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On admissibility for parabolic equations in ℝⁿ

Martino Prizzi (2003)

Fundamenta Mathematicae

We consider the parabolic equation (P) u t - Δ u = F ( x , u ) , (t,x) ∈ ℝ₊ × ℝⁿ, and the corresponding semiflow π in the phase space H¹. We give conditions on the nonlinearity F(x,u), ensuring that all bounded sets of H¹ are π-admissible in the sense of Rybakowski. If F(x,u) is asymptotically linear, under appropriate non-resonance conditions, we use Conley’s index theory to prove the existence of nontrivial equilibria of (P) and of heteroclinic trajectories joining some of these equilibria. The results obtained extend...

On foundations of the Conley index theory

Roman Srzednicki (1999)

Banach Center Publications

The Conley index theory was introduced by Charles C. Conley (1933-1984) in [C1] and a major part of the foundations of the theory was developed in Ph. D. theses of his students, see for example [Ch, Ku, Mon]. The Conley index associates the homotopy type of some pointed space to an isolated invariant set of a flow, just as the fixed point index associates an integer number to an isolated set of fixed points of a continuous map. Examples of isolated invariant sets arise naturally in the critical...

On generalized f -harmonic morphisms

A. Mohammed Cherif, Djaa Mustapha (2014)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we study the characterization of generalized f -harmonic morphisms between Riemannian manifolds. We prove that a map between Riemannian manifolds is an f -harmonic morphism if and only if it is a horizontally weakly conformal map satisfying some further conditions. We present new properties generalizing Fuglede-Ishihara characterization for harmonic morphisms ([Fuglede B., Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978), 107–144], [Ishihara T., A...

On harmonic vector fields.

Jerzy J. Konderak (1992)

Publicacions Matemàtiques

A tangent bundle to a Riemannian manifold carries various metrics induced by a Riemannian tensor. We consider harmonic vector fields with respect to some of these metrics. We give a simple proof that a vector field on a compact manifold is harmonic with respect to the Sasaki metric on TM if and only if it is parallel. We also consider the metrics II and I + II on a tangent bundle (cf. [YI]) and harmonic vector fields generated by them.

On multivortex solutions in Chern-Simons gauge theory

Michael Struwe, Gabriella Tarantello (1998)

Bollettino dell'Unione Matematica Italiana

Motivati dall'analisi asintotica dei vortici nella teoria di Chern-Simons-Higgs, si studia l'equazione - Δ ⁢ u = λ ⁢ e u ∫ Ω e u ⁢ d ⁢ x - 1 Ω , u ∈ H 1 ⁢ Ω dove Ω = R 2 / Z 2 é il toro piatto bidimensionale. In contrasto con l'analogo problema di Dirichlet, si dimostra che per λ ∈ 8 ⁢ π , 4 ⁢ π 2 l'equazione ammette una soluzione non banale. Tale soluzione cattura il carattere bidimensionale dell'equazione, nel senso che, per tali valori di λ , l'equazione non può ammettere soluzioni (periodiche) non banali dipendenti da una sola variabile (vedi [10]).

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