Showing papers in "Annali di Matematica Pura ed Applicata in 1998"
TL;DR: In this article, the authors prove partial regularity for minimizers of quasiconvex integrals with subquadratic growth, where the integral F(ξ) has sub-quadratic growth.
Abstract: We prove partial regularity for minimizers of quasiconvex integrals of the form
$$\int\limits_\Omega {F\left( {Du(x)} \right)} $$
dx where the integral F(ξ) has subquadratic growth, ie
$$\left| {F(\xi )} \right| \le L(1 + |\xi |^p )$$
.
118 citations
TL;DR: In this paper, it was shown that spherically symmetric solutions of the Cauchy problem are asymptotically stable if the initial specific volume is close to a constant in L��∞ and weighted L2.
Abstract: First we prove for the equations of a viscous polytropic ideal gas in bounded annular domains in ℝ
n
(n=2, 3)that (generalized) spherically symmetric solutions decay to a constant state exponentially as time goes to infinity. Then we show that solutions of the Cauchy problem in ℝare asymptotically stable if the initial specific volume is close to a constant in L
∞ and weighted L2, the initial velocity is small in weighted L2 ∩ L4, and the initial temperature is close to a constant in weighted L2.
77 citations
TL;DR: In this article, the classical Sbrana-Cartan theory of isometrically deformable euclidean hypersurfaces is extended to the sphere and hyperbolic space.
Abstract: We first extend the classical Sbrana-Cartan theory of isometrically deformable euclidean hypersurfaces to the sphere and hyperbolic space. Then we construct and characterize a large family of hypersurfaces which admit a unique deformation. This is used to show, by means of explicit examples, that different types of hypersurfaces in the Sbrana-Cartan classification can be smoothly attached. Finally, among other applications, we discuss the existence of complete deformable hypersurfaces in hyperbolic space.
38 citations
TL;DR: A complete classification of semisimple Levi-Tanaka algebras is given in this paper. But this classification is based on the inner derivation of the partial complex structure.
Abstract: After showing that the partial complex structure is defined by an inner derivation, we give a complete classification of semisimple Levi-Tanaka algebra.
31 citations
TL;DR: In this article, generalized Riccati inequalities are employed to establish the results of oscillation criteria for quasilinear elliptic equations of the form (E) below.
Abstract: Oscillation criteria are obtained for quasilinear elliptic equations of the form (E)below. We are mainly interested in the case where the coefficient function oscillates near infinity. Generalized Riccati inequalities are employed to establish our results.
29 citations
TL;DR: In this paper, the Cauchy problem is studied for a class of linear abstract differential equations of hyperbolic type with variable domain, and existence and uniqueness results are proved for (suitably defined) weak solutions.
Abstract: The Cauchy problem is studied for a class of linear abstract differential equations of hyperbolic type with variable domain. Existence and uniqueness results are proved for (suitably defined) weak solutions. Some applications to P.D.E. are also given: they concern linear hyperbolic equations either in non-cylindrical regions or with mixed variable lateral conditions.
14 citations
TL;DR: In this paper, the authors investigated the long-time behavior of the solutions to the Cauchy problem with a nonnegative initial data in L1 (ℝ), when q ∈ (0, 1) and m ⩾ 1.
Abstract: We investigate the long-time behaviour of the solutions to the Cauchy problem
$$u_t + (u^q )_x - (u^m )_{xx} = 0$$
with a nonnegative initial data in L1 (ℝ),when q ∈ (0, 1)and m ⩾ 1. We prove that the long-time profile in L1(ℝ)of these solutions is given by the unique nonnegative entropy sourcetype solution to the conservation law ut}+(uq)x=0 with the same mass. Uniqueness of such a solution is previously established. These results extend previous results obtained for the case q>1 and m⩾1.
13 citations
12 citations
Journal Article•
11 citations
TL;DR: In this paper, the asymptotic behavior of nonlinear second order elliptic equations with Dirichlet boundary conditions in performated domains is studied under very mild assumptions on the capacity of the holes.
Abstract: The asymptotic behaviour of the solutions of nonlinear second order elliptic equations with Dirichlet boundary conditions in performated domains is studied under very mild assumptions on the capacity of the holes
11 citations
TL;DR: In this article, the existence and multiplicity results for light-like geodesics joining a point with a timelike curve on a class of Lorentzian manifolds are proved under intrinsic assumptions.
Abstract: In this paper existence and multiplicity results for lightlike geodesics joining a point with a timelike curve on a class of Lorentzian manifolds are proved under intrinsic assumptions. Such results are obtained using an extension to Lorentzian Geometry of the classical Fermat principle in optics. The results are proved using critical point theory on infinite dimensional manifolds. An application to the gravitational lens effect is presented.
TL;DR: In this paper, the authors derived the solution of the heat equation on a homogeneous tree Γ whose edges have suitable positive conductances and are identified with copies of segments [0, 1] with the codition that the sum of the weighted normal exterior derivatives is 0 at every node (Kirchhoff type condition).
Abstract: We compute explicitly the solution of the heat equation on a homogeneous tree Γ whose edges have suitable positive conductances and are identified with copies of segments [0, 1]with the codition that the sum of the weighted normal exterior derivatives is 0at every node (Kirchhoff type condition). Furthermore we find the expression of the semigroup of linear operators on L2(Γ, c) having Δ as infinitesimal generator. These results derive from the equation governing the spread of the potential along the dendrites of a neuron.
TL;DR: In this article, the authors generalize the notion of finite dimensional system of connections on a fiber bundle to the concept of arbitrary system of connected connections and study the universal connection of a regular system and the universal curvature.
Abstract: Using the theory of smooth spaces, we generalize the notion of finite dimensional system of connections on a fiber bundle to the concept of arbitrary system of connections. Then we study the universal connection of a regular system and the universal curvature.
TL;DR: In this paper, the authors studied boundedness and microlocal properties for a general class of pseudodifferential operators of type 1,1 in the frame of the anisotropic Sobolev spaces.
Abstract: The author studies boundedness and microlocal properties for a general class of pseudodifferential operators of type 1,1 in the frame of the anisotropic Sobolev spaces.
TL;DR: In this article, it was shown that a given pair (A, B) generates a B-bounded semigroup if and only if in a certain extrapolation space related to the operator B, the closure of A generates a semigroup.
Abstract: In [3]A. Bellini-Morante defined and analysed a new one-parameter family of bounded operators which he called a B-bounded semigroup. The definition was motivated by an example from the transport theory where the evolution generated by an operator A was in a certain sense controlled by another operator B. In this paper we show that a given pair (A, B) generates a B-bounded semigroup if and only if in a certain extrapolation space related to the operator B, the closure of A generates a semigroup and we also address some related topics.
TL;DR: In this article, the authors consider a coupling between the Nernst-Planck equations and the Navier-Stokes system, and they study the stationary and the evolution problems.
Abstract: We consider a coupling between the Nernst-Planck equations and the Navier-Stokes system; we study the stationary and the evolution problems. The crucial property turns out to be the existence of an invariant region. An asymptotic result in the case of Neumann boundary conditions is also given.
TL;DR: In this article, the global existence result for one-phase Stefan problems was proved by using energy inequalities, and it was shown that if α > 1 an initial function is sufficiently small, then the free boundary is bounded and decay in exponential order.
Abstract: We consider one-phase Stefan problems for the equationu i =u xx +u 1+a (α>0)in one-dimensional space, which have blow-up solutions for a larger initial data. In this paper, the global existence result for our problem is proved by using energy inequalities. More precisely, if α>1 an initial function is sufficiently small, then the free boundary is bounded and\(|u(t)|_{L^\infty } \) decay in exponential order.
TL;DR: In this article, the authors studied Lagrangian systems with symmetry under the action of a constant generalized force in the direction of the symmetry field, and proved the existence of solutions tending to an orbit of a symmetry group as t→± ∞.
Abstract: We study Lagrangian systems with symmetry under the action of a constant generalized force in the direction of the symmetry field. After Routh's reduction, such systems become nonautonomous with Lagrangian quadratic in time. We prove the existence of solutions tending to an orbit of the symmetry group as t→± ∞. As an example, we study doubly asymptotic solutions for the Kirchhoff problem of a heavy rigid body in an infinite volume of incompressible ideal fluid performing a potential motion.
TL;DR: In this article, a regularity of bounded solutions for some degenerate nonlinear parabolic equations of higher order was studied and it was established the Holder Continuity of solutions by condition that the weighted function belongs to the class A1+q/n.
Abstract: We study a regularity of bounded solutions for some degenerate nonlinear parabolic equations of higher order. It is established the Holder Continuity of solutions by condition that the weighted function belongs to the class A1+q/n.
TL;DR: In this article, the existence of stable vector bundles and stable torsion free sheaves on a reduced projective curve is studied, where the authors assume that the vector bundles are irreducible or stable.
Abstract: Let X be a reduced projective curve; assume X either irreducible or stable. Here we study some geometric properties of stable vector bundles and stable torsion free sheaves on X (essentially, the existence of stable objects with a prescribed order of stability).
TL;DR: In this paper, it was shown that there exists a set A such that A has quasi-minimal enumeration degree, and there are uncountably many sets B such that enumeration reducible to B and B has minimal Turing degree.
Abstract: We show that there exists a set A such that A has quasi-minimal enumeration degree, and there are uncountably many sets B such that A is enumeration reducible to B and B has minimal Turing degree. Answering a related question raised by Solon, we also show that there exists a nontotal enumeration degree which is not e-hyperimmune.
TL;DR: In this article, it was shown that every infinite sequence of elements of a finite semigroup has an infinite factorization of the form x, e, e. e., e. f, e, f, f, f, e f,..., where f is an idempotent of the semigroup.
Abstract: It follows from the Ramsey theorem that every infinite sequence of elements of a finite semigroup has an infinite factorization of the form x, e, e, e, ..., where e is an idempotent of the semigroup. We describe all semigroups with this property and with its analog for two-sided infinite sequences.
TL;DR: In this paper, the existence of a well determined isomorphism of R(M) onto a well defined subgroup of the abelian p-group M = M.
Abstract: Given the abelian p-group M=〈a〉⊕〈b〉⊕C, where ¦a¦=p
n⩾¦b¦=pm > exp C= =ps>1, set R(M) =·ϕ∈P(M)·Hϕ=H, ϕ·ΩS(M)=1}. Our main result is the existence of a well determined isomorphism of R(M) onto a well defined subgroup of
$$\mathop \Pi \limits_{k = 0}^{n - m} PR(p^{n - k - m} R_{n - k} ) \times PR(pR_m )$$
.
TL;DR: In this paper, the authors examined the behavior of the one-dimensional non-homogeneous transport equation of the form ǫut= ux+f, ë1.
Abstract: This study examines the behavior of the one-dimensional non-homogeneous transport equation of the form ɛut= ux+f, ɛ«1. The solution consists of behavior which changes on two different time scales, one rapid and one slow. This time scale behavior is known. Additionally, however, we find here that because of the presence of the non-homogeneous forcing termf, and large wave speed 1/ɛ, there is a component of the solution which will vary only on a very large spatial scale. This large space-scale solution persists throughout all time, even after the source term of the solution has been shut off. The analysis of this large spacescale behavior is the focus of this paper. Numerical experiments highlight some of our results. These results have applications in fields such as meteorology, and other areas where multiple time scales are of interest.
TL;DR: In this article, a generalization of the Prepared Form Theorem for smooth Fredholm maps between Banach spaces is given, which is a basic tool in singularity theory and can be used in many applications.
Abstract: We give a generalization, for smooth Fredholm maps between Banach spaces, of the Preparation Theorem known in finite dimension. As an application we obtain the Prepared Form Theorem which is a basic tool in singularity theory.
TL;DR: The dimension of the Zariski tangent space at a special (symplectic) instanton bundle is 2k(5n−1)+4n2−10n+3, k⩾2 as mentioned in this paper.
Abstract: Let\(MI_{Simp,P^{2n + 1} } (k)\)be the moduli space of stable symplectic instanton bundles on ℙ2n+1 with second Chern class c2=k (it is a closed subscheme of the moduli space\(MI_{P^{2n + 1} } (k)\)).We prove that the dimension of its Zariski tangent space at a special (symplectic) instanton bundle is 2k(5n−1)+4n2−10n+3, k⩾2.
TL;DR: In this article, a relativistic frame of reference, generalized in the polar sense, and the adapted nonholonomic techniques are considered, and extended to non-orthogonal case second-order properties of a standard frame.
Abstract: We consider a relativistic frame of reference, generalized in the polar sense [1]–[2],and the adapted non-holonomic techniques;then, we extend to non-orthogonal case second order propertiesof a standard frame of reference: Riemann tensor decomposition, Lie derivatives of the Ricci rotation coefficients, commutation formulae and spatial Bianchi identity.
TL;DR: In this article, the authors studied radial solutions u=(u1, u2) in an exterior domain of R N (N⩾3) of the elliptic system−Δu+V⊂u)=0, where V is a positive and singular potential.
Abstract: We study radial solutions u=(u1, u2) in an exterior domain ofR N (N⩾3)of the elliptic system−Δu+V⊂u)=0, where V is a positive and singular potential. We look for solutions which satisfy Dirichlet boundary conditions and vanish at infinity. We prove existence of infinitely many radial solutions, which can be topologically classified by their winding numbers around the singularity of V. Furthermore, we study some qualitative properties of such solutions.