Multiple equilibria,periodic solutions and a priori bounds for solutions in superlinear parabolic problems

Consider the Dirichlet problem for the parabolic equation in , where $\Omega$ is a bounded domain in and f has superlinear subcritical growth in u. If f is independent of t and satisfies some additional conditions then using a dynamical method we find multiple (three, six or infinitely many) nontrivial stationary solutions. If f has the form where m is periodic, positive and m,g satisfy some technical conditions then we prove the existence of a positive periodic solution and we provide a locally uniform bound for all global solutions.     

Critical eigenvalues for a nonlinear problem

We study the application, , where is the supremum of positive s such that the problem admits a solution. Where B ₁ is the unit ball in We show that is a decreasing function, with where is the unique solution of the problem . We also give the explicit solutions of the problem , when and show that . We show that the problem doesnt admit a solution. In the end, we give a numerical approximation of , when .     

A singular Moser-Trudinger embedding and its applications

Let Ω be a bounded domain in , we prove the singular Moser-Trudinger embedding: if and only if where and . We will also study the corresponding critical exponent problem.     

On higher order Bessel potentials

The solutions of the equation in , where are investigated, Bessel potentials of higher order are defined, and recurrence relations between these solutions and these Bessel potentials are obtained. It is also proved that these solutions and the solutions of , under certain conditions, are identical. Received: 6 November 2002     

The Laplacian on $$ C(\overline \Omega) $$ with generalized Wentzell boundary conditions

In this note we prove that the Laplacian with generalized Wentzell boundary conditions on an open bounded regular domain in defined by generates an analytic semigroup of angle on for every > 0 and (for the definition of cf. (1.3)).Received: 13 July 2002     

Quasi-linear parabolic problems in L 1 with non homogeneous conditions on the boundary

We are interested in parabolic problems with L¹ data of the type

Categorical Structures Enriched in a Quantaloid: Orders and Ideals over a Base Quantaloid

Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid , which we call ‘ -order’. This requires a theory of semicategories enriched in the quantaloid , that admit a suitable Cauchy completion. There is a quantaloid of -orders and ideal relations, and a locally ordered category of -orders and monotone maps; actually, . In particular is , with Ω a locale, the category of ordered objects in the topos of sheaves on Ω. In general -orders can equivalently be described as Cauchy complete categories enriched in the split-idempotent completion of . Applied to a locale Ω this generalizes and unifies previous treatments of (ordered) sheaves on Ω in terms of Ω-enriched structures.Mathematics Subject Classifications (2000) 06F07, 18B35, 18D05, 18D20.     

Birkhoff coordinates for KdV on phase spaces of distributions

The purpose of this paper is to extend the construction of Birkhoff coordinates for the KdV equation from the phase space of square integrable 1-periodic functions with mean value zero to the phase space of mean value zero distributions from the Sobolev space endowed with the symplectic structure More precisely, we construct a globally defined real-analytic symplectomorphism where is a weighted Hilbert space of sequences supplied with the canonical Poisson structure so that the KdV Hamiltonian for potentials in is a function of the actions alone.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Boundedness of Marcinkiewicz Integral

A weighted norm inequality for the Marcinkiewicz integral operator is proved when belongs to . We also give the weighted L^p-boundedness for a class of Marcinkiewicz integral operators with rough kernels and related to the Littlewood-Paley -function and the area integral S, respectively.     

On the Schwarz reflection principle for monogenic functions

Let ∑ be either an oriented hyperplane or the unit sphere in , let be open and connected and let be an open and connected domain in such that . If in is a null solution of the Dirac operator (also called a monogenic function in ) which is continuously extendable to , then conditions upon are given enabling the monogenic extension of across . In such a way Schwarz reflection type principles for monogenic functions are established in the Spin (1) and Spin cases. The Spin (1) case includes the classical Schwarz reflection principle for holomorphic functions in the plane. The Spin case deals with so-called “half boundary value problems” for the Dirac operator. Received: 2 February 2006     

Invariants and spaces of zero solutions of linear partial differential operators

It is shown that for open convex , d > 1 and a nontrivial polynomial P the space does not have property . If P is elliptic or homogeneous, then this holds for every open Ω. For even cannot occur and if it occurs for some Ω, then P must be hypoelliptic. Received: 18 July 2005     

A Helly-type theorem for countable intersections of starshaped sets

Let k and d be fixed integers, 0kd, and let be a collection of sets in If every countable subfamily of has a starshaped intersection, then is (nonempty and) starshaped as well. Moreover, if every countable subfamily of has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of is at least k-dimensional, too. Finally, dual statements hold for unions of sets.Received: 3 April 2004     

Maximal regularity for evolution equations in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

Let X be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy problem enjoys maximal regularity in weighted L _p -spaces with weights , where , if and only if it has the property of maximal L _p -regularity. Moreover, it is also shown that the derivation operator admits an -calculus in weighted L _p -spaces. Received: 26 February 2003     

On <Emphasis Type="Italic">N</Emphasis>-Summations,II

Given any R-semimodule M equipped with a semitopology we construct an N-protosummation for M. If satisfies certain properties, then a similar construction leads to an unconditional N-summation for M, that is an N-summation for M equipped with the trivial prenorm MD over the N-summation (D^N,_D) for D. Conversely any N-protosummation on M gives rise to a topology . If both and satisfy a certain separation property, then and form a Galois connection. Dedicated to my friend and collegue Nico Pumplün on the occasion of his 70th birthdayMathematics Subject Classifications (2000) 16Y60, 54A05.     

Holomorphic injectivity and the Hopf map

We give sharp conditions on a local biholomorphism which ensure global injectivity. For n 2, such a map is injective if, for each complex line the pre-image F^–1(l) embeds holomorphically as a connected domain into the embedding being unique up to Möbius transformation. In particular, F is injective if the pre-image of every complex line is connected and conformal to The proof uses the topological fact that the natural map associated to the Hopf map admits no continuous sections and the classical Bieberbach–Gronwall estimates from complex analysis.     

{\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pairs}}

Let R be a unital associative ring and two classes of left R-modules. In this paper we introduce the notion of a In analogy to classical cotorsion pairs as defined by Salce [10], a pair of subclasses and is called a if it is maximal with respect to the classes and the condition for all and Basic properties of are stated and several examples in the category of abelian groups are studied. Received: 17 March 2005     

Uniformly summing sets of operators on spaces of vector–valued continuous functions

Let X, Y be Banach spaces. We say that a set is uniformly p–summing if the series is uniformly convergent for whenever (x_n) belongs to . We consider uniformly summing sets of operators defined on a -space and prove, in case X does not contain a copy of c₀, that is uniformly summing iff is, where T (φ x) = (T^#φ) x for all and x∈X. We also characterize the sets with the property that is uniformly summing viewed in . Received: 1 July 2005     

Commutators on Half-Spaces

We study the boundedness and compactness of commutators on , where and are defined by and respectively. If satisfies some upper and lower estimates, then we obtain a necessary and sufficient condition for to be bounded or compact on for . The reproducing kernel of the harmonic Bergman space of can be shown to satisfy all the required estimates. Our result is the real variable analogue of the complex variable one for commutators associated with an analytic reproducing kernel.     

Non-collision periodic solutions for singular Hamiltonian systems

We study the existence of classical (non-collision) T-periodic solutions of the Hamiltonian system where and is a T-periodic function in t which has a singularity at like Under suitable conditions on H, we prove that if then (HS) possesses at least one non-collision solution and if then the generalized solution of (HS) obtained in [5] has at most one time of collision in its period.