Extensions from the Sobolev Spaces H 1 Satisfying Prescribed Dirichlet Boundary Conditions

Extensions from H ¹(_P) into H ¹() (where _P ) are constructed in such a way that extended functions satisfy prescribed boundary conditions on the boundary of . The corresponding extension operator is linear and bounded.     

Rappresentazione con integrali multipli di funzionali dipendenti da funzioni a valori in uno spazio di Banach

Summary We consider a functional J: W _10c ^1,p (, X) ×B() [0, ], where X is a Banach space andB() the class of the Borel subsets of the open R_n, and we assume that J has a suitable semicontinuity property respect its first variable, and depends like a measure from the elements ofB(). We show that under certain conditions such a functional can be represented like a multiple integral of a Caratheodory integrand. The first paragraph is devoted to improve some classical results about Sobolev spaces W^1,p(, R) in the case of W^1,p(, X).     

Canonical contact forms on spherical CR manifolds

We construct the CR invariant canonical contact form can(J) on scalar positive spherical CR manifold (M,J), which is the CR analogue of canonical metric on locally conformally flat manifold constructed by Habermann and Jost. We also construct another canonical contact form on the Kleinian manifold ()/, where is a convex cocompact subgroup of Aut_CRS²ⁿ⁺¹=PU(n+1,1) and () is the discontinuity domain of . This contact form can be used to prove that ()/ is scalar positive (respectively, scalar negative, or scalar vanishing) if and only if the critical exponent ()<n (respectively, ()>n, or ()=n). This generalizes Nayatanis result for convex cocompact subgroups of SO(n+1,1). We also discuss the connected sum of spherical CR manifolds.     

Necessary and sufficient condition of the existence of global Hopf bifurcation for Hamiltonian systems

ForH C ² (,R) where 0 R ²ⁿ ,H (0)=0 and detH(0)0, the paper proves that there is a global Hopf bifurcation fromx=0 for Hamiltonian systemx=JH(x) iffJH(0)possesses purely imaginary eigenvalues. The work improves the corresponding result of J.C.Alexander and J. Yorke (Amer. J. Math., 100 (1978), 263–292).     

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.     

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)¦.     

An embedding theorem for a limiting exponent

We consider the function space B _p ^l () of functionsf(x), defined on the domain of a certain class and characterized by specific differential-difference properties in L_p(). We prove a theorem on the embedding B _p,q ^l () L_q in the case whenl=n/p –n/q >0 and its generalization for vectorl, p, q.Translated from Matematicheski Zametki, Vol. 6, No. 2, pp. 129–138, August, 1969.     

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.     

Barth type vanishing theorems

Let X be a smooth, projective, d-dimensional subvariety of ⁿ (). Barth's theorem says that H ^q (X, ^p _X )=0 when pq and q+p2d–n (if p=0 we must have q>0). It is very interesting to look for analogous vanishing theorems for H ^q (X, ^p _X (m)), m (see [S-S], [F], [S]). In this paper we prove some vanishing theorems for H ^q (X, ^p _X (1)), for H ^q (X, ^p _X (m)) when m–1, and, if dim(X)=n–2, for H ^q (X, ² _X (m)) and H ^q (X, S ^{k 1} _X (m)). We use standard techniques and some of our previous results.     

Asymptotic Formulae and Divisor Problems

We investigate the asymptotic behaviour of the summatory functions of _z(n, ), _k(n, ) z ⁽ⁿ⁾ and _k(n, ) z ⁽ⁿ⁾.     

Irregular but nonfractal drums andn-dimensional weyl conjecture

In this paper, we study the asymptotics of the spectrum of the Dirichlet (or Neumann) Laplacian in a bounded open set R ⁿ (n 1) with irregular but nonfractal boundary. We give a partial resolution of the Weyl conjecture, i.e. for the counting functionN _i ()(i=0 : Dirichlet;i=1 : Neumann), we have got a precise estimate of the remainder term÷ _i ()=() –N _i () for large, where() is the Weyl term. This implies that for the irregular but nonfractal drum , not only the volume || _n is spectral invariant but also the area of boundary || _n–1 might be spectral invariant as well.Partially supported by the National Natural Science Foundation of China and the Grant of Chinese State Education Committee.     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

Variational problems in SBV and image segmentation

We show how it is possible to prove the existence of solutions of the Mumford-Shah image segmentation functional F(u,K) = _\K [u² + (u – g)²]dx + ^{n – 1}(K), u W ^1,2(\K), K closed in .We use a weak formulation of the minimum problem in a special class SBV() of functions of bounded variation. Moreover, we also deal with the regularity of minimizers and the approximation of F by elliptic functionals defined on Sobolev spaces. In this paper, we have collected the main results of Ambrosio and others.     

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.     

G-convergence of periodic parabolic operators with a small parameter by the time derivative

In this paper, we consider a sequenceP ^k of divergent parabolic operators of the second order, which are periodic in time with periodT=const, and a sequenceP ^k of shifts of these operators by an arbitrary periodic vector function X=L²((0,T) × )ⁿ where is a bounded Lipschitz domain in the space ⁿ. The compactness of the family {P ^k ¦ X, k ink with respect to strongG-convergence, the convergence of arbitrary solutions of the equations with the operatorP ^k , and the local character of the strongG-convergence in are proved under the assumptions that the matrix of coefficients ofL ² is uniformly elliptic and bounded and that their time derivatives are uniformly bounded in the space L²(;L²(0,T)).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 525–538, April, 1993.     

Lattice points in high-dimensional spheres

LetN(x, n, ) denote the number of integer lattice points inside then-dimensional sphere of radius (an)^1/2 with center at x. This numberN(x,n, ) is studied for fixed,n , andx varying. The average value (asx varies) ofN(x,n, ) is just the volume of the sphere, which is roughly of the form (2 e, ) ^n/2. it is shown that the maximal and minimal values ofN (x,n, ) differ from the everage by factors exponential inn, which is in contrast to the usual lattice point problems in bounded dimensions. This lattice point problem arose separately in universal quantization and in low density subset sum problems.     

An estimate for the solutions of stokes equations in exterior domains

A boundary value problem for the Stokes equations is examined in an exterior domain ⁿ with a uniform Dirichlet condition on the boundary and a homogeneous condition at infinity. It is shown that estimating the norm L _p() of the second derivatives of the velocity vector field by the same norm of the exterior forces vector field is correct for p < n/2, but not for p n/2. This estimate is valid also for p n/2 if the boundary conditions are modified at infinity. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 105–120, 1990.     

On asymptotics of solutions to semilinear elliptic equations near the first eigenvalue of the nonperturbed problem

We writef=(g) iff(x)cg(x) with some positive constantc for allx from the domain of functionsf andg. We show that at least (n ² /r) entries must be changed in an arbitrary (generalized) Hadamard matrix in order to reduce its rank belowr. This improves the previously known bound (n ²/r ²). If we additionally know that the changes are bounded above in absolute value by some numbern/r, then the number of these entries is bounded below by (n ³/(r ² )), which improves upon the previously known bound (n ² / ² ).Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 535–540, April, 1998.The research of the first author was supported by the Russian Foundation for Basic Research under grants No. 96-01-00094 and No. 96-15-96102 and by the INTAS Foundation under grant No. 93-1376. The research of the second author was supported by the Russian Foundation for Basic Research under grants No. 96-01-01222 and No. 96-15-96090.     

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 .     

Computing a family of reproducing kernels for statistical applications

For an open subset of , an integer,m, and a positive real parameter , the Sobolev spacesH ^m () equipped with the norms: u²=u(t)²dt+(1/^2m u ^(m)(t)² constitute a family of reproducing kernel Hilbert spaces. When is an open interval of the real line, we describe the computation of their reproducing kernels. We derive explicit formulas for these kernels for all values ofm in the case of the whole real line, and form=1 andm=2 in the case of a bounded open interval.This research was partly supported by NSF Grant DMS-9002566.