Showing papers in "Topological Methods in Nonlinear Analysis in 1997"

Hessian measures I

Abstract: Alternatively we may write (1.3) Fk[u] = [Du]k, where [A]k denotes the sum of the k× k principal minors of an n× n matrix A. Our purpose in this paper is to extend the definition of the Fk to corresponding classes of continuous functions so that Fk[u] is a Borel measure and to consider the Dirichlet problem in this setting. A function u ∈ C(Ω) is called k-convex (uniformly k-convex) in Ω if Fj [u] ≥ 0 (> 0) for j = 1, . . . , k. The operator Fk

A Maslov-type index theory for symplectic paths

Abstract: In this paper, we extend the Maslov-type index theory defined in [7], [15], [10], and [18] to all continuous degenerate symplectic paths, give a topological characterization of this index theory for all continuous symplectic paths, and study its basic properties. Suppose τ > 0. We consider an τ -periodic symmetric continuous 2n × 2n matrix function B(t), i.e. B ∈ C(Sτ ,Ls(R)) with Sτ = R/(τZ), L(R2n) being the set of all real 2n×2n matrices, and Ls(R) being the subset of all symmetric matrices. It is well-known that the fundamental solution γ of the linear first order Hamiltonian system

On the fundamental solution of a perturbed harmonic sscillator

Abstract: where H is the Hamiltonian of the system (we choose the units so that ~ = 1). If the operator H does not change in time, then, given ψ(0) = ψ0, we have ψ(t) = eψ0. The fundamental solution of the time-dependent Schrodinger equation is the distributional kernel of the solution operator e−itH . For a non-relativistic quantum particle of mass 1 moving in the space of n dimensions in a potential field V (x), the operator H has the form

Asymptotic analysis of the Navier-Stokes equations in the domains

Abstract: We are interested in this article with the Navier–Stokes equations of viscous incompressible fluids in three dimensional thin domains Let Ωe be the thin domain Ωe = ω × (0, e), where ω is a suitable domain in R and 0 < e < 1 Our aim is to derive an asymptotic expansion of the strong solution u of the Navier–Stokes equations in the thin domain Ωe when e is small, which is valid uniformly in time This study should give a better understanding of the global existence results in thin domains obtained previously; see [15]–[17] and [23], [22] We consider in this work two types of boundary conditions: the Dirichlet-periodic boundary condition and the purely periodic condition For the first type of boundary condition we derive an asymptotic expansion of the solution u in terms of the solution of the associated Stokes problem More precisely, we prove that the solution can be written, for e small, as

Leray-Schauder continuation theorems in the absence of a priori bounds

Abstract: In 1934, Leray and Schauder have published their fundamental paper Topologie et equations fonctionnelles [37], which is the founding father of algebraic topology in infinite-dimensional spaces and a milestone in nonlinear functional analysis and nonlinear differential equations. The style of this paper is still amazingly modern and its influence in contemporary mathematics considerable. This paper was among the thirty-seven most quoted mathematical papers for the period 1950–1965 and its influence still increased in the early seventies, with the development of bifurcation theory, global analysis and the use of topological techniques in critical point theory. The reader can consult the references [53, 32, 40] to get a first idea of the tremendous bibliography related to the consequences and extensions of [37], and the celebrated books of Ladyzhenskaya et al. for striking applications to Navier–Stokes equations [33] and to nonlinear elliptic [34] or parabolic partial differential equations [35]. The central topic of this paper is the study of continuation theorems for proving the existence of a solution to some equations. LetX and Y be topological spaces, A ⊂ X, and f : X → Y , g : X → Y two continuous mappings. The fundamental idea of the continuation method to solve the equation

Einstein-Bianchi hyperbolic system for general relativity

Abstract: We consider the evolution part of the Cauchy problem in General Relativity [8] as the time history of the two fundamental forms of a spacelike hypersurface: its metric g and its extrinsic curvature K. On such a hypersurface, for example an “initial” one, these two quadratic forms must satisfy four initial value or constraint equations. These constraints can be posed and solved as an elliptic system by known methods that will not be discussed here. (See, for example, [8].) The Ricci tensor of the spacetime metric can be displayed in a straightforward 3 + 1 decomposition giving the time derivatives of g and K in terms of the space derivatives of these quantities. These expressions contain also the lapse and shift functions characterizing the threading of the spacelike hypersurfaces by time lines. However, proof of the existence of a causal evolution in local Sobolev spaces into an Einsteinian spacetime does not result directly from these equations, which do not form a hyperbolic system for arbitrary lapse and shift, despite the fact that their characteristics are only the light cone and the normal to the time slices [13].

On asymptotically autonomous differential equations in the plane

Abstract: In this paper, we study qualitative behaviour of trajectories of solutions of perturbed autonomous differential equations in the plane. We work in the framework of an axiomatic theory of solution spaces of ordinary differential equations suggested by V. V. Filippov (see the survey [7] and the references therein). This theory provides a unified approach to the study of solutions of ordinary differential equations, including equations with singularities, as well as of differential inclusions. The theory sets a series of axioms which reflect fundamental properties of solution sets of ordinary differential equations and deals with sets of functions satisfying one or another set of these axioms. Topological structures introduced make it possible to deal with sets of solutions as with elements of a topological space. It is well known that many results in the classical qualitative theory of ordinary differential equations extend to dynamical systems. The theory suggested by Filippov allows one to develop such results in another direction and to extend them, in particular, to differential equations with singularities of various types. There are also many situations where the methods developed lead to new results in the classical realms, even for equations y′ = f(t, y) (y′ = dy/dt) with

Solutions of superlinear at zero elliptic equations via Morse theory

Abstract: (1) { −∆u = f(u) in Ω, u = 0 on ∂Ω, where Ω ⊂ R is an open bounded domain with smooth boundary. We assume that f ∈ C(R,R) satisfies f(0) = 0, so the constant function u ≡ 0 is a trivial solution of (1). We are interested in the existence of nontrivial solutions when f is superlinear at zero, that is near zero it looks like O(u|u|ν−2) for some ν ∈ (1, 2). More precisely, we assume that f and its primitive

A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert spaces

Abstract: A generalized cohomology, similar to Szulkin's cohomology but with more general functorial properties, is constructed. This theory is used to define a relative Morse index and to prove relative Morse relations for strongly indefinite functionals on Hilbert spaces.

Convergence of solutions of quasi-variational inequalities and applications

Abstract: where fn : E × E → R, f : E × E → R, φn : E × E → R ∪ {∞}, φ : E × E → R ∪ {∞}. The aim of this paper is to give suitable conditions on the convergence of (fn)n to f and (φn)n to φ in order to obtain a convergence result for the solutions of (1.1)n to solutions of (1.2). This study was motivated by the increasing interest in the topic of generalized quasi-variational inequalites (in short g.q.v.i.), taking into account that a g.q.v.i. (see (3.2)) can be represented by a q.v.i. (1.2) with appropriate functions f and φ. Moreover, while the problem of the existence of solutions of q.v.i. and g.q.v.i. has been investigated in many papers (see for example [5], [6], [14]), the problem of convergence of solutions of q.v.i. has been studied, in a particular setting, only in [3]. Finally, in Section 4, we consider Nash equilibria with dependent constraints (called in [7] generalized Nash equilibrium), which can be thought

Some new results on global nonexistence for abstract evolution with positive initial energy

Abstract: where P and Q(t) are linear self-adjoint operators, and A(t, u) and F (t, u) are typically a divergence operator in u and a nonlinear driving force. Other versions of (1.1) were considered earlier by Levine [3–6], for which he introduced the important technique of “concavity” analysis of auxiliary second order differential inequalities. In all these papers the principal mechanism of blow–up was the assumption of negative initial energy. In an interesting paper [10], which has just appeared, Ono has also used concavity analysis to study blow–up, but in the more general case when the initial energy is allowed to take appropriately small positive values. His analysis primarily considers linear wave operators, and moreover is restricted to bounded domains in R. (It should, however, be added that Ono also allows Kirchhoff type operators, an added generalization but without serious affect on the principal ideas.) Here we discuss some extentions of Ono’s analysis to the abstract equation (1.1), see Theorem 1. Moreover, in concrete cases, we introduce appropriate methods to treat divergence structure operators in unbounded domains (including but not necessarily restricted to R). Our conclusions also yield a larger class of initial data than in [10] for which blow-up must occur; see Remark 1 in Section 3. In the next section we give a precise meaning to equation (1.1), and give our main abstract theorem. Section 3 discusses a divergence structure equation in R for which blow–up occurs for positive initial energy, even for unbounded domains. Here the primary new idea, in comparison with [7] and [10], is to introduce an appropriate coercive operator associated with the equation. Proofs of the results described here will appear in the forthcoming paper [13].

Morin singularities and global geometry in a class of ordinary differential operators

Abstract: (∗) u′(t) + f(t, u(t)) = g(t), where the unknown u is a real function on S1 and the nonlinearity f : S1×R → R can assume a number of forms. Our approach is to study the global geometry of the operator F : B1 → B0, u → u′ + f(t, u) where the domain is either C1(S1) (the Banach space of periodic functions with continuous derivatives) or the Hilbert space H1(S1) of periodic functions with square integrable derivative. Ideally, we search for global changes of variables in both domain and image taking the operator F to a simple normal form. This goal has been achieved in previous occasions, starting with the seminal work of A. A. Ambrosetti and G. Prodi ([AP]) and its geometric interpretation by M. S. Berger and P. T. Church ([BC]), who showed that the operator associated to a certain nonlinear Dirichlet problem gives rise to a global fold between infinite

On the multivalued Poincaré operators

Abstract: By Poincare operators we mean the translation operator along the trajectories of the associated differential system and the first return (or section) map defined on the cross section of the torus by means of the flow generated by the vector field. The translation operator is sometimes also called as Poincare–Andronov or Levinson or, simply, T -operator. In the classical theory (see [K], [W], [Z] and the references therein), both these operators are defined to be single-valued, when assuming, among other things, the uniqueness of the initial value problems. At the absence of uniqueness one usually approximates the right-hand sides of the given systems by the locally lipschitzian ones (implying uniqueness already), and then applies the standard limiting argument. This might be, however, rather complicated and is impossible for the discontinuous right-hand sides. On the other hand, set-valued analysis allows us to handle effectively also with such classically troublesome situations. In particular, the class of admissible maps in the sense of [G] has been shown to be very useful with this respect, because generalized topological invariants like the Brouwer degree, the fixed point

Existence of multiple solutions of a two-point boundary value problem at resonance

Abstract: (1.1) { − ·· u= nu+ g(t, u), t ∈ (0, π), u(0) = u(π) = 0, where n ∈ N and g : [0, π] × R → R is a continuously differentiable function satisfying the following assumptions: (g1) g(t, 0) ≡ 0. (g2) lim|s|→∞ g′ s(t, s) = 0 uniformly in t ∈ (0, π). (g3) There exists R0 > 0 such that g(t, s)s > 0 for any s ∈ R with |s| ≥ R0 and for a.e. t ∈ (0, π). (g4) There exist some positive numbers C1, C2, R1 and r with 0 < r < 1 such that

Semisimple Frobenius (super)manifolds and quantum cohomology of $\mathbb{P}^r$

Abstract: We introduce and study a superversion of Dubrovin's notion of semisimple Frobenius manifolds. We establish a correspondence between semisimple Frobenius (super)manifolds and special solutions to the (supersymmetric) Schlesinger equations. Finally, we calculate the Schlesinger initial conditions for solutions describing quantum cohomology of projective spaces.

Carathéodory periodic perturbations of the zero vector field on manifolds

Abstract: where λ ≥ 0 is a real parameter. We deal with the problem of the existence of T -periodic solutions of (1.1), with a special attention to the case of small values of λ. Clearly, when λ = 0, any point inM may be regarded as a constant solution of (1.1). Thus, it is natural to think about M as a subset of the set X of all the pairs (λ, x), called T -pairs of (1.1), with λ ≥ 0 and x a T -periodic Caratheodory solution of (1.1) corresponding to the value λ of the parameter. In other words, (λ, x) ∈ X means that x is an absolutely continuous, T -periodic real map into M , such that ẋ(t) = λf(t, x(t)) for almost all t ∈ R. As usual, let CT (R) := CT (R,R) denote the Banach space of all the T -periodic, continuous, R-valued real functions, endowed with the standard norm of uniform convergence. Since any solution of (1.1) is (in particular) continuous, the set X will be considered embedded in the metric space [0,∞) × CT (M), where CT (M) is the subset of CT (R) of those functions whose image lies in M . We will prove that X is

Long time behavior of flows moving by mean curvature, II

Abstract: Here Ω is a bounded domain in R with a smooth boundary ∂Ω, Du = (D1u, . . . . . . , Dnu) is the gradient of u, Diu = ∂u/∂xi, ut = ∂u/∂t, H[u] = div(Du/ √ 1 + |Du|2) is the mean curvature of the graph of u, φ = φ(x) and u0 = u0(x) are given smooth functions. It is well known that, in general, the classical solvability of (1)–(3) fails if Ω is not mean convex. The latter means that the mean curvature H∂Ω(x) of ∂Ω with respect to the inward normal at x ∈ ∂Ω is nonnegative. Similarly, if Ω is not mean convex then the classical solution may not exist for the corresponding stationary problem