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2024-03-29 05:37:17
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Rendiconti Lincei - Matematica e Applicazioni
Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.
RLM
1120-6330
1720-0768
General
10.4171/RLM
http://www.ems-ph.org/doi/10.4171/RLM
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2006)
21
2010
1
Continuous dependence on the data for nonlinear elliptic equations via symmetrization
Maria Francesca
Betta
Università degli Studi di Napoli 'Parthenope', NAPOLI, ITALY
Anna
Mercaldo
Università degli Studi di Napoli “Federico II”, NAPOLI, ITALY
Rearrangements, nonlinear elliptic equations, continuous dependence on the data, uniqueness
We prove the continuous dependence on the data of weak solutions to Dirichlet problem for nonlinear elliptic equations with a first order term and datum in dual spaces of classical Sobolev spaces. We deduce uniqueness results.
Partial differential equations
General
1
14
10.4171/RLM/557
http://www.ems-ph.org/doi/10.4171/RLM/557
Optimal regularity results in spaces of Hölder continuous functions for some infinite dimensional Ornstein−Uhlenbeck semigroup
Giuseppe
Da Prato
Scuola Normale Superiore, PISA, ITALY
PDEs with infinitely many variables, Schauder estimates, Ornstein-Uhlenbeck semigroup
We consider the elliptic equation λ φ – Lφ = f where λ > 0, f is θ-Hölder continuous and L is an Ornstein−Uhlenbeck operator in a Hilbert space H. We show that the mapping D2φ (with values in the space of Hilbert−Schmidt operators on H) is θ-Hölder continuous.
Partial differential equations
General
15
31
10.4171/RLM/558
http://www.ems-ph.org/doi/10.4171/RLM/558
New classes of entire solutions for semilinear elliptic problems in ℝn
Andrea
Malchiodi
Scuola Normale Superiore, PISA, ITALY
Semilinear elliptic equations, entire solutions, Lyapunov-Schmidt reduction, weighted spaces
The goal of this paper is to describe some new results concerning entire solutions of semilinear elliptic equations in ℝn with non trivial asymptotic behavior at infinity. We describe in particular the (focusing, subcritical) nonlinear Schrödinger equation and the Allen-Cahn equation, which enjoy some common features but also present rather di¤erent aspects.
Partial differential equations
General
33
45
10.4171/RLM/559
http://www.ems-ph.org/doi/10.4171/RLM/559
Multiplicity of global minima for parametrized functions
Biagio
Ricceri
Università degli Studi di Catania, CATANIA, ITALY
Multiplicity, global minimum, parametric optimization, minimax inequality
Let X be a topological space, I a real interval and ψ a real-valued function on X x I. In this paper, we prove that if ψ is lower semicontinuous and inf-compact in X, quasiconcave and continuous in I and satisfies supI infX ψ < infX supI ψ, then there exists λ*∈ I such that ψ (∙,λ*) has at least two global minima. An application involving the integral functional of the calculus of variations is also presented.
Calculus of variations and optimal control; optimization
General
47
57
10.4171/RLM/560
http://www.ems-ph.org/doi/10.4171/RLM/560
The ‘‘ergodic limit’’ for a viscous Hamilton−Jacobi equation with Dirichlet conditions
Alessio
Porretta
Università di Roma, ROMA, ITALY
Ergodic limit, blow-up, viscous Hamilton–Jacobi equations
We study the limit, when λ tends to 0, of the solutions uλ of the Dirichlet problem -∆u + λu + |∇u|q = f(x) in Ω u = 0 on ∂ Ω, when 1 < q ≤ 2 and f is bounded. In case the limit problem does not have any solution, we prove that uλ has a complete blow-up (uλ → -∞) and its behaviour is described in terms of the corresponding ergodic problem with state constraint conditions. In particular, λuλ converges to the ergodic constant c0 and uλ + ||uλ||∞ converges to the boundary blow-up solution ν0 of the ergodic problem associated to the stochastic optimal control with state constraint.
Partial differential equations
General
59
78
10.4171/RLM/561
http://www.ems-ph.org/doi/10.4171/RLM/561
Recent results on the stability of time dependent sets and their application to bifurcation problems
Luigi
Salvadori
Università di Trento, TRENTO, ITALY
Francesca
Visentin
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Invariance, first integrals, stability properties of sets, bifurcation
In the first part of the paper we give a short review of our recent results concerning the relationship between conditional and unconditional stability properties of time dependent sets, under smooth differential systems in Rn. More precisely, let M be an ‘‘s-compact’’ invariant set in R x Rn and let Φ be a smooth invariant set in R x Rn containing M. It is assumed that M is uniformly asymptotically stable with respect to the perturbations lying on Φ. The unconditional stability properties of M depend on the stability properties of Φ ‘‘near M’’. This dependence has been analyzed in general, and, in the periodic case, complete characterizations are obtained. In the second part, the above results have been applied to bifurcation problems for periodic differential systems. Some our previous statements on the matter are revisited and enriched.
Ordinary differential equations
Mechanics of particles and systems
General
79
98
10.4171/RLM/562
http://www.ems-ph.org/doi/10.4171/RLM/562
Subelliptic Hamilton–Jacobi equations: the coercive stationary case
Marco
Biroli
Politecnico di Milano, MILANO, ITALY
Partial di¤erential equations in Carnot groups, viscosity solutions of Hamilton−Jacobi equations, regularity
We prove the existence uniqueness and comparison results for a (Lipschitz) viscosity solution for an Hamilton−Jacobi equation on a Carnot group.
Partial differential equations
Calculus of variations and optimal control; optimization
General
99
113
10.4171/RLM/563
http://www.ems-ph.org/doi/10.4171/RLM/563
2
Jenkins−Strebel differentials
Enrico
Arbarello
Università di Roma La Sapienza, ROMA, ITALY
Maurizio
Cornalba
Università di Pavia, PAVIA, ITALY
Quadratic differentials, closed trajectories, Teichmüller space
In this mostly expository paper we revisit a fundamental result of Strebel, asserting the existence and uniqueness, on Riemann surfaces of finite type, of Jenkins−Strebel differentials having double poles with prescribed ‘‘residues’’ at prescribed points. In particular, we give a selfcontained and somewhat shortened proof of Strebel’s result.
Functions of a complex variable
General
115
157
10.4171/RLM/564
http://www.ems-ph.org/doi/10.4171/RLM/564
A Stampacchia-type inequality for a fourth-order elliptic operator on Kähler manifolds and applications
Luca
Lussardi
Politecnico di Torino, TORINO, ITALY
Harmonic forms, Stampacchia-type inequality, Hodge-Kodaira decomposition, Aeppli groups
In this paper we will prove an integral inequality of Stampacchia-type for a fourth-order elliptic operator on complete and connected Kähler manifolds. Our inequality implies a Hodge–Kodaira orthogonal decomposition for the Sobolev-type space Wp,q(X). In particular we will able to prove, under suitable topological conditions on the manifold X, the existence of an isomorphism between the Aeppli groups Λp,q(X) and the groups Hp,q(X) of all global harmonic forms of bidegree (p,q).
Differential geometry
Algebraic geometry
General
159
173
10.4171/RLM/565
http://www.ems-ph.org/doi/10.4171/RLM/565
Regularity results for minimizers of integral functionals with nonstandard growth in Carnot–Carathéodory spaces
Flavia
Giannetti
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Antonia
Passarelli di Napoli
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Nonstandard growth conditions, Carnot–Carathéodory spaces, regularity
We prove regularity results for minimizers of integral functionals of the type ∫Ω f(Xu)dx where f satisfies a nonstandard growth condition and Xu stands for the horizontal gradient of u. More precisely, we obtain regularity in the scale of Campanato spaces without assuming any restriction on the growth exponents and, under a suitable assumption on them, we get the local boundedness as well as an higher integrability result for the gradient.
Calculus of variations and optimal control; optimization
General
175
192
10.4171/RLM/566
http://www.ems-ph.org/doi/10.4171/RLM/566
Remarks on the H Theorem for a non involutive Boltzmann like kinetic model
Giulia
Furioli
Università di Bergamo, DALMINE, ITALY
Elide
Terraneo
Università degli Studi di Milano, MILANO, ITALY
Conservative Boltzmann equations, asymptotic behavior of solutions, kinetic equations
In this paper, we consider a one-dimensional kinetic equation of Boltzmann type in which the binary collision process is described by the linear transformation v* = pv + qw, w* = qv + pw, where (v, w) are the pre-collisional velocities and (v*, w*) the post-collisional ones and p ≥ q > 0 are two positive parameters. This kind of model has been extensively studied by Pareschi and Toscani (in J. Stat. Phys., 124(2–4):747–779, 2006) with respect to the asymptotic behavior of the solutions in a Fourier metric. In the conservative case p2 + q2 = 1, even if the transformation has Jacobian J ≠ 1 and so it is not involutive, we remark that the H Theorem holds true. As a consequence we prove exponential convergence in L1 of the solution to the stationary state, which is the Maxwellian.
Fluid mechanics
Partial differential equations
Statistical mechanics, structure of matter
General
193
213
10.4171/RLM/567
http://www.ems-ph.org/doi/10.4171/RLM/567
Physical significance of the curvature varifold-based description of crack nucleation
Paolo Maria
Mariano
Universita di Firenze, FIRENZE, ITALY
Fracture mechanics, elastic-brittle materials, currents, curvature varifolds, classes of bodies, extended weak diffeomorphisms
The nucleation and/or growth of cracks in elastic-brittle solids has been recently described in [14] in terms of a special class of measures and with a variational technique requiring the minimization of a certain energy over classes of bodies. Here, the physical foundations of the theory and the basic ideas leading to it are described and commented further on. A view on certain possible developments and shifts toward different settings is also given. This article has expository character.
Mechanics of deformable solids
Calculus of variations and optimal control; optimization
General
215
233
10.4171/RLM/568
http://www.ems-ph.org/doi/10.4171/RLM/568
3
Symmetric group actions on the cohomology of configurations in ℝd
Giacomo
d'Antonio
Universität Bremen, BREMEN, GERMANY
Giovanni
Gaiffi
Università di Pisa, PISA, ITALY
Configuration spaces, symmetric group, representations
In this paper we deal with the action of the symmetric group on the cohomology of the con guration space Cn(d) of n points in ℝd. This topic has been studied by several authors and it is well-known that for d even H* (Cn(d);ℂ) ≌ 2IndSnS21 while, for d odd, H* (Cn(d);ℂ) ≌ ℂSn. On the cohomology algebra H* (Cn(d);ℂ) there is, in addition to the natural Sn-action, an extended action of Sn+1; this was shown for the case when d is even by Mathieu, Robinson and Whitehouse and the second author using three di erent methods. For the case when d is odd it was shown by Mathieu (anyway we will give an elementary algebraic construction of the extended action for this case). The purpose of this article is to present some results that can be obtained, in an elementary way, exploiting the interplay between the extended action and the standard action. Among these we will recall a quick proof for the formula cited above for the case when d is even and show how to extend this proof to the case when d is odd. We will also show how to locate among the homogeneous components of the graded algebra H* (Cn(d);ℂ) the copies of the standard, sign and standard tensor sign representations and we will give explicit formulas for both the extended and the canonical actions on the low-degree cohomology modules.
Group theory and generalizations
Algebraic topology
General
235
250
10.4171/RLM/569
http://www.ems-ph.org/doi/10.4171/RLM/569
A Note on Maurin’s Theorem
Enrico
Bombieri
Institute for Advanced Study, PRINCETON, UNITED STATES
P.
Habegger
ETH Zürich, ZÜRICH, SWITZERLAND
David
Masser
Universität Basel, BASEL, SWITZERLAND
Umberto
Zannier
Scuola Normale Superiore, PISA, ITALY
Diophantine geometry, multiplicative dependence, Zilber conjecture
We combine the strategy described in a paper of the first, third and fourth authors with a recent result of the second author to obtain a new proof of Maurin’s Theorem to the effect that the points satisfying two independent multiplicative relations on a fixed algebraic curve form a finite set when there is no natural obstacle.
Number theory
Algebraic geometry
General
251
260
10.4171/RLM/570
http://www.ems-ph.org/doi/10.4171/RLM/570
Holomorphic mappings associated to composition ideals of polynomials
Richard
Aron
Kent State University, KENT, UNITED STATES
Geraldo
Botelho
Universidade Federal de Uberlândia, UBERLANDIA, BRAZIL
Daniel Marinho
Pellegrino
Universidade Federal da Paraíba, JOÃO PESSOA, BRAZIL
Pilar
Rueda
Universitat de Valencia, BURJASSOT (VALENCIA), SPAIN
Holomorphic mappings, polynomial ideals
Many operator ideals ℐ can be naturally associated to polynomial ideals Q. In this paper we initiate a research program whose aim is to relate those holomorphic mappings f that admit factorizations f = u∘g, where u ∈ ℐ and g is holomorphic, with those f whose derivative belongs to the associated composition polynomial ideal Q = ℐ ∘ ℘.
Functional analysis
Operator theory
General
261
274
10.4171/RLM/571
http://www.ems-ph.org/doi/10.4171/RLM/571
La teoria dei perimetri di Caccioppoli–De Giorgi e i suoi più recenti sviluppi
Luigi
Ambrosio
Scuola Normale Superiore, PISA, ITALY
Sets of finite perimeter, Caccioppoli sets, currents
In this paper we illustrate the theory of sets of finite perimeter, starting from the pioneering intuitions of Caccioppoli and De Giorgi’s seminal papers. In the second part of the paper we illustrate some more recent developments and open problems in metric measure spaces, in Carnot groups and in infinite-dimensional Gaussian spaces.
Calculus of variations and optimal control; optimization
General
275
286
10.4171/RLM/572
http://www.ems-ph.org/doi/10.4171/RLM/572
Brody hyperbolicity and homotopy
Simone
Borghesi
Università degli Studi di Milano-Bicocca, MILANO, ITALY
Giuseppe
Tomassini
Scuola Normale Superiore, PISA, ITALY
Hyperbolic spaces, simplicial sheaves, homotopical algebra
The paper’s aim is to develop a theory in which the concept of Brody hyperbolicity of a complex space (cfr. [2]) is interpreted in terms of homotopy-theoretic structures. We contend that this interplay will be particularly useful if implemented by applying homotopy-theoretical techniques and constructions to get information on hyperbolic spaces. Imitating the construction of homotopy groups, we will define holotopy groups that will be able to tell apart different complex structures. From our point of view, the most important feature of these groups is that they vanish in a certain range if evaluated on a Brody hyperbolic complex space (see Theorem 4.1), providing therefore a way to reduce the proof of non hyperbolicity of a complex space to the existence of a nonzero holotopy class in these groups.
Several complex variables and analytic spaces
Category theory; homological algebra
General
287
297
10.4171/RLM/573
http://www.ems-ph.org/doi/10.4171/RLM/573
Mackey convergence and bornological topological modules
Nilson
Bernardes Jr.
Universidade Federal do Rio de Janeiro, RIO DE JANEIRO - RJ, BRAZIL
Dinamérico
Pombo Jr.
Universidade Federal Fluminense - UFF, NITERÓI - RJ, BRAZIL
Mackey convergence, topological modules, bornological topological modules
Functional analysis
General
299
304
10.4171/RLM/574
http://www.ems-ph.org/doi/10.4171/RLM/574
Peano on derivative of measures: strict derivative of distributive set functions
Gabriele
Greco
Università di Trento, POVO (TRENTO), ITALY
Sonia
Mazzucchi
Università di Trento, POVO (TRENTO), ITALY
Enrico
Pagani
Università di Trento, POVO (TRENTO), ITALY
Derivative of measures, strict derivative of set functions, distributive set functions, mass-density paradigm, Peano-Jordan measure
By retracing research on coexistent magnitudes (grandeurs coexistantes) by Cauchy [9, (1841)], Peano in Applicazioni geometriche del calcolo infinitesimale [48, (1887)] defines the "density" (strict derivative) of a "mass" (a distributive set function) with respect to a "volume" (a positive distributive set function), proves its continuity (whenever the strict derivative exists) and shows the validity of the mass-density paradigm: "mass" is recovered from "density" by integration with respect to "volume". It is remarkable that Peano’s strict derivative provides a consistent mathematical ground to the concept of "infinitesimal ratio" between two magnitudes, successfully used since Kepler. In this way the classical (i.e., pre-Lebesgue) measure theory reaches a complete and definitive form in Peano’s Applicazioni geometriche. A primary aim of the present paper is a detailed exposition of Peano’s work of 1887 leading to the concept of strict derivative of distributive set functions and their use. Moreover, we compare Peano’s work and Lebesgue’s La mesure des grandeurs [35, (1935)]: in this memoir Lebesgue, motivated by coexistent magnitudes of Cauchy, introduces a uniform-derivative of certain additive set functions, a concept that coincides with Peano’s strict derivative. Intriguing questions are whether Lebesgue was aware of the contributions of Peano and which role is played by the notions of strict derivative or of uniform-derivative in today mathematical practice.
History and biography
Measure and integration
General
305
339
10.4171/RLM/575
http://www.ems-ph.org/doi/10.4171/RLM/575
Equilibrium configurations of epitaxially strained thin films
Nicola
Fusco
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Free boundary problems, regularity, local minimality, second variation
We present some regularity results on equilibrium configurations for a variational model introduced to describe the epitaxial growth of an elastic film over a thick flat substrate when a lattice mismatch between the two materials is present. We also give a sufficient condition for local minimality based on second variation and apply it to determine analitycally the critical threshold for the local minimality of the flat configuration.
Mechanics of deformable solids
Calculus of variations and optimal control; optimization
General
341
348
10.4171/RLM/576
http://www.ems-ph.org/doi/10.4171/RLM/576
4
Variational methods for singular Liouville equations
Andrea
Malchiodi
Scuola Normale Superiore, PISA, ITALY
Geometric PDEs, variational methods, min-max schemes
In this note we consider a singular Liouville equation on compact surfaces, arising from the study of Chern-Simons vortices. Using improved versions of the Moser-Trudinger inequality and a min-max scheme, we prove existence of solutions in cases with lack of coercivity. Full details and further references can be found in the forthcoming paper [17].
Partial differential equations
Differential geometry
General
349
358
10.4171/RLM/577
http://www.ems-ph.org/doi/10.4171/RLM/577
On a semilinear parabolic equation with inverse-square potential
Fabio
Punzo
Università degli Studi di Milano, MILANO, ITALY
Alberto
Tesei
Università di Roma La Sapienza, ROMA, ITALY
Inverse-square potential, semigroup estimates, nonnegative solutions, wellposedness, instantaneous blow-up
We study existence and uniqueness, nonexistence and nonuniqueness of nonnegative solutions to a semilinear parabolic equation with inverse-square potential. Analogous existence and nonexistence results for the companion elliptic equation were proved in [4]. Concerning nonuniqueness, we extend the results proved in [16] for the case without potential.
Partial differential equations
General
359
396
10.4171/RLM/578
http://www.ems-ph.org/doi/10.4171/RLM/578
Existence for semilinear parabolic stochastic equations
Viorel
Barbu
Romanian Academy, IASI, ROMANIA
Wiener process, mild solution, random differential equation
The boundary value problem for semilinear parabolic stochastic equations of the form dX – ΔX dt+ β (X) dt∋ √QdWt, where Wt is a Wiener process and b is a maximal monotone graph everywhere defined, is well posed.
Partial differential equations
General
397
403
10.4171/RLM/579
http://www.ems-ph.org/doi/10.4171/RLM/579
BV functions in a Hilbert space with respect to a Gaussian measure
Luigi
Ambrosio
Scuola Normale Superiore, PISA, ITALY
Giuseppe
Da Prato
Scuola Normale Superiore, PISA, ITALY
Diego
Pallara
Università degli Studi di Lecce, LECCE, ITALY
Gaussian measures, BV functions, Ornstein-Uhlenbecks semigroups
Functions of bounded variation in Hilbert spaces endowed with a Gaussian measure γ are studied, mainly in connection with Ornstein-Uhlenbeck semigroups for which γ is invariant.
Real functions
Measure and integration
Functional analysis
Probability theory and stochastic processes
405
414
10.4171/RLM/580
http://www.ems-ph.org/doi/10.4171/RLM/580
Involutions, Humbert surfaces, and divisors on a moduli space
Steven
Weintraub
Lehigh University, BETHLEHEM, UNITED STATES
Siegel modular varieties, Humbert surfaces, Keel-Vermeire divisors
Let M *2 be the Igusa compactification of the Siegel modular variety of degree 2 and level 2. In earlier work with R. Lee, we carefully investigated this variety. Subvarieties Dℓ (compactification divisors) and HΔ (Humbert surface of discriminant 1) play a prominent role in its structure; in particular their fundamental classes span H4(M *2; ℤ). We return to this variety and consider another class of subvarieties Kh (Humbert surfaces of degree 4), which we investigate with the help of involutions on M *2. We carefully describe these subvarieties and consider the representations of their fundamental classes in terms of the fundamental classes of the subvarieties Dℓ and HΔ. The space M *2 is also known in a different context. It can also be described as the space M0; 6 of stable curves of genus 2 with ordered Weierstrass points. In this context the divisors Kh are what have come to be known as Keel-Vermeire divisors.
Algebraic geometry
General
415
440
10.4171/RLM/581
http://www.ems-ph.org/doi/10.4171/RLM/581
A sharp Liouville theorem for elliptic operators
Enrico
Priola
Università di Torino, TORINO, ITALY
Feng-Yu
Wang
Swansea University, SINGLETON PARK, UNITED KINGDOM
Liouville theorem, space-time harmonic functions
We introduce a new condition on elliptic operators L = 1/2 Δ + b∙ ∇ which ensures the validity of the Liouville property, i.e., all smooth bounded solutions to Lu = 0 on ℝd are constant. Such condition is sharp when d=1. We extend our Liouville theorem to more general second order operators in non-divergence form assuming a Cordes type condition.
Partial differential equations
Operator theory
General
441
445
10.4171/RLM/582
http://www.ems-ph.org/doi/10.4171/RLM/582