Showing papers in "Mathematische Nachrichten in 1995"
TL;DR: The minimal hypersurfaces in E4 have the following properties: their mean curvature vector field is harmonic as discussed by the authors, which is the only property of E4 hypersurface that is known to be harmonic.
Abstract: The minimal hypersurfaces in E4 are the only hypersurfaces possessing the following property: Its mean curvature vector field is harmonic.
216 citations
144 citations
TL;DR: In this article, the notion of hyperderivability is introduced and discussed, which allows us to obtain, in particular, formulas for the derivatives of the Martinelli-Bochner integrals.
Abstract: The notion of hyperderivability is introduced and discussed which allows us to obtain, in particular, formulas for the derivatives of the Martinelli-Bochner integrals.
104 citations
TL;DR: In this article, the validity question for the Newell-Whitehead equation is treated, i.e., estimates between the exact solutions of the original problem and the associated approximations are given.
Abstract: Modulation equations play an essential role in the description of systems which exhibit patterns of nearly periodic nature, e.g. in Benard's problem. The so called Newell-Whithead equation is derived to describe the envelope of modulated roll-solutions in systems with two large extended or unbounded space directions. Here the validity question for the Newell-Whitehead equation is treated, i.e., estimates between the exact solutions of the original problem and the associated approximations are given. The Newell-Whitehead equation is a good example how important this validity question is. Although the modulation equation can formally be derived, the solutions of the original system can behave in some situations in a completely different manner than predicted by the modulation equation. This happens on a time scale which is very small compared to the natural one.
92 citations
TL;DR: In this article, it was shown that para-multiplication applies to a certain product π(u, v) defined for appropriate u and v in ''IRn'' and the boundedness of π (.,.) is investigated for the anisotropic Besov and Triebel-Lizorkin spaces.
Abstract: It is shown that para-multiplication applies to a certain product π(u, v) defined for appropriate u and v in '(IRn). Boundedness of π(.,.) is investigated for the anisotropic Besov and Triebel-Lizorkin spaces - i.e., for BM,s/p,q and FM,s/p,q with s ∈ R and p and q in [0, ∞] (though p < ∞ in the F-case) - with a treatment of the generic as well as of various borderline cases.
For max (so,s,) > 0 the spaces BM,so/p,qo ⊕ BM,so/p,qo and BM,so/p,qo ⊕ BM,so/p,qo to which π(.,.) applies arc determined. For generic Bso/po,qo ⊕ Bs1/p1,q1 the receiving Fsp,q spaces are characterized.
It is proved that π(f, g) = f · g holds for functions f and g when f · g ∈L1,loc, roughly speaking. In addition, π(f, u) = fu when f ∈ M and u ∈' .
Moreover, for an arbitrary open set Ω⊆ IRn, a product πΩ(., .) is defined by lifting to IRn. Boundedness of n on R′ is shown to carry over to πΩ is general.
64 citations
TL;DR: In this paper, it was shown that for local minima or stationary points w of the functional J, there is no continuous homotopy t H d, O I t < + 0 3, u 0 = u, u r n = u *,
Abstract: where K is a convex subset of some functions space, e.g. W$j'(SZ) or Lp(sl), p > 1, and SZ c R\" is a domain lying symmetrically to the hyperplane {y = 0}, {x = (x', y) , x' E R\"-', y E R). If u E K , we often also have u* E K , where u* denotes the Steiner-symmetrization of u with respect to y , and J(u*) I J(u) . It can be proved for the absolute minimum u of (1) that u = u*. This argumentation fails for local minima or stationary points w of the functional J . Therefore the following question is natural: Is there a (in the norm of X ) continuous homotopy t H d , O I t < + 0 3 , u 0 = u , U r n = u * ,
63 citations
TL;DR: In this paper, the positive cone is preserved when solving a weakly coupled system of elliptic partial differential equations on a bounded domain in IR, with zero Dirichlet boundary condition.
Abstract: In this paper we will study under which conditions the positive cone, or part of the positive cone, is preserved when solving a weakly coupled system of elliptic partial differential equations. Such a system will be as follows: −∆1 0 0 . . . 0 0 −∆k u1 .. uk = c11 · · · c1k .. .. ck1 · · · ckk u1 .. uk + f1 .. fk on a bounded domain in IR, with zero Dirichlet boundary condition. The operators ∆i will be strictly elliptic such as the Laplacian. The system is said to preserve the positive cone if f ≥ 0 implies u ≥ 0. We will classify such systems. For noncooperative systems we need and show pointwise estimates for iterates of the Green function.
63 citations
TL;DR: In this paper, the authors derive linear inequalities which relate the spectrum of a set of Hermitian matrices A1,..., Ar ∈ C n×n with the spectrum spectrum of the sum A1 + · · · + Ar.
Abstract: Using techniques from algebraic topology we derive linear inequalities which relate the spectrum of a set of Hermitian matrices A1 , . . . , Ar ∈ C n×n with the spectrum of the sum A1 + · · · + Ar . These extend eigenvalue inequalities due to Freede-Thompson and Horn for sums of eigenvalues of two Hermitian matrices.
52 citations
TL;DR: In this paper, the Navier-Stokes system is considered and a bounded domain in ℝ3 with a connected Lipschitz boundary ∆ ∆ is considered.
Abstract: Let Ω be a bounded domain in ℝ3 with a connected Lipschitz boundary ∂Ω. Consider the Navier-Stokes system.
44 citations
TL;DR: In this paper, a self-adjoint extension is used to define the singular perturbation of a given selfadjoint operator A by a singular operator T on a Hilbert space.
Abstract: We use the method of self-adjoint extensions to define a self-adjoint operator AT as the singular perturbation of a given self-adjoint operator A by a singular operator T on a Hilbert space.
We also find the structure of a singular operator Q such that the singular perturbation of A2 by Q satisfies (A2)Q = (AT)2. We obtain the explicit form of Q in terms of A and T. A definition of the n-th power for strictly positive symmetric operators is also given.
39 citations
TL;DR: In this paper, a new method of constructing elliptic finite-gap solutions of the stationary KdV hierarchy, based on a theorem due to PICARD, is illustrated in the concrete case of the Lame-Ince potentials -s(s + 1) ρ(z), se (ρ() the elliptic Weierstrass function)
Abstract: A new method of constructing elliptic finite-gap solutions of the stationary Korteweg-de Vries (KdV) hierarchy, based on a theorem due to PICARD, is illustrated in the concrete case of the Lame-Ince potentials -s(s + 1) ρ(z), se (ρ() the elliptic Weierstrass function) Analogous results are derived in the context of the stationary modified Korteweg-de Vries (mKd V) hierarchy for the first time
TL;DR: In this article, the nonlinear eigenvalue problem for p-Laplacian is considered and the existence and C 1,α-regularity of the weak solution is proved.
Abstract: The nonlinear eigenvalue problem for p-Laplacian
is considered. We assume that 1 < p < N and that the function f is of subcritical growth with respect to the variable u. The existence and C1,α-regularity of the weak solution is proved.
TL;DR: In this paper, the boundedness properties of nonregular pseudo-differential operators and their adjoints on Triebel spaces F, 0 < p, q≦ ∞ were studied.
Abstract: We study the boundedness properties of nonregular pseudo-differential operators and their adjoints on Triebel spaces F, 0 < p, q≦ ∞. Special attention is payed to sharp estimates. The second aim of this paper is the compactness properties of these operators on weighted and unweighted Triebel spaces F(w) where the weight w may belong to an arbitrary Muckenhoupt class A∞.
TL;DR: In this article, the basic relations between the stationary Poisson point process and the point process of vertices of the corresponding Voronoi tessellation in IRd and of planar sections through it are given.
Abstract: This paper gives basic relations between the stationary Poisson point process and the point process of vertices of the corresponding Voronoi tessellation in IRd and of planar sections through it. The results are based on a study of the Palm distribution of the point process of vertices. An identity is given connecting the distribution of a Poisson point process and the Palm distribution with respect to the vertices of the corresponding Voronoi tessellation. Distributional properties for the edges are discussed. Finally, identities are given for characteristics of the “typical” edge and an edge chosen at random emanating from the “typical” vertex.
TL;DR: In this paper, it is proved that a weak solution of a noncharacteristic Cauchy problem for linear parabolic equations in divergence form with coefficients in a Holmgren class 2 in time exists if and only if the Cauche data arc functions of a holmgren classification function of a function g(t) defined on (α, β) is said to be of a HOLGEN class 2, if g ϵC∞ (α and β) and for all nonnegative integers n there exist positive constants c and s such that |g(
Abstract: Noncharacteristic Cauchy problems for parabolic equations arc frequently encountered in many areas of heat transfer. These problems are well known to be severely ill-posed. In this paper a solvability criterion for a class of such problems is established. It is proved that a weak solution of a noncharacteristic Cauchy problem for linear parabolic equations in divergence form with coefficients in a Holmgren class 2 in time exists if and only if the Cauchy data arc functions of a Holmgren class 2! A function g(t) defined on (α, β) is said to be of a Holmgren class 2, if g ϵC∞ (α, β) and for all nonnegative integers n there exist positive constants c and s such that |g(n)| < csn(2n)!.
TL;DR: In this paper, the authors considered the problem of complex interpolation of certain Hardy-type subspaces of Kothe function spaces and developed a general framework for such results and their methods apply to many more general situations including the vector-valued case.
Abstract: We consider the problem of complex interpolation of certain Hardy-type subspaces of Kothe function spaces. For example, suppose that X0 and X1 arc Kothe function spaces on the unit circle T. and let HX0 and HX1 be the corresponding Hardy spaces. Under mild conditions on X0. X1 we give a necessary and sufficient condition for the complex interpolation space [HX0. HX1]0 to coincide with HX0 where HX0 = [H0. H1]0. We develop a very general framework for such results and our methods apply to many more general situations including the vector–valued case.
TL;DR: In this paper, a new criterion for starlikeness in the unit disc and an application to a certain class of rational functions are given, based on the star-likeness of the disc.
Abstract: A new criterion for starlikeness in the unit disc and an application to a certain class of rational functions are given.
TL;DR: In this paper, a discrete characterization of Besov and Triebel spaces is presented, which is used to determine various classes Fourier multipliers for these spaces, in particular, results of R. Johnson are recovered.
Abstract: We present a discrete characterization of Besov and Triebel spaces which is used to determine various classes Fourier multipliers for these spaces. In particular, results of R. Johnson are recovered.
Abstract: After proving the Khintchine inequality for the n-Rademacher functions of ARON and GLOBVENIK with constants independent from n e IN, applications are given to the theory of polynomials and holomorphic functions between Banach spaces. In particular, the following result is proved: Every entire mapping from a Banach space into another one of cotype q, vanishing at the origin and of r-dominated type at zero for some r > 0 maps unconditionally 2-summable sequences into absolutely q-summable sequences.
TL;DR: In this article, Ozawa et al. describe self-duality and C*reflexivity of Hilbert A-modules over monotone complete C*-algebras by the completeness of the unit ball of ℳ with respect to two types of convergence being defined, and by a structural criterion.
Abstract: The aim of the present paper is to describe self-duality and C*-reflexivity of Hilbert A-modules ℳ over monotone complete C*-algebras A by the completeness of the unit ball of ℳ with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results ofH. Widom [Duke Math. J. 23, 309-324, MR 17 # 1228] and W. L. Paschke [Trans. Amer. Mat. Soc. 182, 443-468, MR 50 # 8087, Canadian J. Math. 26, 1272-1280, MR 57 # 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. M. Ozawa [J. Math. Soc. Japan 36, 589-609, MR 85 # 46068]). Especially, one derives that for a C*-algebra A the A-valued inner product of every Hilbert A-module ℳ can be continued to an A-valued inner product on it's A-dual Banach A-module ℳ' turning ℳ' to a self-dual Hilbert A-module if and only if A is monotone complete (or, equivalently, additively complete) generalizing a result of M. Hamana [Internat. J. Math. 3 (1992), 185 - 204]. A classification of countably generated self-dual Hilbert A-modules over monotone complete C*-algebras A is established. The set of all bounded module operators End′(ℳ) on self-dual Hilbert A-modules ℳ over monotone complete C*-algebras A is proved again to be a monotone complete C*-algebra. Applying these results a Weyl-Berg type theorem is proved.
TL;DR: In this article, the existence of functions f ∈ Nν assuming prescribed values w1, , wn ∈ C in these points: f(zi) = wi for i = 1, i , n (1) Furthermore, a description of all solutions of (1), through selfadjoint extensions of a certain symmetric operator is given.
Abstract: In this paper we consider an interpolation problem of Nevanlinna-Pick type: If finitely many points z1, , zn of the open upper half plane C + are given, we study the existence of functions f ∈ Nν assuming prescribed values w1, , wn ∈ C in these points: f(zi) = wi for i = 1, , n (1) Furthermore a description of all solutions of (1) through selfadjoint extensions of a certain symmetric operator is given We recall the definition of the classes Nν If f is a complex function denote by ρ(f) the domain of holomorphy of f Definition 1 Let π and ν be nonnegative integers or ∞ Denote by Nν the set of all functions f which are meromorphic in C, such that the kernel
TL;DR: In this article, a very general Opial-type inequalities involving higher-order derivatives of two functions are presented, and extended and improved versions of several recent results are derived from these inequalities.
Abstract: In this paper we shall offer very general Opial-type inequalities involving higher order derivatives of two functions. From these inequalities we then deduce extended and improved versions of several recent results.
TL;DR: In this article, a criterion for the Fredholmness of singular integral operators with Carleman shift in LP(Γ) is obtained, where Γ is either the unit circle or the real line.
Abstract: A criterion for the Fredholmness of singular integral operators with Carleman shift in LP(Γ) is obtained, where Γ is either the unit circle or the real line. The approach allows to consider unbounded coefficients in a class related to that of quasicontinuous functions. Applications to Wiener-Hopf-Hankel type operators and operators with linear fractional Carleman shift on IR are included.
TL;DR: In this paper, a geometric theory of singular perturbations is used to prove convergence of the solutions of the regularized problems towards that of the index 2 problem, and the limits of the present theory are discussed and directions of future research are proposed.
Abstract: The present paper deals with quasilinear differential-algebraic equations with index 2. These equations are approximated by regularization methods. Such methods lead to singularly perturbed differential-algebraic equations. Using a geometric theory of singular perturbations convergence of the solutions of the regularized problems towards that of the index 2 problem is proved. The limits of the present theory are discussed and directions of future research are proposed.
TL;DR: In this paper, the boundary-transmission problems for the two dimensional Helmholtz equation are well-posed in a family of Sobolev spaces with finite energy norms, through a reduction to equivalent systems of boundary integral equations of Wiener-Hopf type in [L2+ (IR)]2.
Abstract: Sommerfeld-type diffraction problems for a half-plane with arbitrary n-th order generalized impedance boundary conditions arc examined in a Sobolev space setting. The corresponding boundary-transmission problems for the two dimensional Helmholtz equation are shown to be well-posed in a family of Sobolev spaces with finite energy norms, through a reduction to equivalent systems of boundary integral equations of Wiener-Hopf type in [L2+ (IR)]2. Formulas for the solutions as well as the so-called edge conditions arc obtained for any n, by explicit canonical generalized factorization of the presymbols of the associated Wiener-Hopf operators.