Showing papers in "Annali di Matematica Pura ed Applicata in 1986"

Compact sets in the spaceL p (O,T; B)

Abstract: A characterization of compact sets in Lp (0, T; B) is given, where 1⩽P⩾∞ and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space Lp (0,T; B) from estimates with values in some spaces X, Y or B where X⊂B⊂Y with compact imbedding X→B. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {fn} is bounded in Lq(0,T; B) and in L loc 1 (0, T; X) and if {∂fn/∂t} is bounded in L loc 1 (0, T; Y) then {fn} is relatively compact in Lp(0,T; B), ∀p

Studies on the Painlevé equations. I: Sixth Painlevé equation PVI

Abstract: In this series of papers, we study birational canonical transformations of the Painleve system ℋ, that is, the Hamiltonian system associated with the Painleve differential equations. We consider also τ -function related to ℋ and particular solutions of ℋ. The present article concerns the sixth Painleve equation. By giving the explicit forms of the canonical transformations of ℋ associated with the affine transformations of the space of parameters of ℋ, we obtain the non-linear representation: G→G*, of the affine Weyl group of the exceptional root system of the type F4 A canonical transformation of G* can extend to the correspondence of the τ -functions related to ℋ. We show the certain sequence of τ -functions satisfies the equation of the Toda lattice. Solutions of ℋ, which can be written by the use of the hypergeometric functions, are studied in details.

Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation

Abstract: The singularly perturbed differential-delay equation $$\varepsilon \dot x(t) = - x(t) + f(x(t - 1))$$ is studied. Existence of periodic solutions is shown using a global continuation technique based on degree theory. For small ɛ these solutions are proved to have a square-wave shape, and are related to periodic points of the mapping f:R→R.When f is not monotone the convergence of x(t) to the square-wave typically is not uniform, and resembles the Gibbs phenomenon of Fourier series.

Nonlinear Stochastic Homogenization

Abstract: In questo lavoro viene studiata l'omogeneizzazione stocastica per funzionali integrali del Calcolo delle Variazioni con integrando dipendente dalla variabile spaziale e convesso nel gradiente, soddisfacente alle usuali ipotesi di uniforme coercitivita e limitatezza. Il risultato generale ottenuto copre un largo spettro di fenomeni riguardanti materiali con disposizione casuale di piu componenti il cui comportamento fisico e retto da equasioni variazionali non lineari.

An identification problem for an elliptic equation in two variables

Abstract: Si studia il problema inverso di determinare il coefficients a nell'equazione ellittica in due variabili $$div (a grad u) = 0$$ (*) ,quando se ne conosce una soluzione u. Si danno un risultato di dipendenza continua di a da u e un metodo di determinazione approssimata di a. Elemento chiave in questi risultati e lo studio di proprieta dei punti critici delle soluzioni u di (*).

A classification of Riemannian manifolds with structure group Spin (7)

Abstract: Riemannian manifolds with structure group Spin (7)are 8-dimensional and have a distinguished 4 -form. In this paper, the covariant derivative of the fundamental 4 -form is studied, and it is shown that there are precisely four classes of such manifolds.

Real hypersurfaces in quaternionic projective space

Abstract: This paper is devoted to make a systematic study of real hypersurfaces of quaternionic projective space using focal set theory. We obtain three types of such real hypersurfaces. Two of them are known. Third type is new and in its study the first example of proper quaternion CR-submanifold appears. We study real hypersurfaces with constant principal curvatures and classify such hypersurfaces with at most two distinct principal curvatures. Finally we study the Ricci tensor of a real hypersurface of quaternionic projective space and classify pseudo-Einstein, almost-Einstein and Einstein real hypersurfaces.

On the homogenization of quasilinear divergence structure operators

Abstract: We study the homogenization of second order quasilinear operators of the form $$A_\varepsilon u = - div a\left( {\frac{x}{\varepsilon },u,Du} \right)$$ in Sobolev spaces H1,p (p>1). An explicit formula of the homogenized operator is given.

Unified boundedness, periodicity, and stability in ordinary and functional differential equations

Abstract: We discuss a unified theory of periodicity of dissipative ordinary and functional differential equations in terms of uniform boundedness. Sufficient conditions for the uniform boundedness are given by means of Liapunov functionals having a weighted norm as an upper bound. The theory is developed for ordinary differential equations, equations with bounded delay, and equations with infinite delay.

On 385-01385-01385-01regularity of the gradient of solutions of degenerate parabolic systems

Abstract: We consider weak solutions u∈Lp([0,T], Wp1(Ω))∩([0,T],L2(Ω))of the degenerate parabolic (model) -system $$\frac{{\partial u^i }}{{\partial t}} - div (| abla u|^{p - 2} abla u^i ) = 0 on \Omega \subset R^N , 1 \mathbin{\lower.3ex\hbox{$\buildrel 2.$$ By local techniques it is proved, using sequences of time-space cylinders, which are adjusted to the alternative whether one is at a point of degeneracy or not, that the spatial gradient of u is α- Holdercontinuous on compact subsets of Ω× [0,T] with some α which depends only on N and p.

A degenerate parabolic equation modelling the spread of an epidemic

Abstract: We consider the Cauchy problem for a degenerate parabolic equation, not in divergence form, representing the diffusive approximation of a model for the spread of an epidemic in a closed population without remotion. We prove existence and uniqueness of the weak solution, defined in a suitable way, and some qualitative properties.

Symbolic dynamics and nonlinear semiflows

Abstract: For a transverse homoclinic orbit γ of a mapping (not necessarily invertible) on a Banach space, it is shown that the mapping restricted to orbits near γ is equivalent to the shift automorphism on doubly infinite sequences on finitely many symbols. Implications of this result for the Poincare map of semiflows are given.

Weak solutions for a system of nonlinear Klein-Gordon equations

Abstract: We prove the existence and uniqueness of weak solutions of the mixed problem for a class of systems of nonlinear Klein-Gordon equations. Uniqueness is proved when the spatial dimension is either n=1, 2or 3.

Stefan problem with a kinetic condition at the free boundary

Abstract: In the one-dimensional two-phase Stefan problem, the standard equilibrium condition θ=0 at the free boundary x=s(t) is replaced by the kinetic law (1) $$s'(t) = \beta \left( {\theta \left( {s\left( t \right),t} \right)} \right);$$ here β is a continuous and increasing functionR→R and β(0)=0. This introduces supercooled and superheated states. Existence of at least one solution is proved. Then (1) is replaced by (2) $$s'_\varepsilon (t): = \beta \left( {\theta _\varepsilon \left( {s_\varepsilon \left( t \right),t} \right)} \right) \left( {\varepsilon constant< 0} \right),$$ and it is shown that as ɛ → 0+ a subsequence of the corresponding solutions (θ e ,s e ) converges to a solution (θ, s) of the reduced problem, which is characterised by the free boundary condition (3) $$\beta \left( {\theta \left( {s\left( t \right),t} \right)} \right) = 0.$$ Then the case of a radially symmetric multidimensional system is dealt with, taking also account of the surface tension effect. Denoting by s(t) the radial co-ordinate of the free boundary, the following linearized kinetics is considered for a water ball surrounded by ice (4) $$ls'(t) + \frac{\lambda }{{s(t)}} = \theta \left( {s\left( t \right),t} \right), where s(t)< 0.$$ An existence result is proved for the problem obtained by coupling (4) with the heat equation.

Tangential Cauchy-Riemann complexes on distributions

Abstract: Dopo aver definito il complesso di Gauchy-Riemann tangenziale su sottovarieta reali generiche, di codimensione qualsiasi, di una varieta eomplessa, per mezzo di una opportuna successione di Mayer-Vietoris se ne studiano i gruppi di coomologia locali. Si ottengono inoltre risultati relativi al problema dell' estensione di «distribuzioni di Cauchy-Riemann» e alla coomologia globale di sottovarieta compatte e di domini compatti con frontiera regolare a tratti.

Second fundamental form of a map

Abstract: This paper is devoted to the study of the 2, fundamental form of a map, which generalizes this notion, well known for isometric immersions. We generalize results by Vilms, Yano, and Ishihara, and study in detail projective and umbilical maps.

Periodic solutions of hamiltonian systems with superquadratic potential

Abstract: In questo lavoro si dimostra un teorema astratto di punti critici per funzionali fortemente indefiniti. Si applica poi tale teorema alla ricerca di soluzioni T-periodiche, con periodo T prefissato, del sistema Hamiltoniano $$\dot p = - H_q (p,q),\dot q = H_p (p,q)$$ dovep,q ∈Rn,e l'Hamiltoniano H e C1(R2n,R) e del tipo $$H(p,q) = \sum\limits_{i,j} {a_{i,j} (q)} p_i p_j + \sum\limits_i {b_i } (q)p_i + V(q)$$ con V(q)/|q|2→+∞ per |q|→+∞.

A diagonal theorem for epireflective subcategories of top and cowellpoweredness

Abstract: For a quotient-reflective subcategoryAof the category Topof topological spaces the following «diagonal theorem» is proved: a topological space (X,τ)belongs toAiff the diagonal Δxis (τ×τ)A-closed, where, for (X, ρ) e Top, σAdenotes the coarsest topology on X which has as closed subsets all the equalizers of pairs of continuous maps with codomain inA.Furthermore an explicit description of τAfor several quotient reflective subcategories defined by means of properties of subspaces is given. It is shown that one of them is not co-(well-powered).

Sur les espaces analytiques quasi-compacts de dimension 1 sur un corps valué complet ultramétrique

Abstract: We study the structure of the one dimensional analytic quasi-compact spaces over a complete non archimedean valued field. An affinoid open subset U of a one dimensional analytic quasi-compact space X is defined by a meromorphic function f on X;i.e. U is the set of all x in X such that f is holomorphic at x and ¦f(x)¦⩽1.The set of the meromorphic functions on X which are holomorphic on U is dense in the ring of all holomorphic functions on U. An irreducible, one dimensional quasi-compact space is either affinoid, or projective. An analytic reduction of X is defined by a meromorphic invertible function f on X;i.e. the reduction is isomorphic to the reduction associated to the covering ¦f(x)¦⩽1and ¦f(x)¦⩾1.

Bifurcation problems with O(2)⊕Z2 symmetry and the buckling of a cylindrical shell

Abstract: In this paper we employ equivariant singularity theory to study the postbuckling behavior of a cylindrical shell under axial compression, obtaining some results about the existence of secondary bifurcations and how they are connected to each other. The basic idea, first employed by Bauer, Keller and Reiss in [1], and then coupled with singularity theory by Schaeffer and Golubitsky in [16] and [17] and by Buzano in [4], consists in unfolding a multiple eigenvalue, obtained by forcing two eigenvalues to coalesce by varying the geometric parameters of the shell. This approah is made possible by a general analysis of bifurcation problems invariant with respect to the symmetries of the cylinder i.e. with respect to the group O(2)⊕Z2.

A steady-state Boussinesq-Stefan problem with continuous extraction

Abstract: One establishes an existence result for the weak solution to a steady-state strongly coupled system between a nonlinear two phases heat equation with convection and the Navier-Stokes equation in the liquid phase. The two phases Rayleigh-Benard problem is included as the particular case corresponding to a zero extraction velocity.

Positive eigenvectors of wedge maps

Abstract: In this paper we investigate the existence of positive eigenvalues and corresponding eigenvectors of nonlinear and noncompact maps defined in a wedge W of a Banach space E. The results are established using the theory of 0 -epi maps introduced by Furi-Martelli-Vignoli [10].We prove a conjecture of I. Massabo - C. Stuart [23]and we obtain a nonlinear version of the celebrated Krein-Rutman [18]theorem, which brings about the different role of the two properties $$f(tx) = tf(x) and f(x + y) = f(x) + f(y)$$ .

Bounce problem with weak hypotheses of regularity

Abstract: In this paper the elastic bounce problem is formulated in very general hypotheses. More precisely we consider the motion of a material point constrained to move in a domain Ω ⊂Rn, bouncing against its boundary, and we suppose that Ω is neither regular nor convex. Assuming that Ω is in the class of p-convex sets introduced in [4] and ∂Ωe C0,1, an existence theorem is stated.

On complex geodesics of balanced convex domains

Abstract: Questions of non-uniqueness for the complex geodesics of a balanced convex domain D in a locally convex Hausdorff vector space E are investigated. The results establish a precise relationship between the shape of the boundary of D at a point y and the structure of the family of complex geodesics «near» ξ ↦ ξy. The case in which E is a Banach space is also considered and a complete description of all the complex geodesics is given for the open unit ball of the space C(X) of all the complex valued continuous functions on a compact Hausdorff space X.

Differentialoperatoren bei partiellen Differentialgleichungen

Abstract: In the present paper those formally hyperbolic differential equations are characterized for which solutions can be represented by means of differential operators acting on holomorphic functions. This is done by a necessary and sufficient condition on the coefficients of the differential equation. These operators are determined simultaneously. By it a general procedure is presented to construct differential equations and corresponding differential operators which map holomorphic functions onto solutions of the differential equations. We also discuss the question under which circumstances all the solutions of a differential equation can be represented by differential operators. For the equations characterized previously we determine the Riemann function. Some special classes of differential equations are investigated in detail. Furthermore the possibility of a representation of pseudoanalytic functions and the corresponding Vekua resolvents by differential operators is discussed.

Asymptotic and oscillatory properties of a class of operator-differential inequalities

Abstract: The present paper studies some asymptotic (including oscillatory) properties of the solutions of operator-differential inequalities of the form $$[(Lx](t)]^{(n)} + (Fx)(t) \leqslant 0,$$ where\(L:\tilde C^{n - 1} [t_0 ,\infty ] \to \tilde C^{n - 1} [t_0 ,\infty ],F:\tilde C^{n - 1} [t_0 ,\infty ] \to \tilde C^{ - 1} [t_1 ,\infty ),t_1 \geqslant t_0 \) (the latter is the spaceof locally summable functions). As an application of the obtained results, theorems are proved]or the asymptotic behaviour of the solutions of certain classes of functional-differential andintegro-differential neutral type equations.

Théorème de Hoffmann-Jorgensen et application aux amarts multivoques

Abstract: On presente ici des versions univoques et multivoques d'un resultat de Hoffmann-Jorgensen ([10]). Ces resultats permettent d'etendre les resultats obtenus anterieurement par Uhl ([23]), Musial ([17]), Luu ([16]) aux amarts multivoques a valeurs convexes compactes d'un espace localement convexe separe complet.

Hyperbolic equations and classes of infinitely differentiable functions

Abstract: We consider the solvability in the Mandelbrojt classes ɛ{Mh} (for the definition see (6), (7), (8), (9) below) of the Cauchy problem for hyperbolic equations of the type utt - - a(t) uxx=0, where a(t) is a strictly positive continuous function. More precisely, we give an example of a function a(t) for which the Cauchy problem is not well-posed in any class ɛ{Mh} containing a non-trivial function with compact support.

Volume on stratified sets

Abstract: In this paper a notion of volume on closed stratified subsets of a riemannian manifold M is defined and the following results are proved: 1) There are compact 2-dimensional stratified suvsets of R3which satisfy strict Whitney condition and have not finite volume. 2) If a closed p-dimensional stratified subset of M satisfies Whitney condition and has strata in dimension p and p — 1 only, then it has locally finite volume. 3) Subanalytic sets have locally finite volume.