Showing papers in "Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze in 2000"

Optimal stability for inverse elliptic boundary value problems with unknown boundaries

Abstract: In this paper we study a class of inverse problems associated to elliptic boundary value problems. More precisely, those inverse problems in which the role of the unknown is played by an inaccessible part of the boundary and the role of the data is played by overdetermined boundary data for the elliptic equation assigned on the remaining, accessible, part of the boundary. We treat the case of arbitrary space dimension n > 2. Such problems arise in applied contexts of nondestructive testing of materials for either electric or thermal conductors, and are known to be ill-posed. In this paper we obtain essentially best possible stability estimates. Here, in the context of ill-posed problems, stability means the continuous dependence of the unknown upon the data when additional a priori information on the unknown boundary (such as its regularity) is available. Mathematics Subject Classification (2000): 35R30 (primary), 35R25, 35R35, 35B60, 31B20 (secondary). 1. Introduction In this paper we shall deal with two inverse boundary value problems. Suppose Q is a bounded domain in W with sufficiently smooth boundary a S2, a part of which, say I (perhaps some interior connected component of a S2 or some inaccessible portion of the exterior component of a03A9), is not known. This could be the case of an electrically conducting specimen, which is possibly defective due to the presence of interior cavities or of corroded parts, which are not accessible to direct inspection. See for instance [K-S-V]. The aim is to detect the presence of such defects by nondestructive methods collecting current and voltage measurements on the accessible part A of the boundary If we assume that the inaccessible part I of a 03A9 is electrically insulated, then, given a nontrivial function 1/1 on A, having zero average (which represents Work supported in part by MURST. Pervenuto alla Redazione il 21 settembre 1999 e in forma definitiva il 24 giugno 2000. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) Vol. XXIX (2000), pp. 755-806 756 the assigned current density on the accessible part A of 8Q), we have that the voltage potential u inside Q satisfies the following Neumann type boundary value problem Here, v is the exterior unit normal to aS2 and cr = denotes the known symmetric conductivity tensor ant it is assumed to satisfy a hypothesis of uniform ellipticity. Let us remark that the solution to ( 1.1 a )-( 1.1 c) is unique up to an undetermined additive constant. In order to specify a single solution, we shall assume, from now on, the following normalization condition Suppose, now, that E is an open subset of which is contained in A, and on which the voltage potential can be measured. Then, the inverse problem consists of determining I provided is known. This is the first object of our study and we shall refer to it as the Inverse Neumann Problem (Neumann case, for short). An allied problem is the one associated to the direct Dirichlet problem Here, as above, I, A are the inaccessible, respectively, accessible, parts of and ar is the conductivity tensor satisfying the same hypotheses. Our second object of study is the inverse problem consisting in the determination of I from the knowledge of orVM’ where E C A is as above. We shall refer to it as the Inverse Dirichlet Problem (Dirichlet case, for short). We believe that also this problem may be of interest for concrete applications of nondestructing testing, for instance in thermal imaging. In this case, the inaccessible boundary I could represent a priviledged isothermal surface, such as a solidification front. Of course, it should be kept in mind that, dealing with thermal processes, the evolutionary model based on parabolic, rather than elliptic, equations is in general more appropriate, for related issues see, for instance, [B-K-W], [Bi], [V 1 ] . However, we trust that also a preliminary study of a stationary model may be instructive. Such two problems, the Neumann and Dirichlet cases, are known to be ill-posed. Indeed there are examples that show that, under a priori assumptions on the unknown boundary I regarding its regularity (up to any finite order of differentiability), the continuous dependence (stability) of I from the measured data in the Neumann case, orVM’ VIE in the Dirichlet case) is, at best, of logarithmic type. See [A12] and also [Al-R]. 757 The main purpose of this paper is to prove stability estimates of logarithmic type (hence, best possible) for both the Neumann and Dirichlet cases, (Theorems 2.1, 2.2), when the space dimension n > 2 is arbitrary. We recall that, for the case n = 2, results comparable to ours have been found in [Be-V] when o~ is homogeneous and in [R], [Al-R] when o~ can be inhomogeneous and also discontinuous. Other related results for the case of dimension two can be found in [Bu-C-Yl], [Bu-C-Y2], [Bu-C-Y3], [Bu-C-Y4], [An-B-J]. Let us also recall that, typically, the above mentioned results are based on arguments related, in various ways, to complex analytic methods, which do not carry over the higher dimensional case. In the sequel of this Introduction, we shall illustrate the new tools we found necessary to develop and exploit when n > 2. But first, let us comment briefly on the connection with another inverse problem which has become quite popular in the last ten years, namely the inverse problem of cracks. On one hand there are similarities, in fact a crack can be viewed as a collapsed cavity, that is a portion of surface inside the conductor, such that a homogeneous Neumann condition like (I,lc) holds on the two sides of the surface. On the other hand there are differences, for the uniqueness in the crack problem at least two appropriate distinct measurements are necessary [F-V], whereas for our problems, either the Neumann or the Dirichlet case, any single nontrivial measurement suffices for uniqueness, see for instance [Be-V] for a discussion of the uniqueness issue. Let us also recall that for the crack problem in dimensions bigger than two, various basic problems regarding uniqueness are still unanswered. See, for the available results and for references [Al-DiB ] . It is therefore clear that a study of the stability for the crack problem in dimensions higher than two shall require new ideas. Nonetheless, the authors believe that the techniques developed here might be useful also in the treatment of the crack problem. The methods we use in this paper are based essentially on a single unifying theme: Quantitative Estimates of Unique Continuation, and we shall exploit it under various different facets, namely the following ones. (a) Stability Estimates of Continuation from Cauchy Data. Since we are given the Cauchy data on E for a solution u to (l.la), we shall need to evaluate how much a possible error on such Cauchy data can affect the interior values of u. Such stability estimates for Cauchy problems associated to elliptic equations have been a central topic of ill-posed problems since the beginning of their modem theory, [H], [Pul], [Pu2]. Here, since one of our underlying aims will be to treat our problems under possibly minimal regularity assumptions, we shall assume the conductivity cr to be Lipschitz continuous (this is indeed the minimal regularity ensuring the uniqueness for the Cauchy problem, [PI], [M]). Our present stability estimates (Propositions 3.1, 3.2, 4.1, 4.2) will elaborate on inequalities due to Trytten [T] who developed a method first introduced by Payne [Pal], [Pa2]. The additional difficulty encountered here will be that we shall need to compare solutions u 1, u 2 which are defined on possibly different domains Q2 758 whose boundaries are known to agree on the accessible part A only. Let us recall that a similar approach, but restricted to the topologically simpler two-dimensional setting, has already been used in [All], [Be-V]. We shall obtain that, if the error on the measurement on the Cauchy data is small, then for the Neumann case, also IVUII I is small, in an L2 average sense, on S21 B Q2, the part of S21 which exceeds Q2. And the same holds for on SZ2 BQI (Propositions 3.1, 3.2). In the Dirichlet case instead we shall prove that u 1 itself is small in S21 B Q2, and the same holds for u2 on S22 (Propositions 4.1, 4.2). (b) Estimates of Continuation from the Interior. We shall also need interior average lower bounds on u and on its gradient (Propositions 3.3, 4.3), on small balls contained inside S2. Bounds of this type have been introduced in [Al-Ros-S, Lemma 2.2] in the context of a different inverse boundary value problem. The tools here involve another form of quantitative unique continuation, namely the following. (c) Three Spheres Inequalities. Also this one is a rather classical theme in connection with unique continuation. Aside from the classical Hadamard’s three circles theorem, in the context of elliptic equations we recall Landis [La] and Agmon [Ag]. Under our assumptions of Lipschitz continuity on a, our estimates (see (5.47) below) shall refer to differential inequalities on integral norms originally due to Garofalo and Lin [G-L], later developed by Brummelhuis [Br] and Kukavica [Ku]. (d) Doubling Inequalities in the Interior. This rather recent tool has been introduced by Garofalo and Lin in the above mentioned paper [G-L]. It provides an efficient method of estimating the local average vanishing rate of a solution to (l.la). Let us recall that it also provides a remarkable bridge to the powerful theory of Muckenhoupt weights [C-F] and that this last connection has been crucially used in [Al-Ros-S] and also in [V2]. The last, fundamental, appearance of quantitative estimates of unique continuation is the following. (e) Doubling Inequalities at the Boundary. For our purposes it will be crucial to evaluate the vanishing rate of Vu (in the Neumann case) or of u (in the Dirichlet case) near the inaccessible boundary I. In particular, the fact that such a rate is not worse than polynomial (Propo

A relative Dobrowolski lower bound over abelian extensions

Abstract: Let a be a non-zero algebraic number, not a root of unity. A wellknown theorem by E. Dobrowolski provides a lower bound for the Weil height h(a) which, in simplified form, reads h(a) » D-l-’, where D = [Q(a) : Q]. On the other hand, F. Amoroso and R. Dvomicich have recently found that if a lies in an abelian extension of the rationals, then h (a) is bounded below by a positive number independent of D. In the present paper we combine these results by showing that, in the above inequality, D may be taken to be the degree of a over any abelian extension of a fixed number field. As an application, we also derive a new lower bound for the Mahler measure of a polynomial in several variables, with integral coefficients. Mathematics Subject Classification (2000):11 G50 (primary), I I Jxx (secondary). 1. Introduction Let a be a non zero algebraic number which is not a root of unity. Then, by a theorem of Kronecker, the absolute logarithmic Weil height h(a) is > 0. More precisely, let K be any number field containing a. By using Northcott’s theorem (see [No]), it is easy to see that h (a) ~: C(K), where C(K) > 0 is a constant depending only on K. In other words, 0 is not an accumulation point for the height in K. In a remarkable paper, Lehmer [Le] asked whether there exists a positive absolute constant Co such that This problem is still open, the best unconditional lower bound in this direction being a theorem of Dobrowolski [Do], who proved: Pervenuto alla Redazione il 11 novembre 1999 e in forma definitiva il 2 maggio 2000. 712 for some absolute constant Cl 1 > 0. However, in some special cases not only the inequality conjectured by Lehmer is true, but it can also be sharpened. Assume for instance that is an abelian extension of the rational field. Then the first author and R. Dvomicich proved in [Am-Dv] the inequality: (notice that such a result was obtained long time ago as a special case of a more general result, by Schinzel (apply [Sch], Corollary 1’, p. 386, to the linear polynomial P (z ) = z a), but with the extra assumption 1). The aim of this paper is to generalise both a result of this type and Dobrowolski’s result, to obtain: THEOREM l.1. Let K be any number field and let L be any abelian extension of K. Then for any nonzero algebraic number a which is not a root of unity, we have where D = [L (a) : IL] and C2 (K) is a positive constant depending only on K. This result implies that heights in abelian extensions behave somewhat specially. In this respect, it may not be out of place to recall the following result recently obtained jointly by E. Bombieri and the second author. Let K be a number field and consider the compositum L of all abelian extensions of K D. Then Northcott’s theorem (see [No]) holds in L, that is, for given T, the number of elements of L with height bounded by T is finite. The main ingredients for the proof of Theorem 1.1 (given in Section 5) are a generalisation of Dobrowolski’s key inequality (Section 3, Proposition 3.4), which follows as an extension of some ideas from [Am-Dv], and a consequence of the absolute Siegel Lemma of Roy-Thunder-Zhang-Philippon-David (Section 4, Proposition 4.2). Let F E xn ] be an irreducible polynomial. Define its Mahler measure as Then it is known (see [Bo], [Law] and [Sm]) that M(F) = 1 if and only if F is an extended cyclotomic polynomial, i.e. if and only if for some and for some cyclotomic polynomial 713 If n = 1 and a is a root of F, then log M ( F ) = deg ( F ) ~ h (a ) ; hence, if F is not a cyclotomic polynomial, by the quoted result of Dobrowolski. Recently the first author and S. David ([Am-Da2]) extended this result in several variables. Assume that F is not an extended cyclotomic polynomial. Then, as a special case of Corollaire 1.8 of [Am-Da2], where C3 is a positive absolute constant and D = deg(F). Using Theorem 1.1 and a density result from [Am-Da2], we can now prove (see Section 6): COROLLARY 1.2. Let F E Z [X 1, , xn] be an irreducible polynomial. Assume that F is not an extended cyclotomic polynomial and that d = degxj ( F) > 1. Then This estimate is stronger than the quoted result of [Am-Da2] if at least one of the partial degrees of F is small. We also notice that the constant in Corollary 1.2 does not depend on the dimension n. 2. Notation and Reductions Throughout this paper, we denote by ~m (m > 3) a primitive m-th root of unity and by ti the set of all roots of unity. We also fix a number field K and we denote by P the set of rational primes p > 3 which split completely in K. For p E P we choose once and for all a prime ideal np of OK lying above p and we identify Jrp with the corresponding place of K. We normalize the corresponding valuation v so that Iplv = p Let L be any abelian extension of K. For a given p E P, we define ep (L) as the ramification index of 7rp in L. Let v be any valuation of L extending np . Since L/K is normal, the completion Lv of L at v depends only on p. Since p splits completely in K we also have Kx = Qp. Then Lv is an abelian extension of Qp and so Lv is contained in a cyclotomic extension of Qp (see e.g. [Wa], p. 320, Theorem 14.2), which we denote by We take m = to be minimal with this property and we define as the maximal power 714 of p dividing m. We remark that = 1 for all but finitely many p E P (if 7rp does not ramify in L, then IL" for some integer m with p f m: see [Wa], p. 321, Lemma 14.4 (a)). We also define We note that if L’ C L are abelian extensions of K, then e’ (IL’ ) e’(L). From now on we let C2(K) be a positive real number sufficiently small to justify the subsequent arguments. Let L be an abelian extension of K and let be a nonzero algebraic number which contradicts Theorem 1.1: where D = [L(a) : L]. We may assume that D is minimal with this property, i.e. that for any JL of degree D’ D over an abelian extension of K, we have: Notice that t ~-+ t (log (2t) / log log (5t)) 13 is an increasing function on [1, +(0) and that the Weil height is invariant by multiplication by roots of unity. Hence (2.1) and (2.2) imply that: Let A be the set of abelian extensions L/K such that for some ~ E JL. We define Replacing if necessary a by Ça for some ~ and L by L n K(a), we may assume that there exists an abelian extension L/K contained in K(a) satisfying the following properties: From now on we fix once and for all an algebraic number a and an abelian extension L/K contained in K(a) which satisfy (2.1), (2.2), (2.3), (2.5), (2.6), (2.7), and (2.8). We also put ep = ep (L) for p E P. The following lemma will be used several times in the next section. 715 LEMMA 2. 1. i) For any integer n we have L (an) = L (a). ii) For any integern such thatgcd(n, [K(a): =1 we also have = K (a). PROOF. We shall use an argument from Rausch (see [Ra], Lemma 3). Assume first for some n E N. The minimal polynomial of a over is a divisor of an. Hence its constant term, say f3 E IL(an), can be written as Çar, where ~ is a n-th root of unity. Moreover, and P V 1L. Hence, by (2.2), Since rD’ D and t H log(2t)/ log log(5t) increases, we deduce that which contradicts (2.1 ) . Assume now r := [K(a) : > 1 and gcd(n, r) = 1. Arguing a: before, we find a n-th root ~ such that E By B6zout’s identity there exist h, it E Z such that hn + J-tr = 1. Hence This contradicts (2.7). 0 3. Congruences The following two lemmas generalise Lemma 2 of [Am-Dv]. We first prove a result which shall be applied to tamely ramified primes. LEMMA 3.1. Let p E P. Then there exists 4$p E Gal(L/K) such that for any integer y E L and for any valuation v extending 7rp. 716 PROOF. Lest 9 be a prime of L above 7rp and let G’ c Gal(L/K) (resp. I c Gal(L/K)) be the decomposition (resp. inertia) group np. Since G’/I is isomorphic to the Galois group of the residue field extension there exists 4Sp E G’ such that for all integers y E L. Let or E Gal(L/K). Putting in place of y in this congruence and applying cr we get On the other hand Gal(L/K) is abelian, and therefore the first displayed congruence holds modulo each prime of L lying over np. Lemma 3.1 follows. D We now consider primes having a large ramification index. LEMMA 3.2. Let p E P. Then there exists a subgroup Hp of order

A descendent relation in genus 2

Abstract: A new codimension 2 relation among descendent strata in the moduli space of stable, 3-pointed, genus 2 curves is found. The space of pointed admissible double covers is used in the calculation. The resulting differential equations satisfied by the genus 2 gravitational potentials of varieties in Gromov-Witten theory are described. These are analogous to the WDVV-equations in genus 0 and Getzler's equations in genus 1. As an application, genus 2 descendent invariants of the projective plane are determined, including the classical genus 2 Severi degrees.

Renormalized entropy solutions of scalar conservation laws.

Abstract: A scalar conservation law ut +div (u) = f is considered with the initial datum u|t=0 = u0 2 L1 loc(RN) and f 2 L1 loc(RN ×(0, T)) only. In this case the classical Krushkov condition can make no sense because of unboundedness of u, if no growth condition on is assumed. This obstacle is overcome by introducing the so-called renormalized entropy solution generalizing the classical one. Existence and uniqueness of such a solution is established.

Blow-up results for nonlinear hyperbolic inequalities

Abstract: We study the nonexistence of global weak solutions for equations and systems of the following types (1) + lu Iq and (II) att u > Lm + where the operators L m and L m are homogeneous linear partial differential operators of order 2m and 2m1. The method relies on a suitable choice of test functions, rescaling techniques and a dimensional analysis. Mathematics Subject Classification (1991): 35L60.

A parabolic quasi-variational inequality arising in a superconductivity model

Abstract: We consider the existence of solutions for a parabolic quasilinear problem with a gradient constraint which threshold depends on the solution itself. The problem may be considered as a quasi-variational inequality and the existence of solution is shown by considering a suitable family of approximating quasilinear equations of p-Laplacian type. A priori estimates on the time derivative of the approximating solutions and on the nonlinear diffusion coefficients are used in the passage to the limit, as well as a suitable sequence of convex sets with variable gradient constraint. The asymptotic behaviour as t -~ oc is also considered, and the solutions of the quasi-variational inequality are shown to converge, at least for subsequences, to a solution of a stationary quasi-variational inequality. These results can be applied to the critical-state model of type-II superconductors in longitudinal geometry. Mathematics Subject Classification (1991): 35K85 (primary), 35K55, 35R35 (secondary). 1. Introduction In a critical-state model of type-II superconductors with a longitudinal geometry, the main unknown is the magnetic field H = (o, 0, u (x, t)), where x = (XI, x2) E S2 c JR2. By Maxwell’s equations, where E denotes the electric field, the unknown density of the induced current J is given by , In some superconductivity models (see [3]), it is assumed that the electric field E inside the superconductor depends on the current density J through This work was partially supported by the project PRAXIS/2/2.1/MAT/125/94. Pervenuto alla Redazione il 15 luglio 1999. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) Vol. XXIX (2000), pp. 153-169 154 a power law (which generalizes the classical Ohm’s law E = p J) with the scalar resistivity given by p = P(Ijl) = po1JIP-2 = where po > 0 is a constant and p > 2. Hence, the Maxwell’s equation for the longitudinal component of the magnetic field implies that u = u (x, t) satisfies the twodimensional quasilinear parabolic equation In Bean’s critical-state model (see [ 11 ]), the current density cannot exceed the critical value jc > 0 and it has been suggested that this threshold may also depend on )H) I (see Kim et al. [6]). The constitutive relation for E is then modified to being ~. > 0 an unknown Lagrange multiplier. Imposing initial and boundary conditions (in a bounded domain S2 in R 2), it may be easily shown that this problem is equivalent to the following quasivariational inequality (for details about this derivation see [II], where only the degenerate case po = 0 was considered) to a new variable h = u he, being where h e = he(t) is related to the density of external currents, as well as f ’ . This model is the main motivation for the study of this new type of quasivariational inequalities. This kind of problems seem not well studied in the literature and are not considered, for instance, in the classical references on quasi-variational inequalities [1] or [2]. Recent works in this area are, for example, the modelling works [10], [11] or a very special case of a onedimensional quasi-variational inequality considered in [7]. In Section 2 we present the main results of this paper, a result on the existence of solutions for the quasi-variational inequality and a result about the asymptotic behaviour in time of these solutions. In Section 3 we consider a family of approximating solutions and we establish apriori estimates. Section 4 includes the passage to the limit in the family of solutions of the approximated problems, concluding with the proof of the existence of at least a solution for the quasi-variational inequality in the N-dimensional case. Section 5 studies the asymptotic behaviour in time of the solutions, which, in particular, also yields the existence of a solution to the corresponding elliptic quasi-variational inequality, extending a result of [8]. 155 2. Main results Let Q be a bounded, open subset of R , with smooth boundary and let Let denote the spatial gradient and suppose that F, f’ and h are ---1 1 given functions such that where denotes the p-Laplacian, and denote the spaces of bounded measures in S2 and Q T , respectively, and We also denote by the space of Lipschitz functions that vanish on Given i we can define a convex set and the quasi-variational problem as follows: To find u in an appropriate class of functions, such that: Our first aim is to prove existence of a solution to the problem (5). For that, we consider, for 8 > 0, fE, h, and F, smooth functions approximating, respectively, f, h and F in the norms of for any q o0 W6, (0), and in , with independent of s) and a CL function R R, non decreasing and such that We consider a sequence of the following approximated problems 156 We noticed that a similar approximation was introduced by Gerhardt (see [5]) in the treatment of the elastic-plastic torsion problem with multiply connected cross-section, which is an elliptic variational inequality and it was also used for studying a parabolic variational inequality with gradient constraint in [12]. THEOREM 1. With the assumptions (1), (2) and (3), for any 1 p 00, the problem (5) has at least a solution u belonging to T ; n C°(QT), such that ut e T; M(Q)). In addition, u is the weak limit in T ; (for any q of a of solutions of the family of approximated problems (7), u£n ~~ u in C°(QT) and also utn -1 n T ; M(Q)) weak-*. REMARK 1. Since _u is continuous and bounded, for each t e [0, T ], C C and the first integral of the left hand side of (5) should be understood in the duality sense between and taking into account that is a.e. a bounded measure. Consider now the corresponding stationary quasi-variational inequality: where we assume F satisfies (1) and Existence of a solution of the problem (8) may be proved as in [3]. We may also consider the problem (5) with T = +00. Then we have the following THEOREM 2. Suppose that the assumption (2) is satisfied for T = +00, i. e., there exists C > 0 such that

Smoluchowski's coagulation equation : probabilistic interpretation of solutions for constant, additive and multiplicative kernels

Abstract: This paper is devoted to the study of the Smoluchowski’s coagulation equation, discrete and continuous version, for the case of constant, additive and multiplicative kernels. Even though, for the discrete case the results stated in this work are not new, our approach allows the simplification of existing proofs. For the continuous case we obtain new results: a connection between the solutions of the additive and multiplicative cases and renormalisation theorems which show that after a convenient scaling, the solution converges to a limit which depends on the initial condition only through its moments of order 1, 2 and 3. Mathematics Subject Classification (2000): 60J80, 44A10.

A relaxation theorem in the space of functions of bounded deformation

Abstract: We obtain an integral representation for the relaxation, in the space of func- tions of bounded deformation, of the energy Z f(Eu(x))dx with respect to L1-convergence. Here Eu represents the absolutely continuous part of the symmetrized distributional derivative Eu and the function f satisfies linear growth and coercivity conditions.

Finite difference approximation of energies in Fracture Mechanics

Abstract: We provide a variational approximation by finite-difference energies of functionals of the type defined for u E SBD(Q), which are related to variational models in ’ fracture mechanics for linearly-elastic materials. We perform this approximation in dimension 2 via both discrete and continuous functionals. In the discrete scheme we treat also boundary value problems and we give an extension of the approximation result to dimension 3. Mathematics Subject Classification (2000): 49J45 (primary), 49M25, 74R10 (secondary). 1. Introduction In this paper we provide a variational approximation by discrete energies of functionals of the type defined for every closed hypersurface K c Q with normal v and u E B K; Ilgn), where Q C is a bounded domain of Here -emu = + . 2 denotes the symmetric part of the gradient of u, [u ] is the jump of u through K along v and is the (n I)-dimensional Hausdorff measure. These functionals are related to variational models in fracture mechanics for linearly elastic materials in the framework of Griffith’s theory of brittle fracture Pervenuto alla Redazione il 10 novembre 1999 e in forma definitiva il 13 aprile 2000. 672 (see [33]). In this context u represents the displacement field of the body, with SZ as a reference configuration. The volume term in (1.1) represents the bulk energy of the body in the \"solid region\", where linear elasticity is supposed to hold, JL, À being the Lame constants of the material. The surface term is the energy necessary to produce the fracture, proportional to the crack surface K in the isotropic case and, in general, depending on the normal v to K and on the jump [u]. The weak formulation of the problem leads to functionals of the type defined on the space SBD(Q) of integrable functions u whose symmetrized distributional derivative Eu is a bounded Radon measure with density Fu with respect to the Lebesgue measure and with singular part concentrated on an (n I)-dimensional set Ju, on which it is possible to define a normal vu in a weak sense and one-sided traces. The description of continuum models in Fracture Mechanics as variational limits of discrete systems has been the object of recent research (see [17], [19], [20], [15] and [36]). In particular, in [19] an asymptotic analysis has been performed for discrete energies of the form where Rs is the portion of the lattice sZ’ of step size > 0 contained in S2 and u : Rs R’ may be interpreted as the displacement of a particle parameterized by x E In this model the energy of the system is obtained by superposition of energies which take into account pairwise interactions, according to the classical theory of crystalline structures. Upon identifying u in (1.3) with the function in L 1 constant on each cell of the lattice the asymptotic behaviour of functionals can be studied in the framework of r-convergence of energies defined on L 1 (see [25], [23]). A complete theory has been developed when u is scalar-valued; in this case the proper space where the limit energies are defined is the space of SB V functions (see for instance [24]). An important model case is when w) In this case we may rewrite He as where R~ is a suitable portion of R, and D! u (x) denotes the difference quotient ~ (u (x + 8~) M(~r)). Functionals of this type have been studied also in [22] in the framework of computer vision. In [22] and, in a general framework, in [19] it has been proved that, if f (t) = min{t, 1 } and p is a positive function with 673 suitable summability and symmetry properties, then R, approximates functionals of the type defined for u E SB V (S2), which are formally very similar to that in (1.2). A similar result holds by replacing min{t, 1 { by any increasing function f with /(0) = 0, > 0 and f (oo) = b -E-oo. Following this approach, in order to approximate (1.2), one may think to \"symmetrize\" the effect of the difference quotient by considering the family of functionals ..----.......---... 11 ....... By letting 8 tend to 0, we obtain as limit a proper subclass of functionals (1.2). Indeed, the two coefficients It and k of the limit functionals are related by a fixed ratio. This limitation corresponds to the well-known fact that pairwise interactions produce only particular choices of the Lame constants. To overcome this difficulty we are forced to take into account in the model non-central interactions. The idea underlying this paper is to introduce a suitable discretization of the divergence, call it div~u, that takes into account also interactions in directions orthogonal to ~, and to consider functionals of the form with 8 a strictly positive parameter (for more precise definitions see Sections 3 and 7). In Theorem 3.1 we prove that with suitable choices of f, p and 9 we can approximate functionals of type (1.2) in dimension 2 and 3 with arbitrary and (D satisfying some symmetry properties due to the geometry of the lattice. Actually, the general form of the limit functional is the following with W explicitly given; in particular we may choose W (~u (x ) ) and c = 2. We underline that the energy density of the limit surface term is always anisotropic due to the symmetries of the lattices The dependence on [u], vu arises in a natural way from the discretizations chosen and the vectorial framework of the problem. To drop the anisotropy of the limit surface energy we consider as well a continuous version of the approximating functionals (1.5) given by 674 where in this case p is a symmetric convolution kernel which corresponds to a polycrystalline approach. By varying f, p and 8, as stated in Theorem 3.8, we obtain as limit functionals of the form for any choice of positive constants ~c, À and y. This continuous model generalizes the one proposed by E. De Giorgi and studied by M. Gobbino in [31], to approximate the Mumford-Shah functional defined for U E SBV(Q). The main technical issue of the paper is that, in the proof of both the discrete and the continuous approximation, we cannot reduce to the 1-dimensional case by an integral-geometric approach as in [ 19], [22], [31], due to the presence of the divergence term. For a deeper insight of the techniques used we refer to Sections 4 and 5; we just underline that the proofs of the two approximations (discrete and continuous) are strictly related. Analogously to [19], in Section 7 we treat boundary value problems in the discrete scheme for the 2-dimensional case and a convergence result for such problems is derived (see Proposition 6.3 and Theorem 6.4). ACKNOWLEDGMENTS. Our attention on this problem was drawn by Andrea Braides, after some remarks by Lev Truskinovsky. We also thank Luigi Ambrosio and Gianni Dal Maso for some useful remarks. This work is part of CNR Research Project \"Equazioni Differenziali e Calcolo delle Variazioni\". Roberto Alicandro and Maria Stella Gelli gratefully acknowledge the hospitality of Scuola Normale Superiore, Pisa, and Matteo Focardi that of SISSA, Trieste. 2. Notation and preliminaries We denote by (., .) the scalar product in I. I will be the usual euclidean norm. For x, y E JRn, [x, y] denotes the segment between x and y. If a, b E R we write a A b and a V b for the minimum and maximum between a and b, respectively. If ~ = (~1, ~z) E II~2, we denote the vector in R2 orthogonal . to ~ defined (-~z, ~ 1). ’ If Q is a bounded open subset of and are the families of open and Borel subsets of Q, respectively. If it is a Borel measure and B 675 is a Borel set, then the measure It L B is defined as n B). We denote by .en the Lebesgue measure in R’ and by Hk the k-dimensional Hausdorff measure. If B c R’ is a Borel set, we will also use the notation IBI ] for L’(B). The notation a.e. stands for almost everywhere with respect to the Lebesgue measure, unless otherwise specified. We use standard notation for Lebesgue spaces. We recall also the notion of convergence in measure on the space L 1 (S2; R\"). We say that a sequence un converges to u in measure if for every 17 > 0 we have limn e ~ : un (x) u(x)] I > = 0. The space L1(Q; when endowed with this convergence, is metrizable, an example of metric being for 2.1. BV and BD functions Let Q be a bounded open set of JRn. If u E L I (Q; we denote by Su the complement of the Lebesgue set of u, i.e. x §t Su if and only if for some .z E If z exists then it is unique and we denote it by E(x). The set Su is Lebesgue-negligible and u is a Borel function equal to u .en a.e. in SZ. Moreover, we say that x E Q is a jump point of u, and we denote by Ju the set of all such points for u, if there exist a, b E R\" and v E sn-1 I such that a ~ b and where j A’ The triplet (a, b, v), uniquely determined by (2.1 ) up to a permutation of (a, b) and a change of sign of v, will be denoted by ( u + (x ) , u (x ) , vu (x ) ) . Notice that Ju is a Borel subset of Su. We say that u is approximately differentiable at a Lebesgue point x if there exists L E such that If u is approximately differentiable at a Lebesgue point x, then L, uniquely determined by (2.2), will be denoted by Vu(x) and will be called the approximate gradient of u at x. 676 Eventually, given a Borel set J C R’ , we say that J is if where ?-~n-1 (N) = 0 and each Ki is a compact subset of a C 1 (n 1) dimensional manifold. Thus, for a 1tn-1-rectifiable set J it is possible to define 1tn-1 a.e. a unitary normal vector field v. 2.1.1. BV functions ’ We recall some definitions and basic results on functions with bounded variation. For a detailed study of the properties of these functions we refer to [9] (see also [26], [30]). DEFINITION 2. l. Let u E L 1 (S2; II~N); we say that u is a function with bounded variation in Q, and we write u E B V (Q; if the distributional derivative Du of u is a N x n matrix-valued measure on Q with finite total variation. If u E BV(Q; R N), then u is approximately differentiable fn a.e. in Q and Ju turns out to be 1tn-1-rectifiable. Let us consider the Lebesgue decomposition of Du with respect to ,Cn, i.e., Du

Lyapunov center theorem for some nonlinear PDE's : a simple proof

Abstract: We give a simple proof of existence of small oscillations in some nonlinear partial differential equations. The proof is based on the Lyapunov-Schmidt decomposition and the contraction mapping principle; the linear frequencies Wj are assumed to satisfy a Diophantine type nonresonance condition (of the kind of the first Melnikov condition) slightly stronger than the usual one. with d > 1, such Diophantine condition will be proved to have full measure in a sense specified below; if d = 1, we will prove that the condition is satisfied in a set of zero measure. Applications to nonlinear beam equations and to nonlinear wave equations with Dirichlet boundary condition are given. The result also applies to more general systems and boundary conditions (e.g. periodic). Mathematics Subject Classification (2000) : 35B 10 (primary), 35B32, 37K55 (secondary).

An eigenvalue problem related to Hardy’s $L^P$ inequality

Abstract: Let S2 be a smooth bounded domain in R’~. We study the relation between the value of the best constant for Hardy’s LP inequality in Q, denoted by J1p(Q), and the existence of positive eigenfunctions in W6’ (Q), for an associated singular eigenvalue problem (EL) for the p-Laplacian. It is known that, in smooth cp = ( 1 and cp is the value of the best constant in the one-dimensional case. In the first part of the paper, we show that, for arbitrary p > 1, J1p (Q) = cp if and only if (EL) has no positive eigenfunction and discuss the behaviour of the positive eigenfunction of (EL) when J1p(Q) cp. This extends a result of [18] for p = 2. In the second part of the paper, we discuss a family of related variational problems as in [5], and extend the results obtained there for p = 2, to arbitrary p > 1. Mathematics Subject Classification (2000): 49R05 (primary), 35J70 (secondary). 1. Introduction Let S2 be a proper subdomain of R’ and p E (I, cxJ). We shall say that Hardy’s LP inequality holds in S2 if there exists a positive constant cH = such that,

The construction of principal spectral curves for Lane-Emden systems and applications

Abstract: In this article we develop a detailed study about the existence of principal eigenvalues for the Lane-Emden system when is a smooth bounded domain, a and f3 are positive numbers with af3 = 1, p and t are non-negative functions on Q and ,Ci is a general elliptic differential operator of second order. We show that the set of the principal eigenvalues of the system above determines a curve in the plane which satisfies several properties such as simplicity, isolation, continuity, asymptotic behaviour. We also furnish a min-max type characterization for this curve. Motivated by these discussions, we investigate the existence and uniqueness of positive solutions for some semilinear elliptic systems in bounded domains and in the whole space. Mathematics Subject Classification (1991): 35P30 (primary), 35J45, 35J55 (secondary.

Decay of Fourier transforms and summability of eigenfunction expansions

Abstract: Fourier coefficients f) of piecewise smooth functions are of the order of and Fourier series exp(2ninx) converge everywhere. Here we consider analogs of these results for eigenfunction expansions f (x) = where {À2} and are eigenvalues and an orthonormal complete system of eigenfunctions of a second order positive elliptic operator on a N-dimensional manifold. We prove that the norms of projections of piecewise smooth functions on subspaces generated by eigenfunctions with A h A + 1 satisfy the estimates I Then we give some sharp results on the Riesz summability of Fourier series. In particular we prove that the Riesz means of order 8 > (N 3)/2 converge. Mathematics Subject Classification (2000): 42C 15. Let M be a smooth manifold of dimension N and let A be a second order positive elliptic operator on M, with smooth real coefficients. Assume that this differential operator with suitable boundary conditions is self adjoint with respect to some positive smooth density d ¡.,¿ and admits a sequence of eigenvalues f;,21 and a system of eigenfunctions orthonormal and complete in JL2(M, Then to every function in one can associate a Fourier transform and a Fourier series, These Fourier series converge in the metric of JL2(M, dit) and more generally in the topology of distributions, but under appropriate conditions the PPrvPnmtn alla R eda7inn« il nttnhre 10QQ

Characterization of homogeneous gradient young measures in case of arbitrary integrands

Abstract: In the case of a continuous integrand L : R n m —> RU {00} and a probability measure v supported in R n m we indicate conditions both necessary and sufficient for this measure to be generated as a homogeneous Young measure by gradients of piece-wise affine functions Uk € I A + Wo'°°(Q) with the property L(Duk) -* (L;v) in L(Q). Here A is the center of mass of v and I A is a linear function with gradient equal to A everywhere. We show also that in the scalar case m = 1 any probability measure with finite action on L has this property. We provide elementary proofs of these results.

Inégalité de Vojta en dimension supérieure

Abstract: Resume On donne une version completement explicite du resultat de Faltings sur la repartition des points algebriques d'une sous-variete d'une variete abelienne, qui entraine la conjecture de Lang. Les methodes generalisent celles utilisees par Bombieri pour la conjecture de Mordell.

Some remarks about a notion of rearrangement

Abstract: We consider an extension to n-dimensions of the notion of increas- ing rearrangement for functions of one variable, and study the behaviour with respect to this operation of some classes of integral functionals. Among other applications, we obtain a simple direct proof of the existence and uniqueness of n-dimensional optimal profiles for transitions in a phase-separation model with non-local interaction energy.

Prym varieties and the canonical map of surfaces of general type

Abstract: Let X be a smooth complex surface of general type such that the image of the canonical map $\phi$ of X is a surface $\Sigma$ and that $\phi$ has degree $\delta\geq 2$. Let $\epsilon\colon S\to \Sigma$ be a desingularization of $\Sigma$ and assume that the geometric genus of S is not zero. Beauville has proved that in this case S is of general type and $\epsilon$ is the canonical map of S. Beauville has also constructed the only infinite series of examples $\phi:X\to \Sigma$ with the above properties that was known up to now. Starting from his construction, we define a {\em good generating pair}, namely a pair $(h:V\to W, L)$ where h is a finite morphism of surfaces and L is a nef and big line bundle of W satisfying certain assumptions. We show that by applying a construction analogous to Beauville's to a good generating pair one obtains an infinite series of surfaces of general type whose canonical map is 2-to-1 onto a canonically embedded surface. In this way we are able to construct more infinite series of such surfaces. In addition, we show that good generating pairs have bounded invariants and that there exist essentially only 2 examples with $\dim |L|>1$. The key fact that we exploit for obtaining these results is that the Albanese variety P of V is a Prym variety and that the fibre of the Prym map over P has positive dimension.

Nonunique continuation for plane uniformly elliptic equations in Sobolev spaces

Abstract: In the half plane x > 0, a Holder continuous, non zero function u (x, y), periodic in y is constructed: u has p 2) second derivatives and it satisfies a.e. a second order, non variational, uniformly elliptic equation Lu = 0; moreover u 0 for x large enough. Mathematics Subject Classification (2000): 35B60 (primary), 35J 15 (secondary).

Smoothness in fractional evolution equations and conservation laws

Abstract: The regularity of solutions of the equation where D~ denotes the fractional derivative, is studied in the case where Q’ > 0. It is also shown that the solution to the Riemann problem for the fractional Burgers equation (where a (r) = !r2) is continuous and has compact support (in the xdirection). A result on the continuity of the interface is established. In order to prove these results it is first shown that if A is an m-accretive operator in a Banach space, k is log-convex with limt jo k(t) = +cxJ, and if u is the solution of then A(u(t)) is continuous when t > 0. Mathematics Subject Classification (1991): 35K99 (primary), 35L99, 45K05 (secondary.

On the angle condition for the perturbation of elliptic systems

Abstract: For perturbed elliptic systems with critical growth we discuss convergence properties of an approximating sequence. We show the strong convergence in the relevant Soblovespace, if a-priori bounds depending on the \"angle condition\" are available for the L 00 norm of the approximations. This condition restricts the angle between the perturbation and the solution as vectors in the target space JRM. Our main tools are testfunctions constructed by projections onto convex sets in the target space. Finally, we present conditions on the inhomogeneity, providing those bounds and consequently the existence of weak solutions. Mathematics Subject Classification (1991): 35J60, 35A35, 49A22.. Introduction On bounded domains Q C R N we consider weak solutions u : S2 -~ Rm of the Dirichlet problem for elliptic systems We are interested in the convergence properties of approximating sequences assuming that the perturbation B(u) is of \"critical\" growth in the gradient, i.e.: the growth exponent is of the same order as the integration exponent of the relevant Sobolev Space. For two reasons these problems are of particular interest. Firstly, the Euler-Lagrange systems of variational problems for vector valued functions are of this type, in the cases where the functional is depending not only on the gradient but also explicitly on the solution. Because of this relation this growth of B(u) is often called \"natural\", too. Secondly, with this growth the usual positivity conditions only provide for the perturbations and heAce there are no compactness results from Functional Analysis available This work was partially supported by SFB 256 Bonn, and by the Research Council of the University of Oklahoma. Pervenuto alla Redazione il 18 febbraio 1999.

Integrable systems and projective images of Kummer surfaces

Abstract: The (-1 )-involution on the Jacobian Jr of an arbitrary Riemann surface r of genus two leads to a singular surface, the Kummer surface Icr of Jr, which, after desingularization, defines a X-3 surface Kr . We consider ample line bundles on Kr coming from the even or odd sections of [n O] with prescribed vanishing at the 2-division points of Jr (0 is the theta divisor of Jr). We use an integrable system to show that in the cases we study the linear system is base-point-free, to determine which curves are contracted to singular points and to compute an explicit equation for the surface in projective space. Our explicit methods apply to the Kummer surface of any Abelian surface, given as the fiber of the moment map of an algebraic completely integrable system. Mathematics Subject Classification (1991): 14J28 (primary), 14H40, 58F07 (secondary). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) Vol. XXIX (2000),

Graded Lie algebras of maximal class IV

Abstract: We describe the isomorphism classes of certain infinite-dimensional graded Lie algebras of maximal class, generated by an element of weight one and an element of weight two, over fields of odd characteristic.