Analysis & Geometry 99

PROCEEDINGS ON ANALYSIS AND GEOMETRY

Sobolev Institute Press, Novosibirsk, 2000. - 692 pages.

This volume contains papers of the participants of the International Conference on Analysis and Geometry in honor of the 70th birthday of the outstanding mathematician and professor, member of the Russian Academy of Sciences, Yurii Grigor'evich Reshetnyak (Novosibirsk, Akademgorodok, August 30 - September 3, 1999). The scope of papers concerns the spaces with bounded curvature in the sense of Alexandrov, quasiconformal mappings and mappings with bounded distortion (quasiregular mappings), nonlinear potential theory, Sobolev spaces and variational problems. The volume reflects modern trends in these areas. Most articles rest on Reshetnyak's original works and demonstrate the vitality of his contribution.

This volume addresses researchers and graduate students.

The publication of these Proceedings was partially supported by the State Program of Support for Leading Scientific Schools (grant 00-15-96165).

Abstracts

Kakutani Theorem for Banach - Kantorovich M-Lattices by N.M. Abasov

The paper is devoted to an operator version of Kakutani theorem on the representation of M-lattices. P.31-35. (in Russian)

The Martin Boundary of Certain Hadamard Manifolds by Werner Ballmann

A complete and simply connected Riemannian manifold with nonpositive sectional curvature is called a Hadamard manifold. Anderson and Schoen showed that the Martin boundary of a Hadamard manifold is homeomorphic to its geodesic boundary if the sectional curvature is pinched between two negative constants. In this paper we show that this conclusion also holds for certain Hadamard manifolds which contain a flat subspace. This is interesting in connection with so-called rank one manifolds. P.36-46.

Ultrametric Spaces by V.N. Berestovskii

In this article, we prove that every compact ultrametric space X can be isometrically embedded into Hilbert space H in the form of a subset X¢ of a sphere S(0,R) with zero center and with minimal radius R depending only on X so that every isometry of X¢ onto itself is uniquely extended to an orthogonal linear transformation of H onto itself. Also there exists a line l in H such that the orthogonal projection of X¢ onto l is a homeomorphism. From this easily follows the known fact that perfect compact ultrametric space is homeomorphic to the classical Cantor set. Also we find some connection between our results and the Jung theorem about the closed ball with the smallest radius which contains a given compact subset of Euclidean space, and we give some corollaries of this connection. Besides, we characterize the topological spaces that admit an ultrametric compatible with their topology, and we indicate some connections between the ultrametric spaces and so-called R-trees. P.47-72.

About Regularity and the Gauss-Kronecker's Curvatures of the Surfaces of Dual Bodies by V.N. Berestovskii, I.A. Zubareva

In this paper we prove a regularity theorem and give the universal lower and upper evaluations for the Gauss-Kronecker's curvatures of the surfaces of dual convex bodies in Euclidean space. P.73-81. (in Russian)

Short Homology Bases and Partitions of Real Riemann Surfaces by Peter Buser and Mika Seppälä

Let X be a compact Riemann surface of genus g ³ 2 and s: X ® X a fixed point free antiholomorphic involution. The pair (X, s) is a real Riemann surface and the quotient X/ásñ by the action of s is a closed Klein surface. We show that for any s-invariant partition P of X into pairs of pants there exists a s-invariant canonical homology basis for X which is combinatorially short with respect to P. In combination with Bers' partition theorem this yields geometrically short homology bases on X and on K = X/ásñ. P.82-102.

Examples of Uniform and NTA Domains in Carnot Groups by Luca Capogna, Nicola Garofalo and Duy-Minh Nhieu

We present a survey of results concerning examples of Non Tangentially Accessible Domains, Uniform and John Domains in the setting of Carnot-Carathéodory manifolds. P.103-121.

Mappings with Bounded Distortion of Two-Step Carnot Groups by N.S. Dairbekov

For mappings with bounded distortion on two-step Carnot groups we establish their main properties such as the morphism of solutions to associated subelliptic equations, local Hölder continuity, almost everywhere differentiability in the Pansu sense, and a change-of-variable formula in the Lebesgue integral. Under the assumption of existence of a C¹-smooth singular solution to the sub-Q-Laplacian we prove openness, discreteness, and preservation of orientation for such mappings and the limit theorem for sequences of mappings with bounded distortion. P.122-155.

On Solvability of the Cauchy Problem for Pseudohyperbolic Equations by G.V. Demidenko

The present is devoted to the study of the Cauchy problem for a class of equations not solved with respect to the highest time derivative. The author establishes well-posedness of the problem in special weighted Sobolev spaces. P.156-172. (in Russian)

Attainable Sets for Left Invariant Control Systems and Carnot-Caratheodory Metrics on Nilpotent Lie Groups by V.M. Gichev

Let G be a semidirect product of a nilpotent Lie group and R. For a left invariant control system on G whose control domain is defined by a convex cone in the Lie algebra of G we prove that the attainable set coincides with a ``half-space'' if the degree of contact of the cone with certain linear subspaces of the Lie algebra is sufficiently high. P.173-185.

On Quasiconformal Mappings of Hermitian Manifolds by S.I. Goldberg and N.C. Petridis

The notion of a quasiconformal function due to M. Lavrentieff is extended to mappings of complex manifolds. They are called pseudoanalytic mappings, and include the holomorphic mappings as a special case. S.-T. Yau showed that a holomorphic map between Kaehler manifolds whose domain is complete and with certain curvature properties is distance-decreasing up to a constant depending on the curvature bounds. This result is extended to pseudoanalytic harmonic mappings of Kaehler manifolds with the same curvature properties. P.186-203.

Extremal Problems on Classes of Quasiconformal Embeddings of Riemann Surfaces by A.A. Golubev, S.Yu. Graf

We prove the existence of weight dilatation, integral weight dilatation, the extremals of the Belinsky functional in the homotopic classes of quasiconformal embedding of finite Riemann surfaces. We obtain also conditions of the uniqueness and properties of the extremal embeddings. P.204-226. (in Russian)

Sobolev Mappings, Co-Area Formula and Related Topics by Piotr Hajlasz

We generalize the classical area and co-area formulas to the setting of Sobolev mappings. In one of the versions of the co-area formula that we obtain, the integral-geometric measure is involved. The proof is based on a Sard type theorem for Borel mappings between Euclidean spaces which is of independent interest. We apply our results to minimizing harmonic mappings. P.227-254.

The Reshetnyak Theorems, Advances and New Perspectives by Tadeusz Iwaniec

I am honored and pleased to have the opportunity to celebrate the 70-th anniversary of Yu.G. Reshetnyak in his town Novosibirsk. It is a great pleasure for me that I can speak on his pioneering ideas and profound work on mappings with bounded distortion. Its fundamental character is attested to by unprecedented implications that go into various areas of analysis, PDEs, nonlinear potential theory, nonlinear elasticity and much more. While dramatic progress has been made on mappings with bounded distortion the quasiconformal analysis in its greater generality is still far from being completely developed, even on the level of basic definitions.

The selection of topics for this lecture reflects years of my interest in Reshetnyak's work. It is clearly impossible to discuss all the contributions of Yu.G. Reshetnyak in one lecture but I can attempt to stress fundamentals of his theory in a wider and perhaps more unifying manner, pointing out its importance to further advancements. Consequently, I will not go into the most remarkable implications of Reshetnyak's work made by many researchers in the field of PDEs, such as the new types of PDEs (the Beltrami-Dirac equations) governing in multidimensional geometric function theory.

I apologize for not discussing much of the others work. Some not mentioned here are included in the references. I simply do not know how to discuss those advances and keep it brief. C.255-272.

Viscosity Solutions of the p-Laplace Equation by Petri Juutinen and Juan J. Manfredi

The objective of this survey is to convince the reader that it is sometimes advantageous to consider viscosity solutions of quasilinear elliptic equations in divergence form like the p-Laplacian.

In the first part we discuss and compare three different notions of weak solution of the p-Laplace equation: Sobolev weak solutions based on distributional derivatives, potential theoretic weak solutions based on the comparison principle, and viscosity solutions based on generalized pointwise derivatives or jets. We have emphasized viscosity solutions because of the essential role they play in the applications that we have selected. Note however that for the p-Laplacian, viscosity supersolutions agree with the potential theoretic weak solutions. This is a non-trivial fact recently obtained in.

In the second part we present applications to ¥-harmonic functions, to the ¥-eigenvalue problem and show the superharmonicity of the p-ground state in a convex domain for p > 2. P.273-284.

Choquet Property for the Sobolev Capacity in Metric Spaces by Juha Kinnunen and Olli Martio

We discuss definitions of first order Sobolev spaces and related capacities on a metric measure space. We show that the natural Sobolev capacity is a Choquet capacity. P.285-290.

On One Idea by Yu.G. Reshetnyak in Measure Theory by S.S. Kutateladze

Yu.G. Reshetnyak proposed to consider majorization of measures for studying inequalities over convex surfaces as far back as in 1954. The talk touches the life and state-of-the art of majorization in convex geometry and subdifferential calculus. P.291-299.

Regularity Properties of a Nonlinear Operator Associated to the Conformal Welding by Massimo Lanza de Cristoforis and Luca Preciso

As it is well known, given a plane simple closed curve z with nonvanishing tangent vector, there exists a pair of suitably normalized Riemann maps (F,G), where F maps the open unit disk D of C onto the domain I[z] interior to z, and where G maps the exterior C cl D of cl D onto the domain E[z] exterior to z. It is also well known that F and G can be extended to boundary homeomorphisms. Thus one can consider the conformal welding homeomorphism F^(-1)°G_|¶D of ¶D to itself, which we denote by w[z]. Now we think both the set of simple closed curves z and the set of welding homeomorphisms as subsets of the Schauder space C^m,a_*(¶D,C ) of the m times continuously differentiable complex-valued functions on ¶D which have m-th order a-Hölder continuous derivative, with a Î ]0,1[, m ³ 1. Then we present some differentiability Theorems for the dependence of w[z] upon z, and a complex analyticity result for a right inverse of w[·]. P.300-317.

Representations of Set-Valued Mappings by Sublinear Operators and Their Applications by Yu.E. Linke

In this paper we introduce the new concept of representation of set-valued mappings by sublinear operators and investigate some properties of four representations. We show that the theory of continuous selections can be considered as some part of the theory of subdifferentiability of sublinear operators. We obtain sublinear analogs of classical convex-valued, compact-valued and zero-dimensional theorems of continuous selections as applications, and prove also two theorems of continuous selections. We discuss some other applications. P.318-349. (in Russian)

The Schrödinger Equation on the Warped Riemannian Products by A.G. Losev

We study properties of bounded solutions of stationary Schrödinger equation on the Riemannian manifolds of a certain type. We prove the exact condition of solvability of the Dirichlet problem at infinity and the condition of validity of the Liouville theorem. P.350-369. (in Russian)

Sufficient Conditions for Change of Variables in Integral by Jan Malý

It is well known that Luzin's property (N) of a mapping f is essential for justifying the change of variables through f in integral. Let us highlight the deep and fundamental contribution of Professor Yurij G. Reshetnyak in investigation of Luzin's property for Sobolev mappings. In this paper we discuss further conditions. One of them is the following n-absolute continuity property of a function f: for any e > 0 there exists d > 0 such that for every disjoint collection {B_j} of balls we have

(diam B_j)ⁿ < dÞ

(diam f(B_j))ⁿ < e.

This condition is satisfied if the gradient of f belongs to the Lorentz space Ú. If Ñf belongs only to Lⁿ, then the Luzin property holds except a singular set, which is the set of all points, where f is not approximately Hölder continuous. Another modification of the result applies to some BV function with nontrivial singular part of the gradient. A large part of referred results stems from joint works with Olli Martio, Pekka Koskela, Janne Kauhanen and Irene Fonseca. P.370-386.

Maximal Algebra of Multipliers Between Fractional Sobolev Spaces by Vladimir Maz'ya and Tatyana Shaposhnikova

It is shown that the maximal Banach algebra A_p^m,l imbedded in the space of multipliers M(W_p^m(Rⁿ)® W_p^l(Rⁿ)) which map the Sobolev space W_p^m(Rⁿ) to W_p^l(Rⁿ) with noninteger m and l, m ³ l, p Î (1,¥), is isomorphic to M(W_p^m(Rⁿ)® W_p^l(Rⁿ))ÇL_¥(Rⁿ). A precise description of all imbeddings A_p^m,l Ì A_p^m,l is given. P.387-400.

A Generalized Maximum Principle for the Differences of p-Harmonic Functions by V.M. Miklyukov and M.K. Vuorinen

We prove a generalized boundary maximum principle and a Liouville type theorem for differences of p-harmonic functions on Riemannian manifolds. P.401-413.

Nonrigid Star-Like Bipyramids of A.D. Alexandrov and S.M. Vladimirova by A.D. Milka

We establish the existence of nonrigid polyhedra with stationary volume. A one-parametric family of such polyhedra (up to homothety) is contained in a class of star-like partyheight bipyramids. We prove the second order rigidity of star-like bipyramids, find model flexors between star-like bipyramids, and discover fractal and dynamical properties of invariant functions of this class of polyhedra. P.414-430. (in Russian)

Second Variation Formula in a Space of Bounded Curvature by Igor G. Nikolaev

We study differential properties of geodesic variations in an Â_K^¢,K domain of a metric space of bounded curvature and obtain generalizations of the classical formula for the second variation of length of a geodesic. If a geodesic variation satisfies a Lipschitz condition on the boundary, then we prove that the variation is of Nikolskii's class S_¥^{( 2,1)}. We show that if a geodesic variation is of class C^1,1 on the boundary, and if the metric tensor of the space possesses the second differential at almost all points of the variation, then the second variation formula for the length l(t) of a geodesic holds for almost all t. Our principal result deals with triangular variations formed by points of a geodesic segment joining a fixed vertex of a triangle and points of the opposite side. We prove that the metric tensor is two-fold differentiable at almost all points of a triangular variation of some dense set of triangular variations, thereby the second variation formula holds true for each variation of this dense set of triangular variations P.431-464.

The Complexified Heisenberg Group by H.M. Reimann and F. Ricci

We show that the generalized contact mappings on the complexified Heisenberg group have to be holomorphic or antiholomorphic. In an analogous way, the contact flows are generated by holomorphic functions. P.465-480.

Metric Space: Classification of Finite Subspaces Instead of Constraints on Metric by Yu.A. Rylov

We suggest a new method of investigation of metric space based on classification of its finite subspaces. It admits to derive information on metric space properties which is encoded in metric and to describe geometry in terms of metric only. The method admits to remove constraints imposed usually on metric (the triangle axiom and nonnegativity of the squared metric). Elimination of the triangle axiom leads to ``tubular generalization'' of metric geometry (T-geometry), when the shortest paths are replaced by hallow tubes. Elimination of the second constraint admits to use the metric space for description of the space-time and other geometries with indefinite metric. P.481-504. (in Russian)

The Estimates of Topological Index of Mappings with Summable Jacobian by V.I. Semenov

We give the effective and exact estimates of the topological index for a wide class of mappings. We establish sufficient conditions for local injectivity of mappings from Sobolev and Besov spaces and mappings with bounded distortion. P.505-513. (in Russian)

On Behavior of Functions from Spaces Defined by Multi-Quasielliptic Operators at Infinity by G.A. Shmyrev

We research the behaviour of functions from spaces defined by multi-quasielliptic operators at infinity. We prove density of finite elements in the considerable spaces. P.514-527. (in Russian)

Geometric Approach in Multivariate Theory of Potential by V.V. Slavski

We study weakly regular conformally flat multivariate metrics of non-negative curvature. These metrics can be considered as multivariate potentials. We study properties ``in the large'' of the conformally flat metrics of non-negative curvature and their local behaviour in singular points. P.528-552. (in Russian)

Embedding Theorems for Sobolev Spaces of Numerical Sequences by E.S. Smailov

We study multiweight spaces of sequences l_q(Z;r) and W_q(Z;a,m). We obtain necessary and sufficient conditions of embedding W_q(Z;a,m)\hookrightarrow l_q(Z;r) at 1 £ q £ q £ +¥ and 1 £ q < q < +¥. We establish also the two-sided estimate for norms of corresponding embedding operators, and necessary and sufficient conditions of compactness of unitary space sphere of W_q(Z;a,m) in l_q(Z;r). P.553-570. (in Russian)

Two-Periodic Maximal Surfaces With Singularities by V.G. Tkachev, V.V. Sergienko

We consider the almost entire solutions to the maximal surface equation which satisfy to certain structure conditions. This approach allows us to construct a large class of periodic solutions in terms of generating matrices. The analytic behaviour of such solutions with mixed type singular points has been established. P.571-584. (in Russian)

Approximately Lipschitz Mappings and Sobolev Mappings Between Metric Spaces by Marc Troyanov

We introduce a notion of Sobolev spaces between metric spaces which extends the Hajasz' notion of Sobolev functions. We then compare this concept to Reshetnyak's definition. P.585-594.

On Automorphic Objects by Vladimir V. Vershinin

A definition of automorphic object in an arbitrary category (not necessary in a category with finite products) is given. This is analogy of group object where instead of the notion of group the notion of automorphic set is taken. An automorphic set is a set with a product for which left (or right) translations are automorphisms. Construction of associated group object of an automorphic object is done and an example of general (not necessary euclidean) root system is considered. P.595-602.

P-Differentiability on Carnot Groups in Different Topologies and Related Topics by Serguei K. Vodop'yanov

The main goal of this paper is to provide a detailed proof of the approximate P-differentiability theorem on Carnot groups extending by this way Pansu's differentiability theorem. More precisely, we prove that a mapping between two Carnot groups which is approximately P-differentiable along horizontal vector fields is also totally approximately P-differentiable almost everywhere. The rest of the paper is a survey in which we discuss results of author's papers dealing with different concepts of differentiability and its applications to geometric measure theory on Carnot groups. Between them, we emphasize the notions of P-differentiability in the uniform topology and in the Sobolev topology. We give some applications of these results to conclude the area formula on Carnot groups. The major part of results is exhibited in more general form by comparison with those in previous author's papers, and accompanied by detailed proofs. P.603-670.

On Measures of Noncompactness by N.A. Yerzakova

We obtain estimates for the measure of noncompactness of bounded subsets of Bochner spaces, discuss a new class of condensing operators, and prove the solvability of Cauchy problem. P.671-687. (in Russian)

On an Inverse Function Theorem by I.V. Zhuravlev

We give an analog of inverse function theorem for a mapping with Sobolev's generalized derivatives of the class L_1,loc(D). P.688-691. (in Russian)

Last modification: March 18, 2001 angeom@math.nsc.ru

PROCEEDINGS ON ANALYSIS AND GEOMETRY

Sobolev Institute Press, Novosibirsk, 2000. - 692 pages.

Contents:

Abstracts