was required. Here we let this last condition fall. In this way the construction becomes much easier to apply when the dimension of S is bigger than one. We also study the functorial properties of base change with respect to the parameter space. PubDate: 2021-11-05 DOI: 10.1007/s00032-021-00344-9

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Abstract: Interpolation inequalities play an essential role in analysis with fundamental consequences in mathematical physics, nonlinear partial differential equations (PDEs), Markov processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view. This paper aims at illustrating some of these recent aspect of entropy-entropy production inequalities, with applications to stability in Gagliardo–Nirenberg–Sobolev inequalities and symmetry results in Caffarelli–Kohn–Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed. PubDate: 2021-10-25 DOI: 10.1007/s00032-021-00341-y

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Abstract: Abstract We prove that small deformations of a projective variety of general type are also projective varieties of general type, with the same plurigenera. PubDate: 2021-10-16 DOI: 10.1007/s00032-021-00339-6

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Abstract: Abstract The paper studies the initial boundary value problem related to the dynamic evolution of an elastic beam interacting with a substrate through an elastic-breakable forcing term. This discontinuous interaction is aimed to model the phenomenon of attachment-detachment of the beam occurring in adhesion phenomena. We prove existence of solutions in energy space and exhibit various counterexamples to uniqueness. Furthermore we characterize some relevant features of the solutions, ruling the main effects of the nonlinearity due to the elastic-breakable term on the dynamical evolution, by proving the linearization property according to Gérard (J Funct Anal 141(1):60–98, 1996) and an asymptotic result pertaining the long time behavior. PubDate: 2021-10-03 DOI: 10.1007/s00032-021-00338-7

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Abstract: Abstract We consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity. PubDate: 2021-09-21 DOI: 10.1007/s00032-021-00335-w

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Abstract: Abstract In this paper, we study the following class of nonlinear equations: $$\begin{aligned} -\Delta u+V(x) u = \left[ x ^{-\mu }*(Q(x)F(u))\right] Q(x)f(u),\quad x\in \mathbb {R}^2, \end{aligned}$$ where V and Q are continuous potentials, which can be unbounded or vanishing at infintiy, f(s) is a continuous function, F(s) is the primitive of f(s), \(*\) is the convolution operation and \(0<\mu <2\) . Assuming that the nonlinearity f(s) has exponential critical growth, we establish the existence of ground state solutions by using variational methods. For this, we establish a version of the Trudinger–Moser inequality for our setting, which was necessary to obtain our main results. PubDate: 2021-08-04 DOI: 10.1007/s00032-021-00334-x