The index of a plane curve and green’s formula

In this paper we compute the line integral of a complex function on a rectifiable cycle homologous to zero obtaining a Green’s formula with multiplicities that involves the of the function and the index of the cycle. We consider this formula in several settings and we obtain a sharp version in terms of the Lebesgue integrability properties of the partial derivatives of the function. This result depends on the proven fact that the index of a rectifiable cycle is square integrable with respect to the planar Lebesgue measure. The work of both authors is partially supported by grants 2000SGR-00059, 2001SGR 00172 of Generalitat de Catalunya and BFM 2002-04072-C02-02 of Ministerio de Ciencia y Tecnologia     

Invariant manifolds of partial functional differential equations

This paper is concerned with the existence, smoothness and attractivity of invariant manifolds for evolutionary processes on general Banach spaces when the nonlinear perturbation has a small global Lipschitz constant and locally C^k-smooth near the trivial solution. Such a nonlinear perturbation arises in many applications through the usual cut-off procedure, but the requirement in the existing literature that the nonlinear perturbation is globally C^k-smooth and has a globally small Lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut-off function. Our general results are illustrated by and applied to partial functional differential equations for which the phase space (where r>0 and being a Banach space) has no smooth inner product structure and for which the validity of variation-of-constants formula is still an interesting open problem.     

On the Existence of Strong Solutions to a Coupled Fluid-Structure Evolution Problem

We consider here a model of fluid-structure evolution problem which, in particular, has been largely studied from the numerical point of view. We prove the existence of a strong solution to this problem.     

Universal fundamental systems in ordered Banach spaces

Let E be a separable Banach space ordered by a reproducing cone with empty interior. We prove the existence of operator functions A : [0, ) P (P the cone of monotone increasing linear operators) and of initial values x₀ such that the solution of x (t) = A(t) x(t), x(0) = x₀, is dense in E.Received: 27 February 2004     

Polar unitals in compact eight-dimensional planes

We determine centralizers and unitals for the polarities of eight-dimensional compact planes with at least 17-dimensional group of automorphisms, and discuss transitivity properties.Received: 7 August 2003     

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c₀ and ?_p for 1?p<∞. We add a new member to this family by showing that there are exactly four closed ideals in for the Banach space E?(⊕?₂ⁿ)_c0, that is, E is the c₀-direct sum of the finite-dimensional Hilbert spaces ?₂¹,?₂²,…,?₂ⁿ,… .     

Continuity properties for modulation spaces, with applications to pseudo-differential calculus—I

Let M^p,q denote the modulation space with parameters p,q∈[1,∞]. If 1/p₁+1/p₂=1+1/p₀ and 1/q₁+1/q₂=1/q₀, then it is proved that . The result is used to get inclusions between modulation spaces, Besov spaces and Schatten classes in calculus of Ψdo (pseudo-differential operators), and to extend the definition of Toeplitz operators. We also discuss continuity of ambiguity functions and Ψdo in the framework of modulation spaces.     

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

We prove the well-posedness of the transmission problem for the Laplacian across a Lipschitz interface, with optimal non-tangential maximal function estimates, for data in Lebesgue and Hardy spaces on the boundary. As a corollary, we show that the spectral radius of the (adjoint) harmonic double layer potential K∗ in is less than , whenever is a bounded convex domain and 1<p?2.     

Large time behavior of the heat kernel

In this paper we study the large time behavior of the (minimal) heat kernel k_P^M(x,y,t) of a general time-independent parabolic operator Lu=u_t+P(x,∂_x)u which is defined on a noncompact manifold M. More precisely, we prove that