The conformal structure of Riemann surfaces with boundary parametrizing minimal surfaces

Let F denote a surface with boundary F, being contained in a Riemann surface R, such that R\F is somedisk. If we vary the boiundary curve _o parametrizing F, we will get a manifold of real dimension 6g–3, such that any bounds some F and any local deformation of F is conformally equivalent to just one F for .This result also implies that none of the conformal invariants of R will be an invariant of this F, since its neighbors {F|} cover all possible deformations of F at all.     

Proper holomorphic mappings between some nonsmooth domains

Sunto In questo lavoro si studia il problema dell'existenza o meno di mappe olomorfe proprie, F: (p,q) (pq) dove (,) Cⁿ sono domini limitati, pseudoconvessi, non lisci, di Reinhardt con centro di simmetria sulla frontiera (per la definizione v. (2), introduzione). In caso affermativo si fornisee esplicitamente l'espressione di tali mappe (v, teoremi I e II nell'introduzione).

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The author, during the redaction of this paper, was supported by MPI (40% and 60%) funds.     

Hasse-Witt matrices of Fermat curves

Let m be an integer with m3. Let K and K be perfect fields of characteristic p and p such that (p,m)=1 and (p,m)=1, respectively. Moreover let A and A be algebraic function fields over K and K defined by x^m+y^m=a(0, ak) and x^m+y^m=a(a0 ak), respectively. Put g=(m–1)(m–2)/2. Denote by M(K,p,a) and M(K,p,a) the Hasse-Witt matrices of A and A with respect to the canonical bases of holomorphic differentials. Then we show that if p+p0(mod.m) then rank M(K,p,a)+rank M(K,p,a)=g and if pp1 (mod.m) then rank M(K,p,a)=rank M(K,p,a).     

Runge property and multiplicity of solutions for elliptic equations in ℝN with a monotone nonlinearity

In this paper, we describe a method for extending (in some approximated sense) solutions of a nonlinear P.D.E. on a domain , to solutions in a domain containing . Such an extension property, the Runge property, is well known for a large class of linear problems including elliptic equations. We prove here the Runge property for semilinear problems of the kind -u+g(u)=f, with f L _loc ¹ (^N). (As a consequence, we get infinitely many solutions for these problems). The proof is based on a homotopy method, and requires a refinement of the linear results: We prove that the Runge extension v on of a solution u in for a linear elliptic equation Lu=f can be choosen in order to depend continuously on u and the coefficients of L.     

Convergence of the highest derivatives in projection methods

Assume that for the approximate solution of an elliptic differential equation in a bounded domain , under a natural boundary condition, one applies the Galerkin method with polynomial coordinate functions. One gives sufficient conditions, imposed on the exact solutionu ^*, which ensure the convergence of the derivatives of order k of the approximate solutions, uniformly or in the mean in or in any interior subdomain. For example, ifu ^*W^k ₂, then the derivatives of order k converge in L₂(), where is an interior subdomain of . Somewhat weaker statements are obtained in the case of the Dirchlet problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 11–18, 1977.The author expresses his gratitude to Yu. K. Dem'yanovich for drawing his attention to [10].     

Diameters of classes of continuous functions in the Lp space

In the L_p(a, b) space the exact values of n-diameters (n=1, 2, ...) are found of the class H[a, b] of the functions f(x) such that ¦f(x)-f(x)¦ (¦x-x¦), where (t) is a given continuity module which is convex upwards.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 493–500, November, 1971.     

Lattice Fully Orderable Groups

Let be a linearly ordered set, A() be the group of all order automorphisms of , and L() be a normal subgroup of A() consisting of all automorphisms whose support is bounded above. We argue to show that, for every linearly ordered set such that: (1) A() is an o-2-transitive group, and (2) contains a countable unbounded sequence of elements, the simple group A()/L() has exactly two maximal and two minimal non-trivial (mutually inverse) partial orders, and that every partial order of A()/L() extends to a lattice one (Thm. 2.1). It is proved that every lattice-orderable group is isomorphically embeddable in a simple lattice fully orderable group (Thm. 2.2). We also state that some quotient groups of Dlab groups of the real line and unit interval are lattice fully orderable (Thms. 3.1 and 3.2).     

Nonbranching weak and starshaped strong solutions for Hele-Shaw dynamics

We consider multidimensional weak and strong Hele-Shaw dynamics (t) of an advancing/receding viscous fluid injected/removed through a single finite point into/from a bounded domain (0). A class of weak solutions is shown to preserve local uniqueness in both directions. Then we also consider strong solutions (t), and show that if (0) is starshaped with respect to a small ball centered on the point of injection, then the evolution (t) exists for all time.Received: 5 March 2004     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

Triangular extension identities

It is shown that if the - -bimodule M generates a category of - -bimodules, then the ideal of identities of the triangular extension of the direct sum of algebras and by means of the bimodule M is equal to the product of ideals of identities of the algebras and .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 5–11, 1983.     

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     

A Proof of the Lewy-Stampacchina's Inequality by a Penalization Method

In this paper we prove the Lewy–Stampacchia's inequality for elliptic variational inequalities with obstacle involving fairly general Leray–Lions operators. The main novelty of the paper is the method of proof, which uses the natural penalization. One of the steps of the proof consists in proving, again thanks to the natural penalization, that the nonnegative cone of W ₀ ^1,p () is dense in the nonnegative cone of W^-1,p().     

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)¦.     

Existence theorems for nonlinear evolution inclusions

Summary In this paper we obtain an existence theorem for the abstract Cauchy problem for multivalued differential equations of the form u–^– f(u)+G(u), u(O)=x₀, where ^– f is the Fréchet subdifferential of a functionf defined on an open subset of a real separable Hilbert space H, taking its values in R {+} and G is a multifunction from C([0, T], ) into the nonempty subsets of L²([0, T], H). As an application we obtain an existence theorem for the multivalued perturbed problem x–^– f(x)+F(t, x), x(0)=x₀, where F:[0, T]×(H) is a multifunction satisfying some regularity assumptions.     

Approximation of a nondifferentiable nonlinear problem related to MHD equilibria

Summary We present a simple method, based on a variant of the implicit function theorem, which leads to the existence of (a part of) a nontrivial solution branch of the nonlinear eigenvalue problem –u=u ⁺ in ,u=–1 on , where is a two-dimensional domain with boundary . The advantage of this method is that we can apply it for analysing the approximation of the above problem by a finite element method; the error analysis of the discrete problem appears immediately. We give also an iteration scheme which allows to solve the approximate problem.     

φ-Factors for Function Seminorms

Let (, A, ) be a measure space, a function seminorm on M, the space of measurable functions on , and M the space {f M : (f) < }. Every Borel measurable function : [0, ) [0, ) induces a function : M M by (f)(x) = (|f(x)|). We introduce the concepts of -factor and -invariant space. If is a -subadditive seminorm function, we give, under suitable conditions over , necessary and sufficient conditions in order that M be invariant and prove the existence of -factors for . We also give a characterization of the best -factor for a -subadditive function seminorm when is -finite. All these results generalize those about multiplicativity factors for function seminorms proved earlier.     

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.     

Convexity theories III. Classification of certain real convexity theories

In this paper we classify all real convexity theories that contain the standard convexity theory _c. For this purpose we consider three subcases: finitary; infinitary and (_sc\_c)Ø; infinitary and _sc=_c. In each of these subcases one encounters a phenomenon resembling bifurcation.This research was supported by the Deutsche Forschungsgemeinschaft.     

On the diameters of certain classes of analytic functions. II

In the spaces E_q(), q1, we consider the classes W^rE_p(), p1, consisting of analytic functions f(z) E_P() the integral moduli of continuity of whose r-th derivatives are majorized by a given nonnegative nondecreasing function .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1135–1138, August, 1992.     

Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. I

In the Banach space of functions analytic in a Jordan domain , we establish order estimates for the Kolmogorov widths of certain classes of functions that can be represented in by Cauchy-type integrals along the rectifiable curve = and can be analytically continued to or to .