Showing papers in "Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze in 2003"

Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces

Abstract: Compactness in the space {L^{p}(0,T;B)}, B being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961, 1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to several abstract time dependent problems related to evolutionary PDEs. In the present paper, the problem is examined in view of Young measure theory: exploiting the underlying principles of “tightness” and “integral equicontinuity”, new necessary and sufficient conditions for compactness are given, unifying some of the previous contributions and showing that the Aubin - Lions condition is not only sufficient but also necessary for compactness. Furthermore, the related issue of compactness with respect to convergence in measure is studied and a general criterion is proved.

Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem

Abstract: In the paper the problem of constructing an optimal urban transportation network in a city with given densities of population and of workplaces is studied. The network is modeled by a closed connected set of assigned length, while the optimality condition consists in minimizing the Monge-Kantorovich functional representing the total transportation cost. The cost of trasporting a unit mass between two points is assumed to be proportional to the distance between them when the transportation is carried out outside of the network, and negligible when it is carried out along the network. The same problem can be also viewed as finding an optimal Dirichlet zone minimizing the Monge-Kantorovich cost of transporting the given two measures. The paper basically studies qualitative topological and geometrical properties of optimal networks. A mild regularity result for optimal networks is also provided.

Connecting topological Hopf singularities

Abstract: Smooth maps between riemannian manifolds are often not strongly dense in Sobolev classes of finite energy maps, and an energy drop in a limiting sequence of smooth maps often is accompanied by the production (or bubbling) of an associated rectifiable current. For finite 2-energy maps from the 3 ball to the 2 sphere, this phenomenon has been well-studied in works of Bethuel-Brezis-Coron and Giaquinta-Modica-Soucek where a finite mass 1 dimensional rectifiable current occurs whose boundary is the algebraic singular set of the limiting map. The relevant algebraic object here is \pi _2(S^2) which provides both the obstruction to strong approximation by smooth maps and the topological structure to the bubbling set and the singular set. With higher homotopy groups, new phenomena occur. For \pi _3(S^2) and the related case of finite 3-energy maps from the 4 Ball to the 2 sphere, there are examples with bubbled objects that no longer have finite mass. We define a new object, a scan, which generalizes a current but still occurs naturally in bubbling while automatically providing the topological connection between the singularities of the limit map. The bubbled scans, which are found via a new compactness theorem, again enjoy a representation using a finite measure 1 rectifiable set and an integer density function which is now however only L^{3/4} (rather than L^1) integrable.

Infinite geodesic rays in the space of Kähler potentials

Abstract: In this paper we prove the existence of solutions of a degenerate complex Monge-Ampere equation on a complex manifold. Applying our existence result to a special degeneration of complex structure, we show how to associate to a change of complex structure an infinite length geodetic ray in the space of potentials. We also prove an existence result for the initial value problem for geodesics. We end this paper with a discussion of a list of open problems indicating how to relate our reults to the existence problem for extremal metrics.

The intersection of a curve with algebraic subgroups in a product of elliptic curves

Abstract: Let C be transverse curve in a power of a C.M. elliptic curve A. We show that the points in the intersection of C and the union of all codimension-two algebraic subgroup of A is a finete set.

Regularity of convex functions on Heisenberg groups

Abstract: We discuss differentiability properties of convex functions on Heisenberg groups. We show that the notions of horizontal convexity (h-convexity) and viscosity convexity (v-convexity) are equivalent and that h-convex functions are locally Lipschitz continuous. Finally we exhibit Weierstrass-type h-convex functions which are nowhere differentiable in the vertical direction on a dense set or on a Cantor set of vertical lines.

Positive knots, closed braids and the Jones polynomial

Abstract: Using the recent Gaus diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY, Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive knots with the same polynomial and no positive knot with trivial polynomial. We also discuss an extension of the Bennequin inequality, showing that the unknotting number of a positive knot is not less than its genus, which recovers some recent unknotting number results of A’Campo, Kawamura and Tanaka, and give applications to the Jones polynomial of a positive knot.

Local and canonical heights of subvarieties

Abstract: Classical results of Weil, Neron and Tate are generalized to local heights of subvarieties with respect to hermitian pseudo-divisors. The local heights are well-defined if the intersection of supports is empty. In the archimedean case, the metrics are hermitian and the local heights are defined by a refined version of the *-product of Gillet-Soule developped on compact varieties without assuming regularity. In the non-archimedean case, the local heights are intersection numbers using methods from rigid and formal geometry to handle non-discrete valuations. To include canonical metrics of line bundles algebraically equivalent to 0, a local Chow cohomology is introduced on formal models over the valuation ring. Using Tate’s limit argument, canonical local heights of subvarieties on an abelian variety are obtained with respect to any pseudo-divisors. By integration over an M-field, we deduce corresponding results for global heights of subvarieties.

Twistor forms on Kähler manifolds

Abstract: Twistor forms are a natural generalization of conformal vector fields on riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We study twistor forms on compact Kahler manifolds and give a complete description up to special forms in the middle dimension. In particular, we show that they are closely related to hamiltonian 2-forms. This provides the first examples of compact Kahler manifolds with non–parallel twistor forms in any even degree.

A Hörmander-type spectral multiplier theorem for operators without heat kernel

Abstract: Hormander’s famous Fourier multiplier theorem ensures the L_p-boundedness of F(-\Delta _{\mathbb{R}} D) whenever F\in \mathcal{H}(s) for some s>\frac{D}{2}, where we denote by \mathcal{H} (s) the set of functions satisfying the Hormander condition for s derivatives. Spectral multiplier theorems are extensions of this result to more general operators A \ge 0 and yield the L_p-boundedness of F(A) provided F\in \mathcal{H}(s) for some s sufficiently large. The harmonic oscillator A=-\Delta _{\mathbb{R}}+x^2 shows that in general s> \frac{D}{2} is not sufficient even if A has a heat kernel satisfying gaussian estimates. In this paper, we prove the L_p-boundedness of F(A) whenever F\in \mathcal{H}(s) for some s>\frac{D+1}{2}, provided A satisfies generalized gaussian estimates. This assumption allows to treat even operators A without heat kernel (e.g. operators of higher order and operators with complex or unbounded coefficients) which was impossible for all known spectral multiplier results.

Self-adjoint extensions by additive perturbations

Abstract: Let AN be the symmetric operator given by the restriction of A to N , where A is a self-adjoint operator on the Hilbert space H and N is a linear dense set which is closed with respect to the graph norm on D(A), the operator domain of A. We show that any self-adjoint extension Aof AN such that D(A� )∩ D(A) = N can be additively decomposed by the sum A� = ¯ A+T� , where both the operators ¯ A and Ttake values in the strong dual of D(A). The operator ¯ A is the closed extension of A to the whole H whereas Tis explicitly written in terms of a (abstract) boundary condition depending on N and on the extension parameter � ,a self-adjoint operator on an auxiliary Hilbert space isomorphic (as a set) to the deficiency spaces of AN . The explicit connection with both Kre˘in's resolvent formula and von Neumann's theory of self-adjoint extensions is given.

The dynamics of holomorphic maps near curves of fixed points

Abstract: Let M be a two-dimensional complex manifold and f:M \rightarrow M a holomorphic map. Let S \subset M be a curve made of fixed points of f, i.e. {\rm {Fix}} (f)=S. We study the dynamics near S in case f acts as the identity on the normal bundle of the regular part of S. Besides results of local nature, we prove that if S is a globally and locally irreducible compact curve such that S\cdot S<0 then there exists a point p \in S and a holomorphic f-invariant curve with p on the boundary which is attracted by p under the action of f. These results are achieved introducing and studying a family of local holomorphic foliations related to f near S.

Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation

Abstract: The null controllability problem for a structurally damped abstract wave equation –often referred to in the literature as a structurally damped equation– is considered with a view towards obtaining optimal rates of blowup for the associated minimal energy function \mathcal{E}_{\min }(T), as terminal time T\downarrow 0. Key use is made of the underlying analyticity of the semigroup generated by the elastic operator \mathcal{A}, as well as of the explicit characterization of its domain of definition. We ultimately find that the blowup rate for \mathcal{E}_{\min }(T), as T goes to zero, depends on the extent of structural damping.

Asymptotic formula for solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary

Abstract: We consider the Dirichlet problem for elliptic equations of arbitrary order and prove an asymptotic formula for a singular solution near a boundary point. The only a priori assumption on the coefficients of the principal part of the equation is the smallness of the local oscillation near the point.

On singular perturbation problems with robin boundary condition

Abstract: We consider the following singularly perturbed elliptic problem 2 u − u + f (u) = 0, u > 0 in , ∂u ∂ν + λu = 0 on ∂ , where f satisfies some growth conditions, 0 ≤ λ ≤ +∞, and ⊂ RN (N > 1) is a smooth and bounded domain. The cases λ = 0 (Neumann problem) and λ = +∞ (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant λ∗ > 1 such that, as → 0, the least energy solution has a spike near the boundary if λ ≤ λ∗, and has an interior spike near the innermost part of the domain if λ > λ∗. Central to our study is the corresponding problem on the half space. Mathematics Subject Classification (2000): 35B35 (primary), 35J40, 92C40 (secondary).

Remarks on the theory of elasticity

Abstract: In compressible Neohookean elasticity one minimizes functionals which are composed by the sum of the L^2 norm of the deformation gradient and a nonlinear function of the determinant of the gradient. Non–interpenetrability of matter is then represented by additional invertibility conditions. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Muller and Spector in 1995. It applies, however, only if some L^p-norm of the gradient with p>2 is controlled (in three dimensions). We first characterize their class of functions in terms of properties of the associated rectifiable current. Then we address the physically relevant p=2 case, and show how their notion of invertibility can be extended to p=2. The class of functions so obtained is, however, not closed. We prove this by giving an explicit construction.

Riemann Maps in Almost Complex Manifolds

Abstract: We prove the existence of stationary discs in the ball for small almost complex deformations of the standard structure. We define a local analogue of the Riemann map and establish its main properties. These constructions are applied to study the local geometry of almost complex manifolds and their morphisms.

Peak solutions for an elliptic system of FitzHugh-Nagumo type

Abstract: The aim of this paper is to study the existence of various types of peak solutions for an elliptic system of FitzHugh-Nagumo type. We prove that the system has a single peak solution, which concentrates near the boundary of the domain. Under some extra assumptions, we also construct multi-peak solutions with all the peaks near the boundary, and a single peak solution with its peak near an interior point of the domain.

Failure of analytic hypoellipticity in a class of differential operators

Abstract: For the hypoelliptic differential operators P = ∂2 x + ( x∂y − x∂t )2 introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of k and l left open in the analysis, the operators P also fail to be analytic hypoelliptic (except for (k, l) = (0, 1)), in accordance with Treves’ conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of eigenvalues of a non-self-adjoint operator. Mathematics Subject Classification (2000): 35B65. 1. – Introduction and result A differential operator P is said to be hypoelliptic (respectively, analytic hypoelliptic) on if for any C∞ (respectively, C) function f on some open set U ⊂ all the solutions u of Pu = f belong to C∞(U ) (respectively, C(U )). The basic result about the hypoellipticity of operators of the type “sum of squares”, P = X2 1 + . . . + X2 n , where X1, . . . , Xn are real vector fields of class C( ), is Hörmander’s theorem [1] which gives necessary and sufficient conditions for hypoellipticity. But these assumptions are not sufficient for analytic hypoellipticity as was first proved by Baouendi and Goulaouic [2]. Other classes of hypoelliptic operators which fail to be analytic hypoelliptic have been found and there are important results on analytic regularity (see Christ [3]-[8], Christ and Geller [9], Derridj and Tartakoff [10]-[13], Derridj and Zuily [14], Francsics and Hanges [15]-[17], Grigis and Sjöstrand [19], Hanges and Himonas [20][22], Helffer [23], Hoshiro [24], Metivier [25]-[26], Pham The Lai and Robert [27], Sjöstrand [28],Tartakoff [29],Treves [30]-[31], see also the survey [18] for more references). Work partially supported by NSF grants 0103807, 0100495 (O.C.) and 0074924 (R.D.C.). Pervenuto alla Redazione il 19 ottobre 2001. 22 OVIDIU COSTIN – RODICA D. COSTIN However, the question of finding a general characterisation of analytic hypoellipticity for sum of squares operators is still open. Christ gave a criterion for analytic hypoellipticity in the two-dimensional case [8]. Treves conjectured a general criterion for analytic hypoellipticity [31]: “For a sum of squares of analytic vector fields to be analytic hypoelliptic it is necessary and sufficient that every Poisson stratum (defined by the symbols of the vector fields) of its characteristic variety be symplectic.” Extending a result of Christ [3], Hoshiro gave new examples of hypoelliptic operators in R3 which fail to be analytic hypoelliptic [24]. They have the form (1.1) P = ∂ 2 ∂x2 + ( x ∂ ∂y − x ∂ ∂t )2 where k < l are non-negative integers. It is shown, through an elegant proof, that P is not analytic hypoelliptic if either of the following assumptions are satisfied: (i) (l + 1)/(l − k) is not a positive integer. (ii) Both l − k and (l + 1)/(l − k) are odd integers. The case k = 0 was studied by Christ [7] and further refined by Yu [35]. In view of Treves’ conjecture, it is interesting to investigate the cases remained open. The purpose of the present paper is to show that in the remaining cases, except for (k, l) = (0, 1), the operator P fails to be analytic hypoelliptic as well. This result is in agreement with Treves’ conjecture. Indeed, the Poisson strata of the characteristic variety = {ξ = xkη−xlτ = 0} of the operators (1.1) with k < l are as follows. For k ≥ 1 then 0 = {ξ = xkη − xlτ = 0, x = 0}, 1 = {x = ξ = 0, η = 0}, and 2 = {x = ξ = η = 0, τ = 0} which is not symplectic since its codimension is odd. For k = 0 and l ≥ 2 the strata are 0 = {ξ = η − xlτ = 0, x = 0} and 1 = {x = ξ = η = 0, τ = 0} which is not symplectic due to its odd codimension. By contrast, for k = 0 and l = 1 the characteristic variety is symplectic, and the operator is analytic hypoelliptic (see also [3]). 2. – Outline of proof The proof uses a standard reduction to ordinary differential equations: Lemma 1 ([24]). Let, for ζ ∈ C, (2.2) Pζ = − d 2 dx2 + (x − xζ ) . FAILURE OF ANALYTIC HYPOELLIPTICITY IN A CLASS OF PDOS 23 If there exist ζ ∈ C and f ∈ L∞(R), f = 0, satisfying

Hörmander systems and harmonic morphisms

Abstract: Given a Hormander system X = \lbrace X_1 , \cdots , X_m \rbrace on a domain \Omega \subseteq {\bf R}^n we show that any subelliptic harmonic morphism \phi from \Omega into a u -dimensional riemannian manifold N is a (smooth) subelliptic harmonic map (in the sense of J. Jost \& C-J. Xu, [9]). Also \phi is a submersion provided that u \le m and X has rank m. If \Omega = {\bf H}_n (the Heisenberg group) and X = \left\lbrace \frac{1}{2}\left( L_\alpha + L_{\overline{\alpha }}\right) , \frac{1}{2i}\left( L_\alpha - L_{\overline{\alpha }}\right)\right\rbrace , where L_{\overline{\alpha }} = \partial /\partial \overline{z}^\alpha - i z^\alpha \partial /\partial t is the Lewy operator, then a smooth map \phi : \Omega \rightarrow N is a subelliptic harmonic morphism if and only if \phi \circ \pi : (C({\bf H}_n ) , F_{\theta _0} ) \rightarrow N is a harmonic morphism, where S^1 \rightarrow C({\bf H}_n ) \overset{\pi }{\rightarrow }{\rightarrow } {\bf H}_n is the canonical circle bundle and F_{\theta _0} is the Fefferman metric of ({\bf H}_n , \theta _0 ). For any $S^1-invariant weak solution to the harmonic map equation on (C({\bf H}_n ) , F_{\theta _0}) the corresponding base map is shown to be a weak subelliptic harmonic map. We obtain a regularity result for weak harmonic morphisms from (C(\lbrace x_1 > 0 \rbrace ), F_{\theta (k)}) into a riemannian manifold, where F_{\theta (k)} is the Fefferman metric associated to the system of vector fields X_1 =\partial /\partial x_1 , X_2 = \partial /\partial x_2 + x_1^k \; \partial /\partial x_3$ \; (k \ge 1) on \Omega = {\bf R}^3 \setminus \lbrace x_1 = 0 \rbrace .

Holder a Priori Estimates for Second Order Tangential Operators on CR Manifolds

Abstract: On a real hypersurface M in \mathbb{C}^{n+1} of class C^{2,\alpha } we consider a local CR structure by choosing n complex vector fields W_j in the complex tangent space. Their real and imaginary parts span a 2n-dimensional subspace of the real tangent space, which has dimension 2n+1. If the Levi matrix of M is different from zero at every point, then we can generate the missing direction. Under this assumption we prove interior a priori estimates of Schauder type for solutions of a class of second order partial differential equations with C^\alpha coefficients, which are not elliptic because they involve second-order differentiation only in the directions of the real and imaginary part of the tangential operators W_j. In particular, our result applies to a class of fully nonlinear PDE’s naturally arising in the study of domains of holomorphy in the theory of holomorphic functions of several complex variables.

Boundary Regularity and Compactness for Overdetermined Problems

Abstract: Let D be either the unit ball B_1(0) or the half ball B_1^+(0), let f be a strictly positive and continuous function, and let u and \Omega \subset D solve the following overdetermined problem: \Delta u(x) = \chi _{_\Omega }(x) f(x) \ \ \text{in} \ \ D, \ \ \ \ 0 \in \partial \Omega , \ \ \ \ u = | abla u| = 0 \ \ \text{in} \ \ \Omega ^c, where \chi _{_\Omega } denotes the characteristic function of \Omega , \Omega ^c denotes the set D \setminus \Omega , and the equation is satisfied in the sense of distributions. When D = B_1^+(0), then we impose in addition that u(x) \equiv 0 \ \ \text{on} \ \ \lbrace \; (x^{\prime}, \; x_n) \; | \; x_n = 0 \; \rbrace \,. We show that a fairly mild thickness assumption on \Omega ^c will ensure enough compactness on u to give us “blow-up” limits, and we show how this compactness leads to regularity of \partial \Omega . In the case where f is positive and Lipschitz, the methods developed in Caffarelli, Karp, and Shahgholian (2000) lead to regularity of \partial \Omega under a weaker thickness assumption

On cubics and quartics through a canonical curve

Abstract: We construct families of quartic and cubic hypersurfaces through a canonical curve, which are parametrized by an open subset in a Grassmannian and a Flag variety respectively. Using G. Kempf’s cohomological obstruction theory, we show that these families cut out the canonical curve and that the quartics are birational (via a blowing-up of a linear subspace) to quadric bundles over the projective plane, whose Steinerian curve equals the canonical curve.

On q-Runge pairs

Abstract: We show that the converse of the aproximation theorem of Andreotti and Grauert does not hold. More precisely we construct a 4-complete open subset D \subset \mathbb{C}^6 (which is an analytic complement in the unit ball) such that the restriction map H^3(\mathbb{C}^6,\mathcal{F}) \rightarrow H^3(D, \mathcal{F}) has a dense image for every \mathcal{F} \in Coh(\mathbb{C}^6) but the pair (D, \mathbb{C}^6) is not a 4-Runge pair.

Families of differential forms on complex spaces

Abstract: On every reduced complex space X we construct a family of complexes of soft sheaves X ; each of them is a resolution of the constant sheaf CX and induces the ordinary De Rham complex of differential forms on a dense open analytic subset of X . The construction is functorial (in a suitable sense). Moreover each of the above complexes can fully describe the mixed Hodge structure of Deligne on a compact algebraic variety. Mathematics Subject Classification (2000): 32C15 (primary), 32S35 (secondary).

A combinatorial approach to singularities of normal surfaces

Abstract: In this paper we study generic coverings of \mathbb{C}^2 branched over a curve s.t. the total space is a normal analytic surface, in terms of a graph representing the monodromy of the covering, called monodromy graph. A complete description of the monodromy graphs and of the local fundamental groups is found in case the branch curve is \lbrace x^n=y^m\rbrace (with n\le m) and the degree of the cover is equal to n or n-1.

On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two

Abstract: In this paper we establish a Liouville type theorem for fully nonlinear elliptic equations related to a conjecture of De Giorgi in R2. We prove that if the level lines of a solution have bounded curvature, then these level lines are straight lines. As a consequence, the solution is one-dimensional. The method also provides a result on free boundary problems of Serrin type.