Volume 15, Issue 1

In this article we investigate solutions to a semilinear partial differential equation with non Lipschitz nonlinearity by using recent theories of generalized functions. To give a meaning to a non Lipschitz characteristic Cauchy problem with irregular data, we replace it by a three parameter family of problems. The first parameter turns the problem into a family of Lipschitz problems, the second one converts the given problem to a non-characteristic one, whereas the third one regularizes the data. Finally, the family of problems is solved in an appropriate three parametric ( C , ɛ , P ) algebra.

We relax the regularity conditions on potentials of higher-dimensional periodic Schrödinger operators while their resolvents may still be defined as compact operators on L 2 . This enables us to define the Bloch varieties locally as the zero locus of a holomorphic map in a more general setting. We also give an asymptotic description of the Fermi curve.

We show that ◊(ℝ, , ∈) together with CH and “all Aronszajn trees are special” is consistent relative to ZFC. The weak diamond for the covering relation of Lebesgue null sets was the only weak diamond in the Cichoń diagramme for relations whose consistency together with “all Aronszajn trees are special” was not yet settled. Our forcing proof gives also new proofs to the known consistencies of several other weak diamonds stemming from the Cichoń diagramme together with “all Aronszajn trees are special” and CH. The main part of our work is an application [Shelah, Proper and Improper Forcing, Springer-Velag, 1998, Chapter V, §§ 1–7] for a special completeness system, such that we have a genericity game. Thus we show new preservation properties of the known forcings.

A smooth variation of constants formula for semilinear hyperbolic systems is established using a suitable Banach space X of continuous functions together with its sun dual space X ⊙ *. It is shown that mild solutions of this variation of constants formula generate a smooth semiflow in X . This proves that the stability of stationary states for the nonlinear flow is determined by the stability of the linearized semigroup.

We analyze term-by-term differentiability of uniformly convergent series of the form , where S m –1 is the unit sphere in , and { Y k } is a sequence of spherical harmonics or even more general functions. Since this class of kernels includes the continuous positive definite kernels on S m –1 , the results in this paper will show that, under certain conditions, the action of convenient differential operators on positive definite (strictly positive definite) kernels on S m –1 generate positive definite kernels.

We consider pairs 〈 A, H ( A )〉 where A is an algebra of sets from some class called the class of algebras of type 〈 κ , λ 〉 and where H ( A ) is the ideal of hereditary sets of A . We characterize which of the above pairs are topological, that is, which are fields of sets with nowhere dense boundary for some topology together with the ideal of nowhere dense sets for this topology. Making use of a theorem of Fichtenholz and Kantorovich which says that in P ( κ ) there is an independent family of cardinality 2 κ , we construct an example of a pair 〈algebra, ideal〉 with complete quotient algebra and the hull property but not topological. This countrexample, given in ZFC, provides the complete solution of a problem posed in [Balcerzak, Bartoszewicz, Ciesielski, Real Anal. Exchange 29: 265–273, 2003/04]. Such an algebra was constructed in [Bartoszewicz, Topology Appl. 149: 9–15, 2005] under some aditional set theoretic assumption.

In this paper, the concept of directional convex extension of the convex valued maps is introduced. Necessary and sufficient conditions for the existence of the directional convex extension of the convex valued map are obtained.

In this paper we describe the W-convergence in C ( X, Y ) where X is a normal topological space, ( Y, d ) is a metric space and C ( X, Y ) is the space of all continuous functions from X to Y . The main theorem of this paper states that the sequence (ƒ n ) n ∈ℕ of functions from C ( X, Y ) is W-convergent to a function ƒ ∈ C ( X, Y ) if and only if it is uniformly convergent to ƒ and there exists a countably compact subset K ⊂ X such that if U is a neighborhood of K then there exists n 0 ∈ ℕ for which ƒ n | ( X \ U ) = ƒ| ( X \ U ) if n ≥ n 0 .