Über eine Verallgemeinerung des Plateauschen Problems

Let T be the domain in ^N defined by the inequalities O < ₁ < ... < _N < +. Put _N+k = /2(1+k) (k=1,2,3), _N+4=₁+2, and denote byF() the set of functions x=x(u,v)=(x¹(u,v),...,x^p(u,v)), (p2) of class , where B is the unit disk u²+v²<1, which maps the circular arcs _k={w=eⁱ:_k<<_K+1} (k=1,..., N+3) into the straight lines containing the edges a_k, a_k+1 (a_N+4=a₁) of a polygon IR^p. Then we show that the function ()= _inf ^xF() D(x) is analytic in T. This generalizes and sharpens an unproved result of I. Marx and M. Shiffman (see [4]).

---

---

Hans Lewy und Charles B. Morrey gewidmet     

Приближение функции ограниченнойp-флукту адии полиномами по мульти пликативным система м

A new discrete modulus of continuity is introduced for functions of boundedp-fluctuation, and direct and converse theorems are proved on the approximation of these functions by polynomials with respect to multiplicative systems. Sufficient conditions for the convergence of Fourier series with respect to multiplicative systems are also obtained and these are the best possible in a certain sense.

---

---

. 60-     

On the domain dependence of solutions to the two-phase Stefan problem

We prove that solutions to the two-phase Stefan problem defined on a sequence of spatial domains _n ^N converge to a solution of the same problem on a domain where is the limit of _n in the sense of Mosco. The corresponding free boundaries converge in the sense of Lebesgue measure on ^N.     

Biclosed Binary Relations and Galois Connections

Given two closure spaces (E,) and (E,), a relation RE×E is said biclosed if every row of its matrix representation corresponds to a closed subset of E, and every column to a closed subset of E. An isomorphism between, on the one hand, the set of all biclosed relations and, on the other hand, the set of all Galois connections between the two lattices of closed sets is established. Several computational applications are derived from this result.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

Weighted Composition Operators on Bergman and Dirichlet Spaces

Let H() denote a functional Hilbert space of analytic functions on a domain . Let w : C and : be such that w f is in H() for every f in H(). The operator wC given by f w f is called a weighted composition operator on H(). In this paper we characterize such operators and those for which (wC )* is a composition operator. Compact weighted composition operators on some functional Hilbert spaces are also characterized. We give sufficient conditions for the compactness of such operators on weighted Dirichlet spaces.     

Spectral pairs in cartesian coordinates

Let ^d have finite positive Lebesgue measure, and let () be the corresponding Hilbert space of -functions on . We shall consider the exponential functionse on given bye (x)=e ^i2·x . If these functions form an orthogonal basis for (), when ranges over some subset in ^d , then we say that (, ) is a spectral pair, and that is a spectrum. We conjecture that (, ) is a spectral pair if and only if the translates of some set by the vectors of tile ^d. In the special case of =I^d, the d-dimensional unit cube, we prove this conjecture, with =I^d, for d3, describing all the tilings by I^d, and for all d when is a discrete periodic set. In an appendix we generalize the notion of spectral pair to measures on a locally compact abelian group and its dual.Acknowledgements and Notes, Work supported by the National Science Foundation.Dedicated to the memory of Irving E. Segal     

Noetherian Equations for Matrices

Let L|K be a finite Galois extension. Using central simple algebras we deal with the crossed representations of G = Gal(L|K) over L which are defined as mappings X of G into GL_n(L) satisfying X = X X. The last equation is the Noetherian equation in case n=1. Furtheron, more general crossed projective representations are considered which obey an equation X X = Xf_, where f_, L.     

Orientation reversing involutions and the first betti number for finite coverings of 3-manifolds

LetMS ³,P ³ be a closed, orientable irreducible 3-manifold which admits an orientation reversing involution :MM. If dim(Fix )=0, suppose ₁ (M) has a subgroup of even index. We show thatM has finite coverMMM} with ₁(M<0). As an application we show that the hyperbolic dodecahedral space has a finite cover with positive 1st betti number.     

The Hodge Filtration and Cyclic Homology

We relate the Hodge filtration of the cohomology of a complex algebraic variety X to the Hodge decomposition of its cyclic homology. If X is smooth and projective, is the quotient of the Betti cohomology by the piece of the Hodge filtration.     

Perturbation of the poles of the scattering matrix under variation of the boundary condition

The behavior of the poles z_n(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |z_n()–z_n|=0(^(1/2)qn), where q_n is the order of the pole of the scattering matrix for the operator ₀u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

The Laplacian with Robin Boundary Conditions on Arbitrary Domains

Using a capacity approach, we prove in this article that it is always possible to define a realization of the Laplacian on L ²() with generalized Robin boundary conditions where is an arbitrary open subset of R ⁿ and is a Borel measure on the boundary of . This operator generates a sub-Markovian C ₀-semigroup on L ²(). If d=d where is a strictly positive bounded Borel measurable function defined on the boundary and the (n–1)-dimensional Hausdorff measure on , we show that the semigroup generated by the Laplacian with Robin boundary conditions has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in L ^p () is independent of p[1,). Our approach constitutes an alternative way to Daners who considers the (n–1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture.     

Projections on Lp-spaces of polyanalytic functions

The fundamental result: for an arbitrary bounded, simply connected domain in , the subspace L_n,m ^p() of the space L^p(, ) ( is the plane Lebesgue measure, p 1), consisting of the (m, n)-analytic functions in , is complemented in LP(, ) (a function f is said to be (m, n)-analytic if (^m+n/¯Z^mZⁿ)f=0 in ). Consequently, by virtue of a theorem of J. Lindenstrauss and A. Pelczyski, the space L_n,m ^P() is linearly homeomorphic to lP. In particular, for m=n=1 we obtain that the space of all harmonic L^P-functions in is complemented in L^P(, ). This result has been known earlier only for smooth domains.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 15–33, 1991.     

Аппроксимация субга рмонических функций

The following statement is proved. Letu be a subharmonic function in the region and _u the associated measure. Then there exists a functionf holomorphic in and such that if _f is the associated measure of the function in ¦f¦, then ¦u(z)–ln¦f(z)¦ A¦ln s¦+B¦ln diam¦+ s(¦lns¦+1)+C. hold at every point z for which the setsD(z, t)={w: ¦w–z¦},t(0,s) lie in and satisfy(D(z, t))t both for= _u and for= _f . In the case where is an unbounded region, In diam should be replaced by ln ¦z¦. The constants, , do not depend on andu.

---

---

. . .     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Total flux estimates for a finite element approximation of the Dirichlet problem using the boundary penalty method

Summary This paper considers a fully practical piecewise linear finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation,Au=f, in a smooth region< ⁿ (n=2 or 3) by the boundary penalty method. Using an unfitted mesh; that is ^h , an approximation of with dist (, ^h )Ch ² is not in general a union of elements; and assuminguH ⁴ () we show that one can recover the total flux across a segment of the boundary of with an error ofO(h ²⁾. We use these results to study a fully practical piecewise linear finite element approximation of an elliptic equation by the boundary penalty method when the prescribed data on part of the boundary is the total flux.Supported by a SERC research studentship     

On Ω-closed sets and Ωs-closed sets in topological spaces

Summary New classes of sets called -closed sets and s-closed sets are introduced and studied. Also, we introduce and study -continuous functions and s-continuous functions and prove pasting lemma for these functions. Moreover, we introduce classes of topological spaces -T_1/2 and -T_s.     

Cohomological characterization of pseudoconvexity in a Banach space

Let X be a Banach space with a countable unconditional basis (e.g., X=₂), X open. We show that is pseudoconvex if and only if for each affine complex line L in X the sheaf cohomology group H ¹ (,I) vanishes, where I is the ideal sheaf of all holomorphic functions on that vanish on L. We also give an example that the condition H ^q (,)=0 for all q1 unlike in finite dimensions does not imply the pseudoconvexity of . Lastly, we prove an interpolation result. Mathematics Subject Classification (2002): 32T05, 46G20.     

Convolution Operators with Fractional Measures Associated to Holomorphic Functions

Let be an open set in the complex plane and let be a holomorphic function on . Let K be a compact subset of with nonempty interior such that 0 K. Let be the Borel measure of R ⁴ C ² given by(E = _K E(z, (z))|z|^–2 d(z)where 0 < 2 and d(x ₁ + ix ₂) = dx ₁ dx ₂ denotes the Lebesgue measure on C. Let T be the convolution operator T f = * f. In this paper we characterize the type set E associated to T .