Showing papers in "Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze in 2011"
TL;DR: In this paper, the authors considered a first order differential equation as a C∞ vector field X on a C ∞ manifold M and showed that there exist unique solutions of X through each point of M.
Abstract: A (first order) differential equation (“autonomous”) may be considered as a C∞ vector field X on a C∞ manifold M (for simplicity, for the moment we take the C∞ point of view; manifolds are assumed not to have a boundary, unless so stated). From the fundamental theorem of differential equations, there exist unique C∞ solutions of X through each point of M. That is, if x ∈ M, there is a curve φt (x), ǀtǀ < e such that, φ0 (x) = x if ǀt0ǀ < e, and φt (x) is C∞ on (t,x) (in a suitable domain).
183 citations
TL;DR: In this article, a nonlinear Neumann eigenvalue problem driven by a possibly nonhomogeneous differential operator is considered, where the right-hand side nonlinearity is (p − 1)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition or to be monotone.
Abstract: We consider a nonlinear Neumann eigenvalue problem driven by a possibly nonhomogeneous differential operator which incorporates as a special case the p-Laplacian We assume that the right-hand side nonlinearity is (p − 1)-superlinear, but need not satisfy the Ambrosetti-Rabinowitz condition or to be monotone We show that, for all values of the parameter λ in an upper half line, the problem has two positive and two negative solutions Subsequently, for the case of the p-Laplacian, we also produce a nodal solution Finally, for the semilinear case we show that the problem has two nodal solutions Mathematics Subject Classification (2010): 35J25 (primary); 35J80, 58E05 (secondary)
44 citations
TL;DR: In this paper, the authors consider closed hypersurfaces which shrink self-similarly under a natural class of fully nonlinear curvature flows and show that the only such surfaces are shrinking spheres.
Abstract: We consider closed hypersurfaces which shrink self-similarly under a natural class of fully nonlinear curvature flows. For those flows in our class with speeds homogeneous of degree 1 and either convex or concave, we show that the only such hypersurfaces are shrinking spheres. In the setting of convex hypersurfaces, we show under a weaker second derivative condition on the speed that again only shrinking spheres are possible. For surfaces this result is extended in some cases by a different method to speeds of homogeneity greater than 1. Finally we show that self-similar hypersurfaces with sufficiently pinched principal curvatures, depending on the flow speed, are again necessarily spheres.
38 citations
TL;DR: In this article, a geometric characterization of BV continuous vector rate independent operators is provided, and the vectorial play operator is shown to have the same continuity properties as the rate independent evolution variational inequalities.
Abstract: We prove a theorem providing a geometric characterization of BV continuous vector rate independent operators We apply this theorem to rate independent evolution variational inequalities and deduce new continuity properties of their solution operator: the vectorial play operator
35 citations
TL;DR: In this article, it was shown that there exists a critical exponent, depending on the homogeneity of the fundamental solution of F, that sharply characterizes the range of p > 1 for which there exist positive supersolutions or solutions in any exterior domain.
Abstract: We study fully nonlinear elliptic equations such as F(Du) = u , p > 1, in Rn or in exterior domains, where F is any uniformly elliptic, positively homogeneous operator. We show that there exists a critical exponent, depending on the homogeneity of the fundamental solution of F , that sharply characterizes the range of p > 1 for which there exist positive supersolutions or solutions in any exterior domain. Our result generalizes theorems of Bidaut-Veron [6] as well as Cutri and Leoni [11], who found critical exponents for supersolutions in the whole space Rn , in case −F is Laplace’s operator and Pucci’s operator, respectively. The arguments we present are new and rely only on the scaling properties of the equation and the maximum principle. Mathematics Subject Classification (2010): 35B53 (primary); 35J60 (secondary).
34 citations
TL;DR: In this paper, the authors studied the local behavior of a solution to the $l$th power of Laplacian with singular coefficients in lower order terms and obtained a bound on the vanishing order of the nontrivial solution.
Abstract: In this paper we study the local behavior of a solution to the $l$th power of Laplacian with singular coefficients in lower order terms. We obtain a bound on the vanishing order of the nontrivial solution. Our proofs use Carleman estimates with carefully chosen weights. We will derive appropriate three-sphere inequalities and apply them to obtain doubling inequalities and the maximal vanishing order.
29 citations
29 citations
TL;DR: In this article, a sharp rearrangement estimate for the nonlinear Havin-Maz'ya potentials is established, which leads to a characterization of those rearrangements invariant spaces between which the potentials are bounded.
Abstract: A sharp rearrangement estimate for the nonlinear Havin-Maz’ya potentials is established. In particular, this estimate leads to a characterization of those rearrangement invariant spaces between which the nonlinear potentials are bounded. In combination with results from [24] and [18], it also enables us to derive local bounds for solutions to quasilinear elliptic PDE’s and for their gradient in rearrangement form. As a consequence, the local regularity of solutions to elliptic equations and for their gradient in arbitrary rearrangement invariant spaces is reduced to one-dimensional Hardy-type inequalities. Applications to the special cases of Lorentz and Orlicz spaces are presented. Mathematics Subject Classification (2010): 31C15 (primary); 35B45 (secondary).
26 citations
TL;DR: In this article, it was shown that the KdV-Burgers equation is well-posed in Sobolev spaces for dispersive-dissipative models.
Abstract: We complete the known results on the local Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in $ H^{-1}(\R) $ with a solution-map that is analytic from $H^{-1}(\R) $ to $C([0,T];H^{-1}(\R))$ whereas it is ill-posed in $ H^s(\R) $, as soon as $ s 0 $ small enough. As far as we know, this is the first result of this type for a dispersive-dissipative equation. The framework we develop here should be very useful to prove similar results for other dispersive-dissipative models
25 citations
TL;DR: In this paper, the authors proved that every quasiconformal harmonic mapping between two plane domains with C 1,α (α < 1) and, respectively, C 1 1 compact boundary is bi-Lipschitz.
Abstract: We prove the following theorem: every quasiconformal harmonic mapping between two plane domains with C1,α (α < 1) and, respectively, C1,1 compact boundary is bi-Lipschitz. This theorem extends a similar result of the author [10] for Jordan domains, where stronger boundary conditions for the image domain were needed. The proof uses distance function from the boundary of the image domain. Mathematics Subject Classification (2010): 58E20 (primary); 30C62 (secondary).
25 citations
TL;DR: The present work extends results of Gustafsson, Kreiss, Sundstrom and a former work of the authors' to the widest possible class of finite difference schemes by dropping some technical assumptions.
Abstract: We study the stability of finite difference schemes for hyperbolic initial boundary value problems in one space dimension. Assuming stability for the dicretization of the hyperbolic operator as well as a geometric regularity condition, we show that the uniform Kreiss-Lopatinskii condition yields strong stability for the discretized initial boundary value problem. The present work extends results of Gustafsson, Kreiss, Sundstrom and a former work of ours to the widest possible class of finite difference schemes by dropping some technical assumptions. We give some new examples of numerical schemes for which our results apply.
TL;DR: In this article, it was shown that the maximum of the L p norm is attained for a given small mass in both 1D and 2D and that the maximizer is unique and obtain a precise estimate of the maximum.
Abstract: Consider the mass-critical nonlinear Schrodinger equations in both focusing and defocusing cases for initial data in L2 in space dimension N . By Strichartz inequality, solutions to the corresponding linear problem belong to a global L p space in the time and space variables, where p = 2 + 4 N . In 1D and 2D, the best constant for the Strichartz inequality was computed by D. Foschi who has also shown that the maximizers are the solutions with Gaussian initial data. Solutions to the nonlinear problem with small initial data in L2 are globally defined and belong to the same global L p space. In this work we show that the maximum of the L p norm is attained for a given small mass. In addition, in 1D and 2D, we show that the maximizer is unique and obtain a precise estimate of the maximum. In order to prove this we show that the maximum for the linear problem in 1D and 2D is nondegenerated. Mathematics Subject Classification (2010): 35Q55 (primary); 35P25, 35B50, 35B45 (secondary).
TL;DR: The relation between cohomology jump loci in a finite Galois cover, formality properties and algebraic monodromy action was explored in this paper. But the relation between the base and total hop loci of the cover is not discussed.
Abstract: We explore the relation between cohomology jump loci in a finite Galois cover, formality properties and algebraic monodromy action. We show that the jump loci of the base and total space are essentially the same, provided the base space is 1-formal and the monodromy action in degree 1 is trivial. We use reduced multinet structures on line arrangements to construct components of the first characteristic variety of the Milnor fiber in degree 1, and to prove that the monodromy action is non-trivial in degree 1. For an arbitrary line arrangement, we prove that the triviality of the monodromy in degree 1 can be detected in a precise way, by resonance varieties.
TL;DR: In this article, the authors analyzed the submarkovian extensions of H, i.e. the selfadjoint extensions which generate sub-Markovian semigroups.
Abstract: Let be an open subset of R d and H = − P d=1 @icij@j a secondorder partial differential operator on L2() with domain C 1 c () where the coefficientscij 2 W 1,1 () are real symmetric and C = (cij) is a strictly positive-definite matrix over . In particular, H is locally strongly elliptic. We analyze the submarkovian extensions of H, i.e. the selfadjoint extensions which generate submarkovian semigroups. Our main result establishes that H is Markov unique, i.e. it has a unique submarkovian extension, if and only if cap(@) = 0 where cap(@) is the capacity of the boundary of measured with respect to H. The second main result establishes that Markov uniqueness of H is equivalent to the semigroup generated by the Friedrichs extension of H being conservative.
TL;DR: In this paper, the authors extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian.
Abstract: In this manuscript we extend De Giorgi’s interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but it does not induce a metric. Assuming the initial condition to be a density function, not necessarily smooth, but solely of bounded first moments and finite “entropy”, we use a variational scheme to discretize the equation in time and construct approximate solutions. Then De Giorgi’s interpolation method is revealed to be a powerful tool for proving convergence of our algorithm. Finally we show uniqueness and stability in L1 of our solutions. Mathematics Subject Classification (2010): 35K59 (primary); 49J40, 82C40, 47J25 (secondary).
TL;DR: In this paper, the Mumford-Shah minimizer is shown to be equivalent to the minimal cone of type P (a plane), Y (three half planes meeting with 120 angles) or T (cone over a regular tetrahedron centered at the origin) in terms of Hausdor distance in B(x;r).
Abstract: We show that if (u;K) is a minimizer of the Mumford-Shah functional in an open set of R 3 , and if x2 K and r > 0 are such that K is close enough to a minimal cone of type P (a plane), Y (three half planes meeting with 120 angles) or T (cone over a regular tetrahedron centered at the origin) in terms of Hausdor distance in B(x;r), then K is C 1; equivalent to the minimal cone in B(x;cr) where c < 1 is an universal constant.
TL;DR: In this paper, the existence of global J-holomorphic discs with boundaries attached to real tori is proved. But it is not yet known whether these discs can be used to remove singularities of almost complex structures.
Abstract: We prove a result on removing singularities of almost complex structures pulled back by a non-diffeomorphic map. As an application we prove the existence of global J -holomorphic discs with boundaries attached to real tori. Mathematics Subject Classification (2010): 32H02 (primary); 53C15 (secondary).
TL;DR: In this article, the question of right equivalence between two local tube realizations of the CR-manifold germ (M,a) was clarified by introducing two different notions of affine equivalence.
Abstract: For every real-analytic CR-manifold M we give necessary and sufficient conditions that M can be realized in a suitable neighbourhood of a given point a ∈ M as a tube submanifold of some C . We clarify the question of the ‘right’ equivalence between two local tube realizations of the CR-manifold germ (M,a) by introducing two different notions of affine equivalence. One of our key results is a procedure that reduces the classification of equivalence classes to a purely algebraic manipulation in terms of Lie theory. Mathematics Subject Classification Primary 32V05; Secondary 32V40, 32M25, 17B66
TL;DR: In this paper, the authors parametrise all linear pfaffian representations of F by an open subset in the moduli space MC(2, KC) for smooth curves.
Abstract: Let C be a smooth curve in P given by an equation F = 0 of degree d. In this paper we parametrise all linear pfaffian representations of F by an open subset in the moduli space MC(2, KC). We construct an explicit correspondence between pfaffian representations of C and rank 2 vector bundles on C with canonical determinant and no sections.
TL;DR: In this article, the authors introduce new definitions of convergence, based on adding stability criteria to Γ-convergence, that are suitable in many cases for studying convergence of local minimizers.
Abstract: We introduce new definitions of convergence, based on adding stability criteria to Γ-convergence, that are suitable in many cases for studying convergence of local minimizers
TL;DR: In this article, the asymptotic behavior of the least energy nodal solution of a problem with a jumping nonlinearity was studied, and the authors studied the behavior of a nonlinear solution of the problem with jumping non-linearity.
Abstract: In this paper we study the asymptotic behavior of the least energy nodal solution of a problem with a jumping nonlinearity. Mathematics Subject Classification (2010):35J10 (primary); 35J65 (secondary).
TL;DR: In this paper, the authors derived a Ck+ 2 Holder estimate for Pφ, where P is either of the two solution operators in Henkin's local homotopy formula for ∂b on a strongly pseudoconvex real hypersurface M in Cn, φ is a (0, q)-form of class Ck on M, and k ≥ 0 is an integer.
Abstract: We derive a Ck+ 2 Holder estimate for Pφ, where P is either of the two solution operators in Henkin’s local homotopy formula for ∂b on a strongly pseudoconvex real hypersurface M in Cn , φ is a (0, q)-form of class Ck on M , and k ≥ 0 is an integer. We also derive a Ca estimate for Pφ, when φ is of class Ca and a ≥ 0 is a real number. These estimates require that M be of class Ck+ 2 , or Ca+2, respectively. The explicit bounds for the constants occurring in these estimates also considerably improve previously known such results. These estimates are then applied to the integrability problem for CR vector bundles to gain improved regularity. They also constitute a major ingredient in a forthcoming work of the authors on the local CR embedding problem. Mathematics Subject Classification (2010): 32V05 (primary); 32A26, 32T15 (secondary).
TL;DR: In this article, it was shown that rational and 1-rational singularities of complex spaces are stable under taking quotients by holomorphic actions of reductive and compact Lie groups.
Abstract: We prove that rational and 1-rational singularities of complex spaces are stable under taking quotients by holomorphic actions of reductive and compact Lie groups. This extends a result of Boutot to the analytic category and yields a refinement of his result in the algebraic category. As one of the main technical tools vanishing theorems for cohomology groups with support on fibres of resolutions are proven. Mathematics Subject Classification (2010): 32M05 (primary), 32S05, 32C36, 14L30 (secondary).
Journal Article•
TL;DR: In this paper, the authors established the W1,p type estimates for weak solutions of a class of degenerate elliptic equations and obtained the optimal estimates by introducing the intrinsic metric associated with the geometry of the operator and then using the compactness method.
Abstract: In the present paper we establish theW1,p type estimates for the weak solutions of a class of degenerate elliptic equations. The optimal estimates are obtained by introducing the intrinsic metric that is associated with the geometry of the operator and then using the compactness method. Mathematics Subject Classification (2010): 35J70 (primary); 35H20 (secondary).
TL;DR: In this paper, the constitutive behavior of a multiaxial viscoelastic material is represented by the nonlinear relation e − A(x) : ∫ t 0 σ (x, τ ) dτ ∈ α(σ, x), which generalizes the classical Maxwell model of visco-elasticity of fluid type.
Abstract: The constitutive behaviour of a multiaxial visco-elastic material is here represented by the nonlinear relation e − A(x) : ∫ t 0 σ (x, τ ) dτ ∈ α(σ, x), which generalizes the classical Maxwell model of visco-elasticity of fluid type. Here α(·, x) is a (possibly multivalued) maximal monotone mapping, σ is the stress tensor, e is the linearized strain tensor, and A(x) is a positive-definite fourth-order tensor. The above inclusion is here coupled with the quasi-static force-balance law, − div σ = # f . Existence and uniqueness of the weak solution are proved for a boundary-value problem. Convergence to a two-scale problem is then derived for a composite material, in which the functions α and A periodically oscillate in space on a short length-scale. It is proved that the coarse-scale averages of stress and strain solve a single-scale homogenized problem, and that conversely any solution of this problem can be represented in that way. The homogenized constitutive relation is represented by the minimization of a time-integrated functional, and is rather different from the above constitutive law. These results are also retrieved via De Giorgi’s notion of %-convergence. These conclusions are at variance with the outcome of so-called analogical models, that rest on an (apparently unjustified) mean-field-type hypothesis. Mathematics Subject Classification (2010): 35B27 (primary); 49J40, 73E50, 74Q (secondary).