Wyniki wyszukiwania

1

On the Conley index in Hilbert spaces in the absence of uniqueness

100%

Izydorek M., Rybakowski K.

Fundamenta Mathematicae

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2002

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tom 171

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nr 1

31-52

EN

Consider the ordinary differential equation (1) ẋ = Lx + K(x) on an infinite-dimensional Hilbert space E, where L is a bounded linear operator on E which is assumed to be strongly indefinite and K: E → E is a completely continuous but not necessarily locally Lipschitzian map. Given any isolating neighborhood N relative to equation (1) we define a Conley-type index of N. This index is based on Galerkin approximation of equation (1) by finite-dimensional ODEs and extends to the non-Lipschitzian case the ℒ𝓢-Conley index theory introduced in [9]. This extended ℒ𝓢-Conley index allows applications to strongly indefinite variational problems ∇Φ(x) = 0 where Φ: E → ℝ is merely a C¹-function.

2

Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory

100%

Izydorek M., Rybakowski K.

Fundamenta Mathematicae

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2003

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tom 176

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nr 3

233-249

EN

Let Ω be a bounded domain in $ℝ^{N}$ with smooth boundary. Consider the following elliptic system: $-Δu = ∂_{v}H(u,v,x)$ in Ω, $-Δv = ∂_{u}H(u,v,x)$ in Ω, u = 0, v = 0 in ∂Ω. (ES) We assume that H is an even "-"-type Hamiltonian function whose first order partial derivatives satisfy appropriate growth conditions. We show that if (0,0) is a hyperbolic solution of (ES), then (ES) has at least 2|μ| nontrivial solutions, where μ = μ(0,0) is the renormalized Morse index of (0,0). This proves a conjecture by Angenent and van der Vorst.

3

Conley index in Hilbert spaces and a problem of Angenent and van der Vorst

100%

Izydorek M., Rybakowski K.

Fundamenta Mathematicae

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2002

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tom 173

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nr 1

77-100

EN

In a recent paper [9] we presented a Galerkin-type Conley index theory for certain classes of infinite-dimensional ODEs without the uniqueness property of the Cauchy problem. In this paper we show how to apply this theory to strongly indefinite elliptic systems. More specifically, we study the elliptic system $-Δu = ∂_{v}H(u,v,x)$ in Ω, $-Δv = ∂_{u}H(u,v,x)$ in Ω, u = 0, v = 0 in ∂Ω, (A1) on a smooth bounded domain Ω in $ℝ^{N}$ for "-"-type Hamiltonians H of class C² satisfying subcritical growth assumptions on their first order derivatives. As shown by Angenent and van der Vorst in [1], the solutions of (A1) are equilibria of an abstract ordinary differential equation ż = f(z) (A2) defined on a certain Hilbert space E of functions z = (u,v). The map f: E → E is continuous, but, in general, not differentiable nor even locally Lipschitzian. The main result of this paper is a Linearization Principle which states that whenever z₀ is a hyperbolic equilibrium of (A2) then the Conley index of {z₀} can be computed by formally linearizing (A2) at z₀. As a particular application of the Linearization Principle we obtain an elementary, Conley index based proof of the existence of nontrivial solutions of (A1), a result previously established in [1] via Morse-Floer homology. Further applications of our method to existence and multiplicity results for strongly indefinite systems appear in [3] and [10].

4

Continuation of the connection matrix for singularly perturbed hyperbolic equations

100%

Carbinatto M., Rybakowski K.

Fundamenta Mathematicae

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2007

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tom 196

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nr 3

253-273

EN

Let $Ω ⊂ ℝ^{N}$, N ≤ 3, be a bounded domain with smooth boundary, γ ∈ L²(Ω) be arbitrary and ϕ: ℝ → ℝ be a C¹-function satisfying a subcritical growth condition. For every ε ∈ ]0,∞[ consider the semiflow $π_{ε}$ on H¹₀(Ω) × L²(Ω) generated by the damped wave equation ε∂ₜₜu + ∂ₜu = Δu + ϕ(u) + γ(x), x ∈ Ω, t > 0, u(x,t) = 0, x ∈ ∂Ω, t > 0 Moreover, let π' be the semiflow on H¹₀(Ω) generated by the parabolic equation ∂ₜu = Δu + ϕ(u) + γ(x), x ∈ Ω, t > 0, u(x,t) = 0, x ∈ ∂Ω, t > 0 Let Γ: H²(Ω) → H¹₀(Ω) × L²(Ω) be the imbedding u ↦ (u,Δu+ϕ(u)+γ). We prove that whenever K' is a compact isolated π'-invariant set and $(Mp')_{p∈P}$ is a partially ordered Morse decomposition of K' then the imbedded sets K = Γ(K') and $M_{p,0} = Γ(Mp')$, p ∈ P, continue, for ε > 0 small, to an isolated $π_{ε}$-invariant set $K_{ε}$ a Morse decomposition $(M_{p,ε})_{p∈P}$ of $K_{ε}$, relative to $π_{ε}$, such that the homology index braid of $(π_{ε},K_{ε},(M_{p,ε})_{p∈P}) is isomorphic to the homology index braid of $(π',K',(M'_{p})_{p∈P})$. This, in particular, implies a connection matrix continuation principle.

Ograniczanie wyników

4 Fundamenta Mathematicae

4 Rybakowski K.

3 Izydorek M.

1 Carbinatto M.

1 2007

1 2003

2 2002

On the Conley index in Hilbert spaces in the absence of uniqueness

Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory

Conley index in Hilbert spaces and a problem of Angenent and van der Vorst

Continuation of the connection matrix for singularly perturbed hyperbolic equations