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In relazione con sue precedenti ricerche, l'A. dimostra che se logB(x) è a variazione limitata trax 0 e+∞, eC(x) assolutamente integrabile nello stesso intervallo, per ogni integrale dell' equazione differenzialey′′+[B(x)+C(x)]y=0 si ha: $$y\sqrt {B(x)} = Re^{\varepsilon 1} \cos \left( {\int\limits_{x_0 }^x {\sqrt {B(t)} dt - \gamma - \varepsilon _2 } } \right),{\text{ }}y' = Re^{\varepsilon 1} \sin \left( {\int\limits_{x_0 }^x {\sqrt {B(t)} dt - \gamma - \varepsilon _2 } } \right)$$ conR e γ costanti,E 1,E 2 infinitesime perx → ∞ Segue la determinazione dell'integrale di data forma asintotica ed un' applicazione ad un caso più generale.

In this paper we consider the uniformly elliptic equation: $$ - \sum\limits_{i = 1}^3 {a_i (x)D_{x_i }^2 } u(x) + \lambda u(x) = f(x), x \in R^3 ,$$ assuming the coefficients locally belonging to the Sobolev-Slobodeckij class W4/3,4. Such an equation is nonvariational and has (in general) discontinuous coefficients. We prove, with the aid of C. Miranda technique, that for anyf ε W2/1, 2R 3) there is at least one solution u of (*) in the space W2/5,2(R 3).

Let Ω denote a bounded domain in some Riemannian manifold X with smooth boundary δω and consider a submanifold Y of Euclidean space RL with or without boundary. We show that if U: Ω → Y minimizes the penergy functional $$E_p (U,\Omega ): = \int\limits_\Omega {\left\| {DU} \right\|^p dVol}$$ for smooth boundary data g: δω → Y, then U is continuous in a neighborhood of δω. This completes the interior partial regularity results of Part I. As an application we obtain an existence theorem concerning small solutions of the Dirichlet problem for pharmonic maps.

We describe inertial endomorphisms of an abelian group $$A$$ , that is endomorphisms $$\varphi $$ with the property $$|(\varphi (X)+X)/X|<\infty $$ for each $$X\le A$$ . They form a ring $$IE(A)$$ containing the ideal $$F(A)$$ formed by the so-called finitary endomorphisms, the ring of power endomorphisms and also other non-trivial instances. We show that the quotient ring $$IE(A)/F(A)$$ is commutative. Moreover, inertial invertible endomorphisms form a group, provided $$A$$ has finite torsion-free rank. In any case, the group $$IAut(A)$$ they generate is commutative modulo the group $$FAut(A)$$ of finitary automorphisms, which is known to be locally finite. We deduce that $$IAut(A)$$ is locally-(center-by-finite). Also, we consider the lattice dual property, that is $$|X/(X\cap \varphi (X))|<\infty $$ for each $$X\le A$$ and show that this implies the above one, provided $$A$$ has finite torsion-free rank.

Sotto ampie ipotesi è dimostrato un teorema di esistenza relativo alla soluzione (in senso generalizzato) di un problema misto per il sistema semilineare (I) $$\sum\limits_{j = 1}^m {\alpha _{ij} \left( {x,y} \right)\left\{ {\frac{{\partial z_j }}{{\partial x}} + \rho _i \left( {x,y} \right)\frac{{\partial z_j }}{{\partial y}}} \right\} = f_i \left( {x, y, z_1 , ..., z_m } \right), \left( {i = 1, .. , m} \right).} $$ La soluzione é ricercata nel campo funzionale costituito dalle m-ple di funzioni zi(x, y), (i=1, ..., m), le quali nel proprio campo di definizione sono assolutamente continue in x e lipschitziane in y, e soddisfano il sistema (I) quasi ovunque.

In this paper, we study the well-posedness of weakly hyperbolic systems with time-dependent coefficients. We assume that the eigenvalues are low regular, in the sense that they are Hölder with respect to t. In the past, these kinds of systems have been investigated by Yuzawa (J Differ Equ 219(2):363–374, 2005) and Kajitani and Yuzawa (Ann Sc Norm Super Pisa Cl Sci (5) 5(4):465–482, 2006) by employing semigroup techniques (Tanabe–Sobolevski method). Here, under a certain uniform property of the eigenvalues, we improve the Gevrey well-posedness result of Yuzawa (2005) and we obtain well-posedness in spaces of ultradistributions as well. Our main idea is a reduction of the system to block Sylvester form and then the formulation of suitable energy estimates inspired by the treatment of scalar equations in Garetto and Ruzhansky (J Differ Equ 253(5):1317–1340, 2012).

In two earlier papers, we presented a perturbation theory for laminated, or foliated, invariant sets $\mathcal{K}^o$ for a given finite-dimensional system of ordinary differential equations, see [20,21]. The main objective in that perturbation theory is to show that: if the given vector field has a suitable exponential trichotomy on $\mathcal{K}^o$ , then any perturbed system that is C1-close to the given vector field near $\mathcal{K}^o$ has an invariant set $\mathcal{K}^n$ , where $\mathcal{K}^n$ is homeomorphic to $\mathcal{K}^o$ and where the perturbed vector field has an exponential trichotomy on $\mathcal{K}^n$ . In this paper we present a dual-faceted extension of this perturbation theory to include: (1) a class of infinite-dimensional evolutionary equations that arise in the study of reaction diffusion equations and the Navier–Stokes equations and (2) nonautonomous evolutionary equations in both finite and infinite dimensions. For the nonautonomous problem, we require that the time-dependent terms in the problem lie in a compact, invariant set M. For example, M may be the hull of an almost periodic, or a quasiperiodic, function of time.