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2024-03-28 16:50:35
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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)
20
2009
1
Continuous dependence on the constitutive functions for a class of problems describing fluid flow in porous media
Iacopo
Borsi
Universita di Firenze, FIRENZE, ITALY
Angiolo
Farina
Universita di Firenze, FIRENZE, ITALY
Roberto
Gianni
Università di Roma La Sapienza, ROMA, ITALY
Mario
Primicerio
Universita di Firenze, FIRENZE, ITALY
Flows in porous media, continuous dependence on parameters
In this paper we consider the PDE describing the fluid flow in a porous medium, focusing on the solution’s dependence upon the choice of the saturation curve and the hydraulic conductivity. Basically, we consider two different saturation curves (say θ1 and θ2) and two different hydraulic conductivities (K1 and K2) which are both “close” in the L∞loc-norm. Then we find estimates to prove a constitutive stability for the solutions of the corresponding problems with the same boundary and initial conditions.
Partial differential equations
Fluid mechanics
General
1
24
10.4171/RLM/531
http://www.ems-ph.org/doi/10.4171/RLM/531
Multipartite entanglement in qubit systems
Paolo
Facchi
Università degli Studi di Bari, BARI, ITALY
Entanglement, quantum nonlocality, quantum information, Hilbert spaces
We introduce a potential of multipartite entanglement for a system of n qubits, as the average over all balanced bipartitions of a bipartite entanglement measure, the purity. We study in detail its expression and look for its minimizers, the maximally multipartite entangled states. They have a bipartite entanglement that does not depend on the bipartition and is maximal for all possible bipartitions. We investigate their structure and consider several examples for small n.
Quantum theory
General
25
67
10.4171/RLM/532
http://www.ems-ph.org/doi/10.4171/RLM/532
Efficient representation in spaces of plane curves
Kathryn
Leonard
California State University, CAMARILLO, UNITED STATES
Shape approximation, epsilon-entropy, medial axis
This paper evaluates the Blum medial axis representation of embeddings of S1 into ℝ2 from the perspective of efficiency, using a C1-type metric. For compact classes of curves with Lipschitz tangent angle, we compute the ε-entropy and compare that efficiency benchmark with uniform approximation using the Blum medial axis. In the compact setting, the boundary curve is more efficient. For noncompact classes of embeddings, we establish a geometric criterion for when the medial axis will be more efficient in an adaptive approximation.
Approximations and expansions
Global analysis, analysis on manifolds
General
69
93
10.4171/RLM/533
http://www.ems-ph.org/doi/10.4171/RLM/533
On the number of terms of a power of a polynomial
Andrzej
Schinzel
Polish Academy of Sciences, WARSZAWA, POLAND
Umberto
Zannier
Scuola Normale Superiore, PISA, ITALY
Algebra of polynomials
Let f(x) be a polynomial with complex coefficients. Rényi and independently Erdős in 1949 conjectured that a bound for the number of terms of f(x)2 implies a bound for the number of terms of f(x). In 1987 Schinzel found a proof of this conjecture, actually for all powers f(x)l, and he gave some explicit bounds. The aim of this paper is to improve such inequalities in a substantial way.
Field theory and polynomials
General
95
98
10.4171/RLM/534
http://www.ems-ph.org/doi/10.4171/RLM/534
2
Systems of nonlinear Schrödinger equations. A survey
Antonio
Ambrosetti
SISSA, TRIESTE, ITALY
Nonlinear Schrödinger equations and systems, variational methods, perturbation methods
In this paper we survey some recent advances on various kind of systems of nonlinear Schrödinger equations. The arguments rely on critical point theory, the concentration compactness and perturbation methods.
Partial differential equations
General
99
110
10.4171/RLM/535
http://www.ems-ph.org/doi/10.4171/RLM/535
The limit as p → ∞ for the p-Laplacian with mixed boundary conditions and the mass transport problem through a given window
Jesús
García-Azorero
Universidad Autónoma de Madrid, MADRID, SPAIN
Juan
Manfredi
University of Pittsburgh, PITTSBURGH, UNITED STATES
Ireneo
Peral
Universidad Autónoma de Madrid, MADRID, SPAIN
Julio
Rossi
Universidad de Buenos Aires, BUENOS AIRES, ARGENTINA
Quasilinear elliptic equations, Neumann boundary conditions
In this paper we study the limit as p → ∞ in a PDE problem involving the p-Laplacian with a right hand side, − div(|Du|p − 2Du) = f, with mixed boundary conditions, u = 0 on Γ and |Du|p − 2 ∂u⁄∂ν = 0 on ∂Ω \ Γ. We find that this limit is related to an optimal mass transport problem, where the total mass given by f is transported outside the domain through a given window on the boundary Γ.
Partial differential equations
General
111
126
10.4171/RLM/536
http://www.ems-ph.org/doi/10.4171/RLM/536
Existence and multiplicity results for a weighted p-Laplace equation involving Hardy potentials and critical nonlinearities
Roberta
Musina
Università di Udine, UDINE, ITALY
Variational methods, critical growth, weighted Lp-Laplace operator, Hardy inequalities, Caffarelli–Kohn–Nirenberg inequalities, breaking symmetry.
We study a class of elliptic problems involving weighted p-Laplace operators, critical growths and Hardy potentials. The main motivation lies in some Hardy–Sobolev type inequalities that were proved by Caffarelli–Kohn–Nirenberg in 1984.
Differential geometry
Calculus of variations and optimal control; optimization
General
127
143
10.4171/RLM/537
http://www.ems-ph.org/doi/10.4171/RLM/537
Uniqueness in the Cauchy problem for a class of hypoelliptic ultraparabolic operators
Chiara
Cinti
Università di Bologna, BOLOGNA, ITALY
Hörmander operators, ultraparabolic operators, Cauchy problem, uniqueness theorems, homogeneous Lie groups
We consider a class of hypoelliptic ultraparabolic operators in the form \begin{equation*} \L=\sum_{j=1}^m X_j^2+X_0-\partial_t, \end{equation*} under the assumption that the vector fields $X_1,\ldots,X_m$ and $X_0-\partial_t$ are invariant with respect to a suitable homogeneous Lie group $\mathbb{G}$. We show that if $u,v$ are two solutions of $\L u = 0$ on $\RN \times ]0,T[$ and $u(\cdot,0)=\varphi$, then each of the following conditions: $|u(x,t)-v(x,t)|$ can be bounded by $M \exp (c|x|_{\mathbb{G}}^2)$, or both $u$ and $v$ are non negative, implies $u\equiv v$. We use a technique which relies on a pointwise estimate of the fundamental solution of $\L$.
Partial differential equations
General
145
158
10.4171/RLM/538
http://www.ems-ph.org/doi/10.4171/RLM/538
Plücker formulae for curves in high dimensions
C.T.C.
Wall
University of Liverpool, LIVERPOOL, UNITED KINGDOM
Plücker relations, orrespondences, Weierstrass points
The classical relations of Plücker between the invariants and singularities of a plane curve can be expressed as two linear relations and two involving quadratic terms. The linear relations were generalised to curves in n-space already in the nineteenth century, but true generalisations of the others were obtained only in 3-space. In this article, using the classical method of correspondences, we obtain formulae in n-space corresponding to the original ones in the plane.
Algebraic geometry
General
159
177
10.4171/RLM/539
http://www.ems-ph.org/doi/10.4171/RLM/539
Gradient estimates in non-linear potential theory
Frank
Duzaar
Universität Erlangen-Nünberg, ERLANGEN, GERMANY
Giuseppe
Mingione
Università di Parma, PARMA, ITALY
p-Laplacian, Wolff potential, regularity
We present pointwise gradient bounds for solutions to p-Laplacian type non-homogeneous equations employing non-linear Wolff type potentials, and then prove similar bounds, via suitable caloric potentials, for solutions to parabolic equations. A method of proof entails a family of non-local Caccioppoli inequalities, together with a DeGiorgi’s type fractional iteration.
Partial differential equations
General
179
190
10.4171/RLM/540
http://www.ems-ph.org/doi/10.4171/RLM/540
3
Editorial note
Antonio
Ambrosetti
SISSA, TRIESTE, ITALY
Editorial Note On the occasion of the thirtieth anniversary of the death of Guido Stampacchia (1922-1978), the International Conference “Recent trends in nonlinear partial differential equations” has been organised by some of his former colleagues and students at the Accademia dei Lincei in Roma, November 6, 2008. Stampacchia was the President of the Unione Matematica Italiana in the period of 1967-1973, and a very active member of the Accademia dei Lincei. One of the leaders of the Italian school of partial differential equations, his contributions to differential equations, calculus of variations and variational inequalities are well known. His popularity in the Italian mathematical community explains very well the participation of so many researchers in the Conference. After the opening addresses by Enrico Magenes and David Kinderlehrer, plenary lectures were given by Louis Nirenberg, Haïm Brezis, Lucio Boccardo, Vieri Benci, Henry Berestycki and Luis Caffarelli. Some of the speakers have gladly agreed to write up their lectures and Rendiconti Lincei - Matematica e Applicazioni is proud to publish them. The paper by Brezis already appeared in vol. XIX (2008), No 4, pp. 335-338 while others are collected in the present issue. The Editor-in-Chief Antonio Ambrosetti
General
191
192
10.4171/RLM/541
http://www.ems-ph.org/doi/10.4171/RLM/541
Apertura dei lavori
Enrico
Magenes
, PAVIA, ITALY
General
193
194
10.4171/RLM/542
http://www.ems-ph.org/doi/10.4171/RLM/542
A Calderon–Zygmund theory for infinite energy minima of some integral functionals
Lucio
Boccardo
Università di Roma La Sapienza, ROMA, ITALY
Calderon–Zygmund theory, infinite energy minima
A Calderon–Zygmund theory in Lebesgue and Marcinkiewicz spaces for infinite energy minima of some integral functionals is proved.
Calculus of variations and optimal control; optimization
Partial differential equations
General
195
205
10.4171/RLM/543
http://www.ems-ph.org/doi/10.4171/RLM/543
Some (big) irreducible components of the moduli space of minimal surfaces of general type with pg = q = 1 and K2 = 4
Roberto
Pignatelli
Università di Trento, TRENTO (TN), ITALY
Surfaces of general type, fibrations, moduli
This paper is devoted to the irregular surfaces of general type having the smallest invariants, pg = q = 1. We consider the still unexplored case K2 = 4, classifying those whose Albanese morphism has general fibre of genus 2 and such that the direct image of the bicanonical sheaf under the Albanese morphism is a direct sum of line bundles. We find 8 unirational families, and we prove that all are irreducible components of the moduli space of minimal surfaces of general type. This is unexpected because the assumption on the direct image bicanonical sheaf is a priori only a closed condition. One more unexpected property is that all these components have dimension strictly bigger than the expected one.
Algebraic geometry
General
207
226
10.4171/RLM/544
http://www.ems-ph.org/doi/10.4171/RLM/544
The relative power and its invariance
Paolo Maria
Mariano
Universita di Firenze, FIRENZE, ITALY
Relative power, invariance, mutant bodies
The relative power of actions in Cauchy bodies su¤ering mutations due to defect evolution is introduced. It is shown that its invariance under the action of the Euclidean group over the ambient space and the material space allows one to obtain (i) the balance of standard and configurational actions and (ii) the identification of configurational ingredients from a unique source.
Mechanics of deformable solids
General
227
242
10.4171/RLM/545
http://www.ems-ph.org/doi/10.4171/RLM/545
Existence of hylomorphic solitary waves in Klein–Gordon and in Klein–Gordon–Maxwell equations
Vieri
Benci
Università di Pisa, PISA, ITALY
Donato
Fortunato
Università degli Studi di Bari, BARI, ITALY
Q-balls, hylomorphic solitons, vortices, Abelian gauge theories
This paper is devoted to the study of solitary waves whose existence is related to the ratio energy/charge. These solitary waves are called hylomorphic. This class includes the Q-balls, which are spherically symmetric solutions of the nonlinear Klein–Gordon equation (NKG), as well as solitary waves and vortices which occur, by the same mechanism, in the nonlinear Schrödinger equation and in gauge theories. It is proved an abstract theorem which allows to show the existence of hylomorphic solitary waves and vortices in the (NKG) and in the nonlinear Klein–Gordon–Maxwell equations (NKGM).
Operator theory
Partial differential equations
Mechanics of deformable solids
Quantum theory
243
279
10.4171/RLM/546
http://www.ems-ph.org/doi/10.4171/RLM/546
Surfaces minimizing nonlocal energies
Luis
Caffarelli
University of Texas at Austin, AUSTIN, UNITED STATES
Diffusion, minimal surface, phase field, long range correlations
In this lecture, we discuss what we understand by a non local diffusion equation and explain the particular case of surface evolution by non local mean curvature and the corresponding minimal surfaces.
Partial differential equations
Calculus of variations and optimal control; optimization
General
281
299
10.4171/RLM/547
http://www.ems-ph.org/doi/10.4171/RLM/547
4
Orlicz–Sobolev regularity of mappings with subexponentially integrable distortion
Albert
Clop
Universidad Autonoma de Barcelona, BELLATERRA, SPAIN
Pekka
Koskela
University of Jyväskylä, JYVÄSKYLÄ, FINLAND
Mappings of finite distortion, higher integrability, subexponential distortion
We study regularity properties of mappings of finite distortion. We show that some sort of self-improvement phenomena hold also when only subexponential integrability is assumed for the distortion function. We extend to this setting results by Faraco, Koskela and Zhong [9] and Bildhauer, Fuchs and Zhong [6].
Functions of a complex variable
Real functions
Partial differential equations
General
301
326
10.4171/RLM/548
http://www.ems-ph.org/doi/10.4171/RLM/548
Recognizing the Farey–Stern–Brocot AF algebra
Daniele
Mundici
Università degli Studi di Firenze, FIRENZE, ITALY
Approximately finite dimensional algebra, Elliott classification, Grothendieck group, lattice-ordered group, Farey sequence
In his 2008 paper published in the Canadian Journal of Mathematics, F. Boca investigates an AF algebra A, whose Bratteli diagram arises from the Farey–Stern–Brocot sequence. It turns out that A coincides with the AF algebra M1 introduced in 1988 by the present author in a paper published in Advances in Mathematics. We give a procedure to recognize A among all finitely presented AF algebras whose Murray–von Neumann order of projections is a lattice. Further: (i) A is a *-subalgebra of Glimm universal algebra; (ii) tracial states of A correspond to Borel probability measures on the unit real interval; (iii) all primitive ideals of A are essential; (iv) the automorphism group of A has exactly two connected components.
Functional analysis
Order, lattices, ordered algebraic structures
Number theory
Dynamical systems and ergodic theory
327
338
10.4171/RLM/549
http://www.ems-ph.org/doi/10.4171/RLM/549
The level 1 case of Serre’s conjecture revisited
Luis Victor
Dieulefait
Universitat de Barcelona, BARCELONA, SPAIN
Galois representations, modular forms, modularity conjectures
We prove existence of conjugate Galois representations, and we use it to derive a simple method of weight reduction. As a consequence, an alternative proof of the level 1 case of Serre’s conjecture follows.
Number theory
General
339
346
10.4171/RLM/550
http://www.ems-ph.org/doi/10.4171/RLM/550
Stability-Instability criteria for nonautonomous systems
Salvatore
Rionero
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Nonautonomous systems, Liapunov Direct Method, stability
Nonautonomous binary systems of O.D.Es are considered. Apart from a critical case, it is shown that a temporal uniform validity of the Hurwitz conditions appear to be a basic condition to require for guaranteing the stability. Stability-instability criteria are obtained. Applications to the equation d2/dt2 x + p(t) d/dt x + q(t)x = 0 and in particular to the Hill equation, are furnished. The Hill equation associated to the (linear) stability of the nonautonomous Lotka–Volterra system is considered.
Ordinary differential equations
Dynamical systems and ergodic theory
Systems theory; control
General
347
367
10.4171/RLM/551
http://www.ems-ph.org/doi/10.4171/RLM/551
Alternative Forms of the Harnack Inequality for Non-Negative Solutions to Certain Degenerate and Singular Parabolic Equations
Emmanuele
DiBenedetto
Vanderbilt University, NASHVILLE, UNITED STATES
Ugo
Gianazza
Università di Pavia, PAVIA, ITALY
Vincenzo
Vespri
Universita di Firenze, FIRENZE, ITALY
Degenerate and singular parabolic equations, Harnack estimates
Non-negative solutions to quasi-linear, degenerate or singular parabolic partial differential equations, of p-Laplacian type for p > 2N/(N+1), satisfy Harnack-type estimates in some intrinsic geometry ([2, 3]). Some equivalent alternative forms of these Harnack estimates are established, where the supremum and the infimum of the solutions play symmetric roles, within a properly redefined intrinsic geometry. Such equivalent forms hold for the non-degenerate case p = 2 following the classical work of Moser ([5, 6]), and are shown to hold in the intrinsic geometry of these degenerate and/or parabolic p.d.e.’s. Some new forms of such an estimate are also established for 1 < p < 2.
Partial differential equations
General
369
377
10.4171/RLM/552
http://www.ems-ph.org/doi/10.4171/RLM/552
On a Sobolev-type inequality
Angelo
Alvino
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Sobolev inequality, isoperimetric inequalities, one-dimensional calculus of variations
A new proof of the classical Sobolev inequality in ℝn with the best constant is given. The result follows from an intermediate inequality which connects in a sharp way the Lp norm of the gradient of a function u to Lp* and Lp*-weak norms of u, where p ∈ ]1; n[ and p* = np/(n-p) is the Sobolev exponent.
Calculus of variations and optimal control; optimization
Real functions
Difference and functional equations
General
379
386
10.4171/RLM/553
http://www.ems-ph.org/doi/10.4171/RLM/553
On Brooks–Jewett, Vitali–Hahn–Saks and Nikodým convergence theorems for quasi-triangular functions
Paola
Cavaliere
Università di Salerno, FISCIANO (SA), ITALY
Paolo
de Lucia
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Non-additive functions, convergence theorems
Connections are established between Brooks–Jewett, Vitali–Hahn–Saks and Nikodym type convergence theorems for quasi-triangular functions on Boolean rings satisfying the Subsequential Completeness Property and valued into Hausdorff topological spaces.
Measure and integration
General
387
396
10.4171/RLM/554
http://www.ems-ph.org/doi/10.4171/RLM/554
A sharp isoperimetric inequality in the plane involving Hausdorff distance
Angelo
Alvino
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Vincenzo
Ferone
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Carlo
Nitsch
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Isoperimetric inequality, Bonnesen-style inequality, Hausdorff distance, isoperimetric deficit
We show that among all the convex bounded domain in ℝ2 having an assigned asymmetry index related to Hausdorff distance, there exists only one convex set (up to a similarity) which minimizes the isoperimetric deficit. We also show how to construct this set. The result can be read as a sharp improvement of the isoperimetric inequality for convex planar domain.
Convex and discrete geometry
Calculus of variations and optimal control; optimization
General
397
412
10.4171/RLM/555
http://www.ems-ph.org/doi/10.4171/RLM/555
Deep foundations
Piero
Villaggio
Università di Pisa, PISA, ITALY
Soil mechanics, plane elasticity
The stress transmission between a rigid foundation and the ground below is traditionally formulated into mathematical terms as the elastic problem of finding the stress state in a half-plane loaded by a rigid indentor. But this model is not realistic since foundations are not built on the surface of the ground but below its level, at the bottom of an excavation. We here suggest a solution for a notched elastic half-plane loaded by a rigid punch applied at the throat of the cavity.
Mechanics of deformable solids
General
413
420
10.4171/RLM/556
http://www.ems-ph.org/doi/10.4171/RLM/556