University Jena ,   Department of Mathematics & Computer Science ,   Mathematical Institute ,   Research group "Function spaces"

International Conference

Function Spaces, Differential Operators,
and Nonlinear Analysis


in honour of Hans Triebel on his 65th birthday

                                                June, 28 - July, 4, 2001, in Teistungen (Germany)

Invited Talks

Oleg V. Besov Gérard Bourdaud Djairo G. De Figueiredo Ron De Vore David E. Edmunds
Vladimir Mazya Akihiko Miyachi Stanislav I. Pohozaev Michael Solomyak Gunther Uhlmann

  Oleg V. Besov Steklov Mathematical Institute, Moscow / Russia

Spaces of differentiable functions

Famous books by Professor Hans Triebel (FSU, Jena), to whom the Conference is dedicated contain an excellent exposition of and detailed historical remarks on the theory of function spaces. This theory was essentially developed by mathematicians of Russia (= fSU): S.L. Sobolev (spaces $W^k_p$ ), S.M. Nikol'skiy (classes $H^r_p$), P.I. Lizorkin (Liouville spaces $L^r_p$), O.V. Besov, V.P, Il'in, L.D. Kudryavtzev, G.A. Kalyabin and many others. The results of these investigations are summarized in their books. The aim of this talk is to give a survey of new achievements related to various aspects of function spaces: imbeddings and equivalent normings, description in terms of smoothness properties; decompositions and approximations; interpolation via real and complex methods; trace problems; extension operators for regular and irregular domains; capacity estimates, etc. We shall be focused mainly on the two most important and useful scales of function spaces - $B^r_{pq}$ and $F^r_{pq}, 1 \le p,q \le \infty, r>0 $ - intensively studied by H. Triebel. The spaces of generalized smoothness will be also under consideration. This is joint work with Gennadiy A. Kalyabin.

  Gérard Bourdaud Université Paris 6, Paris / France

Functional Calculus on Zygmund and BMO spaces

We give a full characterization of functions $f$ such that the superposition operator $g\mapsto f\circ g$ acts on the Zygmund spaces $B^m_{\infty\infty}$, for $m$ positive integer, and on $BMO({\bf R}^n$. The continuity and differentiability of such operator is also discussed.

  Djairo G. De Figueiredo Universidade Estadual de Campinas, Campinas / Brazil

On an inequality of Trudinger-Moser and related elliptic equations

It has been shown by Trudinger and Moser that for normalized functions $u$ of the Sobolev space $W^{1,N}(\Omega)$, where $\Omega$ is a domain in $\R^{N}$, the integral $\int_{\Omega}exp(u^{\alpha_{N}\frac{N}{N-1}})dx$ remains bounded. Here $\alpha_{N}=N\omega_{N-1}^{\frac{1}{N-1}}$, where $\omega_{N-1}$ is the $(N-1)$-dimensional surface of the unit sphere in $\R^{N}$. Carleson and Chang proved that there exists a corresponding extremal function in the case that $\Omega$ is the unit ball in $\R^{N}$. In this joint work with J.M. do O' and B. Ruf we give a new proof, a generalization, and a new interpretation of this result. In particular, we give an explicit sequence which is maximizing for the above integral among all normalized "concentrating sequences".

  Ron De Vore University of South Carolina, Columbia / U.S.A.

Some new ideas in interpolation of operators

A common method for finding interpolation spaces by the real method of interpolation is to use retracts. We introduce new ideas on how to use retracts in conjunction with wavelet decompositions in order to establish new results on interpolation spaces. These new results are then used to prove new versions of the averaging lemma in kinetic theory and new Gagliardo-Nirenberg inequalities. This is joint work with Albert Cohen, Wolfgang Dahmen, and Ingrid Daubechies.

  David E. Edmunds University of Sussex, Brighton / U.K.

Entropy, embeddings and equations

A survey will be given of some of Hans Triebel's many contributions to Analysis. Particular attention will be given to entropy numbers, decompositions in function spaces, embeddings between function spaces and sharp inequalities, with applications to linear and semilinear elliptic equations.

  Vladimir Mazya Linköping University, Linköping / Sweden

The Schrödinger and the relativistic Schrödinger operators on the energy space : boundedness and compactness criteria

This is a joint work with I. Verbitsky. We give a complete characterization of the class of functions (or, more generally, complex-valued distributions) $Q$ such that the following inequality holds:

$$ |\int_{R^n} |u(x)|^2 Q(x) dx| \leq const \int_{R^n} |\nabla u(x)|^2 dx,$$

where the constant is independent of $u\in C^{\infty}_0(R^n)$. Similar inequalities are proved for the inhomogeneous Sobolev space $W^1_2(R^n)$. In other words, we establish a criterion for form-boundedness of $Q$ relative to the Laplacian $\Delta$ under no a priori assumptions on $Q$. For the Schrödinger operator $L=-\Delta +Q$, our criterion describes the class of admissible perturbations $Q$ such that $L:\ring L^1_2(R^n)\to L _2^{-1}(R^n)$. We also establish similar boundedness and compactness criteria for the relativistic Schrödinger operator.

  Akihiko Miyachi Tokyo Woman's Christian University, Tokyo / Japan

Hardy type function spaces and its application

We first show that the purely real variable method can be applied to the theory of the weighted Hardy space. We show that the theory can be developped on arbitrary open subset of the Euclidean space and with respect to general doubling measures. We next use the same method to study the Herz-type Hardy spaces. The theory of Herz-type Hardy space known so far relied on the basic $L^{q}$ theory with $1 < q < \infty$. We show that the theory can be extended to the range $q \leqq 1$. Finally, as an application, we consider the transplantation theorem for Jacobi series.

  Stanislav I. Pohozaev Steklov Mathematical Institute, Moscow / Russia

The general blow-up theory for nonlinear PDE's

We present the new approach to investigation the nonexistence of global solutions to nonlinear partial differential equations. This approach is based on the introduction of a nonlinear capacity that is appropriate to the initial nonlinear problem and, further, on the use of standard inequalities with parameters for functions in corresponding function spaces. The asymptotic analysis of obtained inequalities gives us unimprovable conditions of absence of global solutions for the nonlinear problems considered.
These results were obtained together with Prof. E. Mitidieri (Italy), A. Tesei (Italy), and L. Veron (France).

  Michael Solomyak The Weizmann Institute of Science, Rehovot / Israel

Laplacian and Schrödinger operator on regular metric trees

Metric tree $\Gamma$ is a tree whose edges are regarded as line segments of finite length. The Laplacian $\Delta$ on $\Gamma$ is defined as $\Delta u=u''$ on each edge, with the matching conditions at each vertex which come from the Kirchhoff laws.
We suppose that a vertex $o\in\Gamma$ ({\it{root}} of $\Gamma$) is singled out, whose degree is $1$. At this point a boundary condition, say $u(o)=0$, should be specified. For a point $x\in\Gamma$, the number $|x|$ is defined as the distance between $x$ and $o$. If $x$ is a vertex, then its {\it{generation}} is the number of edges connecting $x$ and $o$. Generation of an edge is defined as the generation of its initial point. A tree is called {\it{regular}} if all the vertices of the same generation are of the same degree and all the edges of the same generation are of the same length. Evidently, all the infinite paths starting at any vertex of generation $j$ have the same length, say $L_j\le\infty$.
It turns out that the Laplacian on a regular tree admits the decomposition into orthogonal sum of an infinite family of simpler operators. Each of them can be reduced to a differential operator acting in $\mathcal L_2(0,L_j)$. A similar decomposition holds for the Schrödinger operator, provided its potential $V(x)$ is symmetric, that is depends only on $|x|$. Sometimes this decomposition allows one to give the complete description of the spectrum of the initial operator.
Let in particular all the vertices $x\not=o$ are of the same degree and all the edges are of the same length. Then the spectrum of the Laplacian has the band-gap structure. If the Laplacian is perturbed by a real-valued decaying potential $\alpha V$ (where $\alpha$ is a large parameter), then eigenvalues in the gaps may appear. Study of the behaviour of the number of these eigenvalues as $\alpha\to\infty$ has much in common with the study of some spectral problems related to fractals. Several examples will be presented.

  Gunther Uhlmann University of Washington, Seattle / U.S.A.

Inside-Out: Inverse Boundary Problems

We survey recent progress in inverse boundary problems. In these problems one attempts to determine the internal properties of a medium by making measurements at the boundary of the medium. In particular we will consider the problem of determining a Riemannian metric on a Riemannian manifold with boundary by measuring the geodesic distance between boundary points. We shall consider also the inverse problem of determining a potential or a conductivity on a bounded domain in Euclidean space from partial information on the Dirichlet-to-Neumann map for the associated Schrödinger equation or conductivity equation.

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Dr. D. Haroske; 2001-05-07