This paper is devoted to prove the existence of strong solutions for a brittle fracture model of quasi-static crack propagation in the two dimensional antiplane setting. As usual, the time continuous evolution is obtained as the limit of a discrete in time evolution by letting the time step tend to zero. The analysis rests on a density lower bound estimate for quasi-minimizers of Mumford-Shah type functionals, under a homogeneous Dirichlet boundary condition on a part of the boundary. In contrast with (...) the previous results, since boundary cracks may be obtained as limits of interior cracks, such a density lower bound has to be established also on balls centered inside the domain but possibly intersecting the Dirichlet boundary. Thanks to a 2D geometrical argument, the discrete in time crack turns out to satisfy a uniform density lower bound which can pass to the limit, leading to the closedness of the continuous in time crack. We also establish better convergence properties of the discrete in time displacement/crack pair towards its time continuous counterpart. (shrink)

We study the error in quadrature rules on a compact manifold. Our estimates are in the same spirit of the Koksma-Hlawka inequality and they depend on a sort of discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative estimates for quadrature rules of functions in Sobolev classes.

We establish local higher integrability and differentiability results for minimizers of variational integrals F = ∫ΩF)dx over W1,p-Sobolev mappings v: Ω ⇢ Rn → RN satisfying a Dirichlet boundary condition. The integrands F are assumed to be autonomous, convex and of growth, but are otherwise not subjected to any further structure conditions, and we consider exponents in the range 1 < p ≤ q < p*, where p* denotes the Sobolev conjugate exponent of p.

We present a new proof of a recent ε-regularity of G. Tian and J. Viaclovsky. Our idea also provides a new proof of a classical result of M. Anderson about volume rigidity of Einstein manifolds. Eventually, we also obtain new rigidity results for critical metrics.

In this paper we study the asymptotic behavior for a nonlocal heat equation in an inhomogenous medium: put = J * u - u in RN x, where p is a continuous positive function, u is non-negative and J is a probability measure having finite second-order momentum. Depending on integrability conditions on the initial data u0 and p, we prove various isothermalisation results, i.e. u converges to a constant state in the whole space.

The central theme in this paper is the Hopf-Laplace equation, which represents stationary solutions with respect to the inner variation of the Dirichlet integral. Among such solutions are harmonic maps. Nevertheless, minimization of the Dirichlet energy among homeomorphisms often leads to mappings which are neither harmonic nor homeomorphism. We prove that such mappings are harmonic outside of a singular set with small image. On the singular set they are locally Lipschitz, but not necessarily differentiable.

We give effective bounds for the uniformity of the Iitaka fibration. These bounds follow from an effective theorem on the birationality of some adjoint linear series. In particular we derive an effective version of the main theorem in [19].

We prove stability estimates for teh Brunn-Minkowski inequality for convex sets. As opposed to previous stability results, our estimates improve as the dimension grows. In particular, we obtain a non-trivial conclusion for high dimensions when Voln ≤ 5√Voln Voln. Our results are equivalent to a think shell bound, which is one of the central ingredients in the proof of the central limit theorem for convex sets.

We present a geometric model of social choice when the latter takes place among bundles of interdependent elements, that we will call objects. We show that the outcome of the social choice process is highly dependent on the way these bundles are formed. By bundling and unbundling the same set of constituent elements an authority enjoys a vast power of determining the social outcome. We provide necessary and sufficient conditions under which a social outcome may be a local or global (...) optimum for a set of objects, and we show that, by appropriately redefining the set of objects, intransitive cycles may be broken and the median voter may be turned into a loser. (shrink)

We study the ergodic theory of a one-parameter family of interval maps Tα arising from generalized continued fraction algorithms. First of all, we prove the dependence of the metric entropy of Tα to be Hölder-continuous in the parameters α. Moreover, we prove a central limit theorem for possibly unbounded observables whose bounded variation grows moderately. This class of functions is large enough to cover the case of Birkhoff averages converging to the entropy.

We prove that C1as-minimal surfaces are of class C∞. For this, we develop a new bootstrap regularity theory for solutions of integro-differential equations of very general type, which we believe is of independent interest.

The main goal of this paper is to give a unified proof of the corona theorem for weighted Hardy spaces and for Morrey spaces. We use a technique that reduces the problem to the weighted Hardy spaces H2.

We construct a duality for F-manifolds with eventual identities and certain special families of connections and we study its interactions with several well-known constructions from the theory of Frobenius and F-manifolds.

Let Ω be a bounded domain in R2 with smooth boundary; we study the following Neumann problem where v is the outer normal vector of δΩ, λ > 0 is a small parameter and 0 < p < 2. We construct bubbling solutions to problem by a Lyapunov-Schmidt reduction procedure.

For a smooth projective variety X over an algebraically closed field of characteristic p > 0, we show that all irreducible stratified bundles on X have rank 1 if and only if the commutator [π1, π1] of the étale fundamental group π1 of X is a pro-p-group, and we prove that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if π1 is an Abelian without p-power quotient. This answers positively a conjecture by (...) Gieseker [4, page 8]. (shrink)

In this paper we study normal forms of Levi-flat hypersurfaces with singularities. We prove a result analogous to the Burns-Georg theorem for the existence of rigid normal forms of Levi-flat hypersurfaces which are defined by the vanishing of the real part of Ak, Dk, Ek singularities.

For a standard Finsler F on a manifold M, the domain is the whole tangent bundle T M and the fundamental tensor g is positive-definite. However, in many cases, these two conditions hold in a relaxed form only, namely one has either a psuedo-Finsler metric or a conic Finsler metric. Our aim is twofold. First, we want to give an account of quite a few subtleties that appear under such generalizations, say, for conic pseudo-finsler metrics. Second, we aim to provide (...) some criteria that determine when a pseudo-Finsler metric F obtained as a general homogeneous combination of Finsler metrics and one-forms is again a Finsler-metric -- or, more precisely, that the conic domain on which g remains positive-definite. Such a combination generalizes the known -metrics in different directions. Remarkably, classical examples of Finsler metrics are reobtained and extended, with explicit computations of their fundamental tensors. (shrink)

We prove that the solution of the wave equation associated to the Grushin operator G = -Δ - |x|2δt2 is bounded on Lp(Rn+1_, with 1 < p < ∞ when |1/p - 1/2| < 1/n+2.

The theme of this work is the study of the Nekovar-Selmer group H1f attached to a twisted Hida family T† of Galois representations and a quadratic number field K. The results that we obtain have the following shape: if a twisted L-function of a suitable modular form in the Hida family has order of vanishing r ≤ 1 at the central critical point then the rank of H1f as a module over a certain local Hida-Hecke algebra is equal to R. (...) Under the above assumption, we also show that infinitely many twisted L-functions of modular forms in the Hida family have the same order of vanishing at the central critical point. Our theorems extend to more general arithmetic situations results obtained by Howard when K is an imaginary quadratic field and all the primes dividing the tame level of the Hida family split in K. (shrink)

Using the notion of H-subdifferential, we characterize both first and second order differentiability of h-convex functions in stratified groups. Besides some new results involving the h-subdifferential of h-convex functions, we show that all h-differentiability points of an h-convex function, the existence of a second order expansion coincides with a suitable differentiability of its horizontal gradient.

In this work, we present "infinite" multiplicative formulae for countable collections of sums of squares. our formulae generalize the classical Pfisters's ones concerning the representation as a sum of 2r squares of the product of two elements of a field K which are sums of 2r squares. As a main application, we reduce the representation of a positive semidefinite analytic function on Rn as a sum of squares to the representation as sums of squares of its special factors. Recall that (...) roughly speaking a special factor is an analytic function on Rn which has just one complex irreducible factor and whose zeroset has dimension between 1 and n-2. (shrink)

We show that the Moser functional J = ∫Ω dx on the set B = {u ϵ H10: ||∆u||2 ≤ 1}, where Ω R2 is a bounded domain, fails to be weakly continuous only in the following exceptional case. Define gsw = s-1/2w for s > 0. If uk → u in B while lim inf J > J, then, with some sk → 0, uk = gsk [-1/2 min {1,log 1/|x|}], up to translations and up to a remainder vanishing (...) in the Sobolev norm. In other words, the weak continuity fails only on translations of concentrating Moser functions. The proof is based on a profile decomposition similar to that of Solimini [16], but with different concentration operators, pertinent to the two-dimensional case. (shrink)

Let Ω Rn be a bounded open set satisfying the uniform exterior cone condition. Let A be a uniformly elliptic operator given by Au = nΣi,j=1 aij∂iju + nΣj=1 bj∂ju + cu where aji=aij ϵC, and bj, c ϵ L ∞, C ≤ ). We show that the realization A0 of A in C0: = {u ϵ C: u|∂Ω=0} given by D:= {u ϵ C0 ∩ W2,n loc } a0u:= Au generates a bounded holomorphic C0-semigroup on C0. The result is (...) in particular true if Ω is a Lipschitz domain. so far the best known result seems to be the case where Ω has C2-boundary [12, Secton 3.1.5]. We also study the elliptic problem -Au = f u|∂Ω = g. (shrink)

We initiate the study of the finiteness condition ∫Ωu-Bdx≤C 0 for which the aforementioned condition holds under various hypotheses on the smoothness of Ω and demands on the nature of the constant C. Classes of domains for which our analysis applies include bounded piecewise C1 domains in Rn, n ≥ 2, with conical singularities, polyhedra in R3, and bounded domains which are locally of class C2 and which have outwardly pointing cusps. For example, we show that if uN is the (...) solution of the Saint Venant problem in the regular polygon ΩN with N sides circumscribed by the unit disc in the plane, then for each B ϵ the following asymptomatic formula holds: ∫ΩNuN-Bdx=4Bπ/1/B+O as N→∞. One of the original motivations for addressing the aforementioned issues was the study of sublevel set estimates for functions v satisfying v = 0, Vv and ∆v≥C>0. (shrink)

We show that any terminal 3-fold divisiorial contraction to a point of index>1 with non-minimal discrepancy may be factored into a sequence of flips, flops and divisorial contractions to points with minimal discrepancies.

We give a monodromy representation of Lauricella's system of differential equations annihilating the hypergeometric series FA, ; x) of k-variables; its rank is 2k. Under some non-integral conditions for parameters a, =, =, we find circuit matrices with respect to solutions represented by integrals. We make use of the intersection numbers of the domains of the integrals.

We give a monodromy representation of Lauricella's system of differential equations annihilating the hypergeometric series FA, ; x) of k-variables; its rank is 2k. Under some non-integral conditions for parameters a, =, =, we find circuit matrices with respect to solutions represented by integrals. We make use of the intersection numbers of the domains of the integrals.

An element g of a group is called reversible if it is conjugate in the group to its inverse. This paper is about reversibles in the group B = B2 of formally invertible pairs of formal power series in two variables, with complex coefficients. The main result is a description of the generic reversible elements of B2. We list two explicit sequences of reversibles which between them represent all the conjugacy classes of such reversibles. We show that each such element (...) is reversible by some element of finite order, and hence is the product of two elements of finite even order. The elements that may be reversed by an involution are called strongly reversible. We also characterize them. We draw some conclusions about generic reversibles in the group G = G2 of biholomorphic germs in two variables, and about the factorization of formal maps as products of reversibles. Specifically, we show that each product of reversibles reduces to the product of five. (shrink)

We consider the problem of finding the optimal constant for the embedding of the space W∆2,1:={uϵ W01,1 | ∆u ϵL1} into the space L1, where ΩRN is a bounded convex domain, or a bounded domain with boundary of class C1,a. This is equivalent to finding the first eigenvalue of the 1-biharmonic operator under Navier boundary conditions. In this paper we provide an interpretation for the eigenvalue problem, we show some properties of the first eigenfunction, we prove an inequality of Faber-Krahn (...) type, and we compute the first eigenvalue and the associated eigenfuction explicitly for a ball, and in terms of the torsion function for general domains. (shrink)

We review the relations between compact complex manifolds carrying various types of Hermitian metrics and those satisfying topological properties such as the ∂∂-lemma, or the degeneration at E1 of the Frölicher spectral sequence. We also review the behaviour of these properties under holomorphic deformations. The emphasis will be placed on the notion of strongly Gauduchon manifold that we introduced recently in the study of deformation limits of projective and Moishezon manifolds. Besides its expository aspect, the paper presents new results such (...) as exhibiting various examples of sG and non-sG manifolds. These are then used to review and reinterpret from this new standpoint a range of constructions already known in the literature. (shrink)

This paper investigates a specific smooth quintic surface suggested by Barth for it contains the current record of 75 lines over the complex numbers. Our main incentive is to prove that the complex quintic has Picard number 41, and to compute the Néron-Severi group up to a 2-power index. We also compute Picard numbers for reductions to positive characteristic and verify the Tate conjecture.

We consider, in a bounded domain Ω 1, and the function H grows at most like |Du|q+f, with p-1 (...) approach, generalizing the method introduces in our note [29]. We complete the results known in the previous literature where either q ≤ p - 1 or m ≥ N/p. (shrink)

Given a connected Lipschitz domain Ω we let Λ be the set of functions in W2,2 with u=0 on ∂Ω and whose gradient satisfies Vu.nx=1, where nx is the inward pointing unit normal to ∂Ω at x. The functional Iϵ=1/2 ∫Ωϵ-1|1 -|Vu|2|2+ϵV2u|2dz, minimised over Λ, serves as a model in connection with problems in liquid crystals and thin film blisters. It is also the most natural higher order gerealisation of the Modica and Mortola functional. in [16] Jabin, Otto and Perthame (...) characterised a class of functions which includes all limits of sequences un ϵ Λ with Iϵπ→ 0 as ϵn→ 0. A corollary to their work is that if there exists such a sequence for a bounded domain Ω, then Ω must be a ball and u := limn→∞un is equal dist. We prove a quantative generlisation of this corollary for the class of bounded convex sets. Namely we show that there exists a positive constant γ1 such that, if Ω is a convex set of diameter 2 and u ϵ Λ with Iϵ = B, then |B1ΔΩ|≤cBγ1 for some x and ∫Ω|Vu+z-x/|z-x|2 dz≤cBγ1. A corollary of this result is that there exists a positive constant γ2 <γ1 such that if Ω is convex with diameter 2 and C2 boundary with curvature bounded by ϵ-1/2, then for any minimiser v of Iϵ over Λ we have ||v-ζ||W1,2 ≤ c|)γ2, where ζ=dist. Niether of the constands γ1 or γ2 are optimal. (shrink)

We consider semilinear and quasilinear parabolic systems with a non-smooth rate-independent and a viscous dissipation term in the limit of very slow loading rates, or equivalently with fixed loading and vanishing viscosity ε > 0. Because for nonconvex energies the solutions will develop jumps, we consider the vanishing-viscosity limit for the graphs of the solutions in the extended state space in arclength parameterization. Here the choice of the viscosity norm for parametrization. here the choice of the viscoscity norm for parametrization (...) is crucial to keep the subdifferential structure of the problem. A crucial point in the analysis are new a priori estimates that are rate independent and that allow us to show that the total length of the graph remains bounded in the vanishing-viscosity limit. To derive these estimates we combine parabolic regularity estimates with ideas from rate-independent systems. (shrink)

We construct a 4-dimensional family of surfaces of general type with pg=0 and K2=3 and fundamental z/2xqg is the quaternion group. The family constructed contains the Burniat surfaces with K2=3. Additionally, we construct the universal coverings of the surfaces in our family as complete intersections on 4 and we also give an action of Z/2xQ8 on 4 lifting the natural action on the surfaces. The strategy is the following. We consider an étale 3-cover T of a surface with pg=0 and (...) K2=3 and assume that it may be embedded in a Fano 3-fold V. We construct V by using the theory of parallel unprojection. Since V is an Enriques-Fano 3-fold, considering its Fano cover yields the simple description of the above universal covers. (shrink)

In this paper we shall show that the diameter of a complete Sasaki manifold whose transverse Ricci curvature is bounded from below by a positive constant has a universal upper bound. This gives another proof of the result of Hasegawa and Seino in [9] which asserts that such manifolds are always compact with finite fundamental group.