<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> norm estimates of the cost of the controllability for heat equations

This paper is concerned with the bound of the cost of approximate controllability and null controllability of heat equations, i.e., the minimal Lp norm and L∞ norm of a control needed to control the system approximately or a control needed to steer the state of the system to zero. The methods we use combine observability inequalities, energy estimates for heat equations and the dual theory.     

Square functions and <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> calculus on subspaces of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and on Hardy spaces

Let X be a (closed) subspace of L^p with 1≤p<∞, and let A be any sectorial operator on X. We consider associated square functions on X, of the form and we show that if A admits a bounded H^∞ functional calculus on X, then these square functions are equivalent to the original norm of X. Then we deduce a similar result when X=H¹(ℝ^N) is the usual Hardy space, for an appropriate choice of || ||_F. For example if N=1, the right choice is the sum for h ∈ H¹(ℝ), where H denotes the Hilbert transform.     

An <Emphasis Type="Italic">L</Emphasis>-approach for packing (<Emphasis Type="Italic">ℓ</Emphasis>, <Emphasis Type="Italic">w</Emphasis>)-rectangles into rectangular and <Emphasis Type="Italic">L</Emphasis>-shaped pieces

This paper presents an approach using a recursive algorithm for packing (?, w)-rectangles into larger rectangular and L-shaped pieces. Such a problem has actual applications for non-guillotine cutting and pallet/container loading. Our motivation for developing the L-approach is based on the fact that it can solve difficult pallet loading instances. Indeed, it is able to solve all testing problems (more than 20 000 representatives of infinite equivalence classes of the literature), including the 18 hard instances unresolved by other heuristics. We conjecture that the L-approach always finds optimum packings of (?, w)-rectangles into rectangular pieces. Moreover, the approach may also be useful when dealing with cutting and packing problems involving L-shaped pieces.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">μ</Emphasis>) for vector-valued measure <Emphasis Type="Italic">μ</Emphasis>

Let μ be a measure from a σ-algebra of subsets of a set T into a sequentially complete Hausdorff topological vector space X. Assume that μ is convexly bounded, i.e., the convex hull of its range is bounded in X, and denote by L ¹(μ) the space of scalar valued functions on T which are integrable with respect to the vector measure μ. We study the inheritance of some properties from X to L ¹(μ). We show that the bounded multiplier property passes from X to L ¹(μ). Answering a 1972 problem of Erik Thomas, we show that for a rather large class of F-spaces X the non-containment of c ₀ passes from X to L ¹(μ).     

Fibers of the <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> algebra and disintegration of measures

It is shown that Gelfand transforms of elements \({f\in L^{\infty} (\mu)}\) are almost constant at almost every fiber \({\Pi^{-1}(\{x\})}\) of the spectrum of L ^∞(μ) in the following sense: for each \({f\in L^{\infty} (\mu)}\) there is an open dense subset U = U(f) of this spectrum having full measure and such that the Gelfand transform of f is constant on the intersection \({\Pi^{-1}(\{x\})\cap U}\). As an application a new approach to disintegration of measures is presented, allowing one to drop the usually taken separability assumption.     

Harmonic maps from manifolds of <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>-Riemannian metrics

For a bounded domain Ω ⊂ R ⁿ endowed with L ^∞-metric g, and a C ⁵-Riemannian manifold (N, h) ⊂ R ^k without boundary, let u ∈ W ^1,2(Ω, N) be a weakly harmonic map, we prove that (1) u ∈ C ^α (Ω, N) for n = 2, and (2) for n ≥ 3, if, in additions, g ∈ VMO(Ω) and u satisfies the quasi-monotonicity inequality (1.5), then there exists a closed set Σ ⊂ Ω, with H ^n-2(Σ) = 0, such that for some α ∈ (0, 1). C. Y. Wang Partially supported by NSF.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Poincaré inequality for general symmetric forms

Lp Poincare inequalities for general symmetric forms are established by new Cheeger＇s isoperimetric constants. Lp super-Poincare inequalities are introduced to describe the equivalent conditions for the Lp compact embedding, and the criteria via the new Cheeger＇s constants for those inequalities are presented. Finally, the concentration or the volume growth of measures for these inequalities are studied.     

<Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> Estimates of solution for <Emphasis Type="Italic">m</Emphasis>-Laplacian parabolic equation with a nonlocal term

In this paper, we consider the global existence, uniqueness and L ^∞ estimates of weak solutions to quasilinear parabolic equation of m-Laplacian type u _t − div(|∇u| ^m−2∇u) = u|u| ^β−1 ∫_Ω |u| ^α dx in Ω × (0,∞) with zero Dirichlet boundary condition in tdΩ. Further, we obtain the L ^∞ estimate of the solution u(t) and ∇u(t) for t > 0 with the initial data u ₀ ∈ L ^q (Ω) (q > 1), and the case α + β < m − 1.     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

On blocks with <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">B</Emphasis>)−<Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">B</Emphasis>)=1

We use the method of local representation and original method of Brauer to study the block with K(B)−L(B)=1, and get some properties on the defect group and the structure of this kind of blocks. Then, we show that K(B) conjecture holds for this kind of blocks.     

Optimal <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Riemannian Gagliardo–Nirenberg inequalities

Let (M, g) be a compact Riemannian manifold of dimension n ≥  2 and 1 <  p ≤  2. In this work we prove the validity of the optimal Gagliardo–Nirenberg inequality

<Emphasis Type="Italic">C</Emphasis><Superscript>∞</Superscript> regularity of solutions of the Levi equation in<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">2n+1</Emphasis></Superscript>

We study the regularity of the solutions of the Levi equation in ?²ⁿ⁺¹. It is a second order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every functionu∈C ². We show that the operator associated to the equation can be represented as a sum of squares of non linear vector fields. Then, by using a freezing method, we prove theC ^∞ regularity of the solutions.     

Bases of exponents in weighted spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Italic">−π, π</Emphasis>)

The system of exponents $ \left\{ {e^{i\lambda _n t} } \right\}_{n \in \mathbb{Z}} $ \left\{ {e^{i\lambda _n t} } \right\}_{n \in \mathbb{Z}} is considered. A sufficient condition for a Riesz-property basis in the weighted space L ^p (−π, π) is obtained.     

Difference sets with <Emphasis Type="Italic">n</Emphasis> = 5<Emphasis Type="Italic"> p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>

Let D be a (v, k, λ)-difference set in an abelian group G, and (v, 31) = 1. If n = 5p ^r with p a prime not dividing v and r a positive integer, then p is a multiplier of D. In the case 31|v, we get restrictions on the parameters of such difference sets D for which p may not be a multiplier.      

<Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>-estimates for Sobolev functions in terms of ∇ and Δ

Several L ^∞-estimates are obtained for in terms of and , where are determined by m. If p = 2, then the estimates are given with explicit constants. However, if p ≠ 2, it is difficult to derive explicit constants except in two simple cases. Applicability to PDE’s is illustrated.     

Free involutions on <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Topological free involutions on S ¹ × S ⁿ are classified up to conjugation. We prove that this is the same as classifying quotient manifolds up to homeomorphism. There are exactly four possible homotopy types of such quotients, and surgery theory is used to classify all manifolds within each homotopy type.     

Cycles in <Emphasis Type="Italic">G</Emphasis>-orbits in <Emphasis Type="Italic">G</Emphasis><Superscript><Emphasis Type="Italic">ℂ</Emphasis></Superscript>-flag manifolds

There is a natural duality between orbits of a real form G of a complex semisimple group G on a homogeneous rational manifold Z=G /P and those of the complexification K of any of its maximal compact subgroups K: (,) is a dual pair if is a K-orbit. The cycle space C() is defined to be the connected component containing the identity of the interior of {g:g() is non-empty and compact}. Using methods which were recently developed for the case of open G-orbits, geometric properties of cycles are proved, and it is shown that C() is contained in a domain defined by incidence geometry. In the non-Hermitian case this is a key ingredient for proving that C() is a certain explicitly computable universal domain.Research of the first author partially supported by Schwerpunkt Global methods in complex geometry and SFB-237 of the Deutsche Forschungsgemeinschaft.The second author was supported by a stipend of the Deutsche Akademische Austauschdienst.     

<Emphasis Type="Italic">L</Emphasis>log <Emphasis Type="Italic">L</Emphasis> Condition for Supercritical Branching Hunt Processes

In this paper we use the spine decomposition and martingale change of measure to establish a Kesten–Stigum Llog L theorem for branching Hunt processes. This result is a generalization of the results in Asmussen and Hering (Z. Wahrscheinlichkeitstheor. Verw. Geb. 36:195–212, 1976) and Hering (Branching Processes, pp. 177–217, 1978) for branching diffusions.     

Existence of ℋ-matrix approximants to the inverse FE-matrix of elliptic operators with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">∞</Emphasis></Superscript>-coefficients

This article deals with the existence of blockwise low-rank approximants — so-called -matrices — to inverses of FEM matrices in the case of uniformly elliptic operators with L-coefficients. Unlike operators arising from boundary element methods for which the -matrix theory has been extensively developed, the inverses of these operators do not benefit from the smoothness of the kernel function. However, it will be shown that the corresponding Green functions can be approximated by degenerate functions giving rise to the existence of blockwise low-rank approximants of FEM inverses. Numerical examples confirm the correctness of our estimates. As a side-product we analyse the -matrix property of the inverse of the FE mass matrix. Mathematics Subject Classification (1991):35C20, 65F05, 65F50, 65N30