Showing papers in "Annali di Matematica Pura ed Applicata in 2020"

Removable sets in non-uniformly elliptic problems

Abstract: We analyze fine properties of solutions to quasilinear elliptic equations with double-phase structure and characterize, in the terms of intrinsic Hausdorff measures, the size of the removable sets for Holder continuous solutions.

Essentially isospectral transformations and their applications

Abstract: We define and study the properties of Darboux-type transformations between Sturm–Liouville problems with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of eigenvalues and norming constants, oscillation of eigenfunctions, regularized trace formulas, and inverse uniqueness and existence theorems.

Pretorsion theories, stable category and preordered sets

Abstract: We show that in the category of preordered sets there is a natural notion of pretorsion theory, in which the partially ordered sets are the torsion-free objects and the sets endowed with an equivalence relation are the torsion objects. Correspondingly, it is possible to construct a stable category factoring out the objects that are both torsion and torsion-free.

Logarithmic difference lemma in several complex variables and partial difference equations

Abstract: In this paper, we mainly propose improvements of the logarithmic difference lemma for meromorphic functions in several complex variables and then investigate meromorphic solutions of partial difference equations from the viewpoint of Nevanlinna theory.

The periodic version of the Da Prato–Grisvard theorem and applications to the bidomain equations with FitzHugh–Nagumo transport

Abstract: In this article, the periodic version of the classical Da Prato–Grisvard theorem on maximal $${{L}}^p$$ -regularity in real interpolation spaces is developed, as well as its extension to semilinear evolution equations. Applying this technique to the bidomain equations subject to ionic transport described by the models of FitzHugh–Nagumo, Aliev–Panfilov, or Rogers–McCulloch, it is proved that this set of equations admits a unique, strong T-periodic solution in a neighborhood of stable equilibrium points provided it is innervated by T-periodic forces.

Halfspaces minimise nonlocal perimeter: a proof via calibrations

Abstract: We consider a nonlocal functional $$J_K$$ that may be regarded as a nonlocal version of the total variation. More precisely, for any measurable function $$u:\mathbb {R}^d\rightarrow \mathbb {R}$$ , we define $$J_K(u)$$ as the integral of weighted differences of u. The weight is encoded by a positive kernel K, possibly singular in the origin. We study the minimisation of this energy under prescribed boundary conditions, and we introduce a notion of calibration suited for this nonlocal problem. Our first result shows that the existence of a calibration is a sufficient condition for a function to be a minimiser. As an application of this criterion, we prove that halfspaces are the unique minimisers of $$J_K$$ in a ball, provided they are admissible competitors. Finally, we outline how to exploit the optimality of hyperplanes to recover a $$\varGamma $$ -convergence result concerning the scaling limit of $$J_K$$ .

Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory

Abstract: For nonnegative even kernels K, we consider the K-nonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated K-nonlocal mean curvature equation in an open set $$\Omega \subset \mathbb {R}^n$$ , we built a calibration for the nonlocal perimeter inside $$\Omega \subset \mathbb {R}^n$$ . The calibrating functional is a nonlocal null-Lagrangian. As a consequence, we conclude the minimality in $$\Omega $$ of each leaf of the foliation. As an application, we prove the minimality of K-nonlocal minimal graphs and that they are the unique minimizers subject to their own exterior data. As a second application of the calibration, we give a simple proof of an important result from the seminal paper of Caffarelli, Roquejoffre, and Savin, stating that minimizers of the fractional perimeter are viscosity solutions.

Stratified periodic water waves with singular density gradients

Abstract: We consider Euler’s equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. For this problem we establish three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients. Based upon this equivalence we then construct two-dimensional symmetric periodic traveling waves that are monotone between each crest and trough. Our analysis uses, to a large extent, the availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis.

On a class of generalized solutions to equations describing incompressible viscous fluids

Abstract: We consider a class of viscous fluids with a general monotone dependence of the viscous stress on the symmetric velocity gradient. We introduce the concept of dissipative solution to the associated initial boundary value problem inspired by the measure-valued solutions for the inviscid (Euler) system. We show the existence as well as the weak–strong uniqueness property in the class of dissipative solutions. Finally, the dissipative solution enjoying certain extra regularity coincides with a strong solution of the same problem.

Existence and concentration of positive solutions for p -fractional Schrödinger equations

Abstract: We deal with the existence and concentration of positive solutions for the following p-fractional Schrodinger equation: $$\begin{aligned} \varepsilon ^{sp}(-\Delta )_{p}^{s}u+V(x)|u|^{p-2}u=f(u)+\gamma |u|^{p^{*}_{s}-2}u \quad \text{ in } \mathbb {R}^{N}, \end{aligned}$$where $$\varepsilon >0$$ is a small parameter, $$s\in (0, 1)$$, $$p\in (1, \infty )$$, $$N>sp$$, $$\gamma \in \{0, 1\}$$, $$p^{*}_{s}=\frac{Np}{N-sp}$$ is the fractional critical Sobolev exponent, $$(-\Delta )_{p}^{s}$$ is the fractional p-Laplacian operator, V is a continuous positive potential having a local minimum and f is a superlinear continuous function with subcritical growth. The main results are obtained by using penalization techniques and suitable variational arguments.

Flat morphisms of finite presentation are very flat

Abstract: Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring R, R-modules built from the rings of functions on principal affine open subschemes in $${{\,\mathrm{Spec}\,}}R$$ using ordinal-indexed filtrations and direct summands are called very flat. The related class of very flat quasi-coherent sheaves over a scheme is intermediate between the classes of locally free and flat sheaves, and has serious technical advantages over both. In this paper, we show that very flat modules and sheaves are ubiquitous in algebraic geometry: if S is a finitely presented commutative R-algebra which is flat as an R-module, then S is a very flat R-module. This proves a conjecture formulated in the February 2014 version of the first author’s long preprint on contraherent cosheaves (Positselski in Contraherent cosheaves, arXiv:1209.2995 [math.CT]). We also show that the (finite) very flatness property of a flat module satisfies descent with respect to commutative ring homomorphisms of finite presentation inducing surjective maps of the spectra.

Porosity and differentiability of Lipschitz maps from stratified groups to Banach homogeneous groups

Abstract: Let f be a Lipschitz map from a subset A of a stratified group to a Banach homogeneous group. We show that directional derivatives of f act as homogeneous homomorphisms at density points of A outside a $$\sigma $$-porous set. At all density points of A, we establish a pointwise characterization of differentiability in terms of directional derivatives. These results naturally lead us to an alternate proof of almost everywhere differentiability of Lipschitz maps from subsets of stratified groups to Banach homogeneous groups satisfying a suitably weakened Radon–Nikodym property.

Leray–Hirsch theorem and blow-up formula for Dolbeault cohomology

Abstract: We prove a theorem of Leray–Hirsch type and give an explicit blow-up formula for Dolbeault cohomology on (not necessarily compact) complex manifolds. We give applications to strongly q-complete manifolds and the $$\partial \bar{\partial }$$ -lemma.

Asymptotic behavior and existence of solutions for singular elliptic equations

Abstract: We study the asymptotic behavior, as $$\gamma $$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated with singular semilinear elliptic equations whose model is $$\begin{aligned} -\Delta u=\frac{f(x)}{u^\gamma }\,\text { in }\Omega , \end{aligned}$$ where $$\Omega $$ is an open, bounded subset of $${\mathbb {R}}^{N}$$ and f is a bounded function. We deal with the existence of a limit equation under two different assumptions on f: either strictly positive on every compactly contained subset of $$\Omega $$ or only nonnegative. Through this study, we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated with $$\begin{aligned} -\Delta v + \frac{| abla v|^2}{v} = f\,\text { in }\Omega . \end{aligned}$$

Fractional Laplacian, homogeneous Sobolev spaces and their realizations

Abstract: We study the fractional Laplacian and the homogeneous Sobolev spaces on $${{\mathbb {R}}}^d$$, by considering two definitions that are both considered classical. We compare these different definitions, and show how they are related by providing an explicit correspondence between these two spaces, and show that they admit the same representation. Along the way, we also prove some properties of the fractional Laplacian.

Weak formulation of elastodynamics in domains with growing cracks

Abstract: In this paper, we formulate and study the system of elastodynamics on domains with arbitrary growing cracks. This includes homogeneous Neumann conditions on the crack sets and mixed general Dirichlet–Neumann conditions on the boundary. The only assumptions on the crack sets are to be $$(n-1)$$ -rectifiable with finite surface measure, and increasing in the sense of set inclusions. In particular, they might be dense; hence, the weak formulation must fall outside the usual context of Sobolev spaces and Korn’s inequality. We prove existence of a solution for both the damped and undamped systems, while in the damped case we are also able to prove uniqueness and an energy balance.

Asymptotic expansions in a general system of decaying functions for solutions of the Navier–Stokes equations

Abstract: We study the long-time dynamics of the Navier–Stokes equations in the three-dimensional periodic domains with a body force decaying in time. We introduce appropriate systems of decaying functions and corresponding asymptotic expansions in those systems. We prove that if the force has a large-time asymptotic expansion in Gevrey–Sobolev spaces in such a general system, then any Leray–Hopf weak solution admits an asymptotic expansion of the same type. This expansion is uniquely determined by the force, and independent of the solutions. Various applications of the abstract results are provided which particularly include the previously obtained expansions for the solutions in case of power decay, as well as the new expansions in case of the logarithmic and iterated logarithmic decay.

A class of weighted Hardy inequalities and applications to evolution problems

Abstract: We state the following weighted Hardy inequality: $$\begin{aligned} c_{o, \mu }\int _{{\mathbb {R}}^N}\frac{\varphi ^2 }{|x|^2}\, \mathrm{d}\mu \le \int _{{\mathbb {R}}^N} | abla \varphi |^2 \, \mathrm{d}\mu + K \int _{\mathbb {R}^N}\varphi ^2 \, \mathrm{d}\mu \quad \forall \, \varphi \in H_\mu ^1, \end{aligned}$$in the context of the study of the Kolmogorov operators: $$\begin{aligned} Lu=\Delta u+\frac{ abla \mu }{\mu }\cdot abla u, \end{aligned}$$perturbed by inverse square potentials and of the related evolution problems. The function $$\mu $$ in the drift term is a probability density on $$\mathbb {R}^N$$. We prove the optimality of the constant $$c_{o, \mu }$$ and state existence and nonexistence results following the Cabre–Martel’s approach (Cabre and Martel in C R Acad Sci Paris 329 (11): 973–978, 1999) extended to Kolmogorov operators.

Fine properties of the curvature of arbitrary closed sets

Abstract: Given an arbitrary closed set A of $$ {\mathbf {R}}^{n} $$ , we establish the relation between the eigenvalues of the approximate differential of the spherical image map of A and the principal curvatures of A introduced by Hug–Last–Weil, thus extending a well-known relation for sets of positive reach by Federer and Zahle. Then, we provide for every $$ m = 1, \ldots , n-1 $$ an integral representation for the support measure $$ \mu _{m} $$ of A with respect to the m-dimensional Hausdorff measure. Moreover, a notion of second fundamental form $$Q_{A}$$ for an arbitrary closed set A is introduced so that the finite principal curvatures of A correspond to the eigenvalues of $$ Q_{A} $$ . Finally, we establish the relation between $$Q_A$$ and the approximate differential of order 2 for sets introduced in a previous work of the author, proving that in a certain sense the latter corresponds to the absolutely continuous part of $$Q_A$$ .

The tangential $k$-Cauchy-Fueter complexes and Hartogs' phenomenon over the right quaternionic Heisenberg group

Abstract: We construct the tangential k-Cauchy–Fueter complexes on the right quaternionic Heisenberg group, as the quaternionic counterpart of $$\overline{\partial }_b$$-complex on the Heisenberg group in the theory of several complex variables. We can use the $$L^2$$ estimate to solve the nonhomogeneous tangential k-Cauchy–Fueter equation under the compatibility condition over this group modulo a lattice. This solution has an important vanishing property when the group is higher dimensional. It allows us to prove the Hartogs’ extension phenomenon for k-CF functions, which are the quaternionic counterpart of CR functions.

Kodaira dimension of almost Kähler manifolds and curvature of the canonical connection

Abstract: The notion of Kodaira dimension has recently been extended to general almost complex manifolds. In this paper we focus on the Kodaira dimension of almost Kahler manifolds, providing an explicit computation for a family of almost Kahler threefolds on the differentiable manifold underlying a Nakamura manifold. We concentrate also on the link between Kodaira dimension and the curvature of the canonical connection of an almost Kahler manifold and show that in the previous example (and in another one obtained from a Kodaira surface) the Ricci curvature of the almost Kahler metric vanishes for all the members of the family.

Heat conservation for generalized Dirac Laplacians on manifolds with boundary

Abstract: We consider a notion of conservation for the heat semigroup associated with a generalized Dirac Laplacian acting on sections of a vector bundle over a noncompact manifold with a (possibly noncompact) boundary under mixed boundary conditions. Assuming that the geometry of the underlying manifold is controlled in a suitable way and imposing uniform lower bounds on the zero-order piece (Weitzenbock potential) of the Dirac Laplacian, and on the endomorphism defining the mixed boundary condition, we show that the corresponding conservation principle holds. A key ingredient in the proof is a domination property for the heat semigroup which follows from an extension to this setting of a Feynman–Kac formula recently proved by the author de Lima (Pac J Math 292(1):177–201, 2018) in the context of differential forms. When applied to the Hodge Laplacian acting on differential forms satisfying absolute boundary conditions, this extends previous results by Vesentini (Ann Math Pura Appl 182:1–19, 2003) and Masamune (Atti Accad Naz Lincei Rend Lincei Mat Appl 18(4):351–358, 2007) in the boundaryless case. Along the way, we also prove a vanishing result for $$L^2$$ harmonic sections in the broader context of generalized (not necessarily Dirac) Laplacians. These results are further illustrated with applications to the Dirac Laplacian acting on spinors and to the Jacobi operator acting on sections of the normal bundle of a free boundary minimal immersion.

Characterization of manifolds of constant curvature by spherical curves

Abstract: It is known that the so-called rotation minimizing (RM) frames allow for a simple and elegant characterization of geodesic spherical curves in Euclidean, hyperbolic, and spherical spaces through a certain linear equation involving the coefficients that dictate the RM frame motion (da Silva, da Silva in Mediterr J Math 15:70, 2018). Here, we shall prove the converse, i.e., we show that if all geodesic spherical curves on a Riemannian manifold are characterized by a certain linear equation, then all the geodesic spheres with a sufficiently small radius are totally umbilical, and consequently, the given manifold has constant sectional curvature. We also furnish two other characterizations in terms of (i) an inequality involving the mean curvature of a geodesic sphere and the curvature function of their curves and (ii) the vanishing of the total torsion of closed spherical curves in the case of three-dimensional manifolds. Finally, we also show that the same results are valid for semi-Riemannian manifolds of constant sectional curvature.

Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a subcontinuum

Abstract: Let $$\Omega \subset {\mathbb {R}}^{N}$$ ( $$N\ge 1$$ ) be a smooth bounded domain, $$a\in C({\overline{\Omega }})$$ a sign-changing function, and $$0\le q<1$$ . We investigate the Robin problem where $$\alpha \in [-\infty ,\infty )$$ and $$ u $$ is the unit outward normal to $$\partial \Omega $$ . Due to the lack of strong maximum principle structure, this problem may have dead core solutions. However, for a large class of weights a we recover a positivity property when q is close to 1, which considerably simplifies the structure of the solution set. Such property, combined with a bifurcation analysis and a suitable change of variables, enables us to show the following exactness result for these values of q: $$(P_{\alpha })$$ has exactly one nontrivial solution for $$\alpha \le 0$$ , exactly two nontrivial solutions for $$\alpha >0$$ small, and no such solution for $$\alpha >0$$ large. Assuming some further conditions on a, we show that these solutions lie in a subcontinuum. These results partially rely on (and extend) our previous work (Kaufmann et al. in J Differ Equ 263:4481–4502, 2017), where the cases $$\alpha =-\infty $$ (Dirichlet) and $$\alpha =0$$ (Neumann) have been considered. We also obtain some results for arbitrary $$q\in \left[ 0,1\right) $$ . Our approach combines mainly bifurcation techniques, the sub-supersolutions method, and a priori lower and upper bounds.

The classification of ERP G2-structures on Lie groups

Abstract: A complete classification of left-invariant closed $$G_2$$ -structures on Lie groups which are extremally Ricci pinched (i.e., $$ {d}\tau = \tfrac{1}{6}|\tau |^2\varphi + \tfrac{1}{6}*(\tau \wedge \tau )$$ ), up to equivalence and scaling, is obtained. There are five of them, they are defined on five different completely solvable Lie groups and the $$G_2$$ -structure is exact in all cases except one, given by the only example in which the Lie group is unimodular.

Improved multipolar Poincaré–Hardy inequalities on Cartan–Hadamard manifolds

Abstract: We prove a family of improved multipolar Poincare–Hardy inequalities on Cartan–Hadamard manifolds. For suitable configurations of poles, these inequalities yield an improved multipolar Hardy inequality and an improved multipolar Poincare inequality such that the critical unipolar singular mass is reached at any pole.

Stabilization in a two-dimensional attraction–repulsion Stokes system with consumption of chemoattractant

Abstract: This paper is concerned with the following attraction–repulsion Stokes system $$\begin{aligned} \left\{ \begin{array}{ll} n_t+u\cdot abla n=\Delta n-\chi abla \cdot (n abla c)+\xi abla \cdot (n abla v),&{}x\in \Omega ,~t>0,\\ c_t+u\cdot abla c=\Delta c-nc,&{}x\in \Omega ,~t>0,\\ v_t+u\cdot abla v=\Delta v-v+n,&{}x\in \Omega ,~t>0,\\ u_t+ abla P=\Delta u+n abla \phi ,&{}x\in \Omega ,~t>0,\\ abla \cdot u=0,&{}x\in \Omega ,~t>0, \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^2$$ is a general bounded domain with smooth boundary. It is shown that for any properly regular initial data, the above system with homogeneous Neumann boundary condition admits a unique classical solution which is globally bounded and particularly, if the uniform $$L^{\infty }$$ -module of n fulfills $$\begin{aligned} \Vert n\Vert _{L^{\infty }(\Omega \times (0,\infty ))}<\frac{2}{K_{\Omega }m\xi ^2} \end{aligned}$$ with $$m:=\int _{\Omega }n_0$$ and $$K_{\Omega }>0$$ only depending on $$\Omega ,$$ then $$\begin{aligned} \Vert n(\cdot ,t)-{\bar{n}}_0\Vert _{L^{\infty }(\Omega )} +\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\Omega )}+\Vert v(\cdot ,t) -{\bar{n}}_0\Vert _{W^{1,\infty }(\Omega )}+\Vert u(\cdot ,t)\Vert _{L^{\infty }(\Omega )}\rightarrow 0\quad \text {as}\quad t\rightarrow \infty , \end{aligned}$$ where $${\bar{n}}_0:=\frac{1}{|\Omega |}\int _{\Omega }n_0.$$

Powers of conjugacy classes in a finite group

Abstract: Part of this paper was written during the stay of C.Melchor at the University of Cambridge in autumn 2017, which was financially supported by the grant E-2017-02, Universitat Jaume I of Castellon. C. Melchor would like to thank R. Camina and the Department of Mathematics for their warm hospitality. A. Beltran, M.J. Felipe and C. Melchor are supported by Proyecto PGC2018-096872-B-100, Ministerio de Ciencia, Innovacion y Universidades.

Invariant translators of the solvable group

Abstract: We classify the translators to the mean curvature flow in the three-dimensional solvable group $$Sol_3$$ that are invariant under the action of a one-parameter group of isometries of the ambient space. In particular, we show that $$Sol_3$$ admits graphical translators defined on a half plane, in contrast with a rigidity result of Shahriyari (Geom Dedicata 175:57–64, 2015) for translators in the Euclidean space. Moreover, we exhibit some nonexistence results.

Wave-breaking phenomena and persistence properties for the nonlocal rotation-Camassa–Holm equation

Abstract: Consideration in the present paper is a mathematical model proposed as an equation of long-crested shallow water waves propagating in one direction with the effect of Earth’s rotation. This model equation is analogous to the Camassa–Holm approximation of the two-dimensional incompressible and irrotational Euler equations, and its solution corresponding to physically relevant initial perturbations is more accurate on a much longer timescale. The effects of the Coriolis force caused by the Earth rotation and nonlocal higher nonlinearities on the blow-up criteria and wave-breaking phenomena in the periodic setting are investigated. Moreover, working with moderate weight functions that are commonly used in time–frequency analysis, some persistence results to the equation are illustrated.