Band 31, Heft 2 -

In this note we establish a Brunn–Minkowski inequality for the first eigenvalue of the Finsler–Laplacian.

In a recent paper a new direct proof for the irrationality of Euler´s number e  = ∑ k  = 0 ∞ 1/ k ! and on the same lines a simple criterion for some fast converging series representing irrational numbers was given. In the present paper, we give some generalizations of our previous results.

We consider the Nitsche functional, which is a linear combination of the area, the Willmore functional and the total Gauß curvature, on a class of surfaces of revolution with Dirichlet boundary data. We give sufficient conditions on the boundary data for the existence of a regular minimizer, and obtain thereby a solution of the corresponding Euler–Lagrange equation. Moreover we prove that above some threshold boundary value the optimal Nitsche energy is monotonically increasing as a function of the boundary values. Considering symmetric profile curves, we find a minimizer, whose profile curve is monotonically decreasing on the left half, and monotonically increasing on the right half of the interval.

For a monic polynomial with a given uniform norm on an arbitrary quasidisk a “detailed one-sided” version of the Erdős–Turán-type theorem on the distribution of its zeros is proved. In the case of the unit disk it coincides (up to inessential constants) with a recent result by Erdélyi.

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by universal series of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie on a prescribed surface outside of – Ω, known as the pseudo-boundary . The domains under consideration satisfy a rather mild boundary regularity requirement, namely, the segment condition. We study approximations with respect to the norms of the spaces C ℓ ( – Ω)and we establish the existence of universal series. Analogous results are obtainable with respect to the norms of Hölder spaces C ℓ , ν ( – Ω). The sequence a  = { a n } n  ∈ ℕ of coefficients of the universal series may be chosen in  ∩  p  > 1 l p (ℕ) but it can not be chosen in l 1 (ℕ).

In the present paper we prove two different theorems to exclude boundary branch points for minimal surfaces X in ℝ n . The statements roughly read as follows: A minimal surface X has no branch points on the boundary ∂Ω, (1) if for any P  ∈  X (∂Ω) there exists some strictly two-convex C 2 -subdomain U  =  U ( P ) ⊂ ℝ n whose boundary ∂ U contains P and such that X ( – Ω) ⊂  – U ; (2) or if for each point P  ∈  X (∂Ω) there exists some strictly two-convex C 2 -subdomain U  =  U ( P ) ⊂ ℝ n with P  ∈ ∂ U , X (∂Ω) ⊂  – U and such that ℝ n  ∖  U can be foliated by the boundaries of a family of strictly two-convex C 2 -subdomains of ℝ n ; (3) and in particular if X maps ∂Ω into the boundary of some strictly two-convex, star-shaped C 2 -subdomain of ℝ n .

We investigate the convergence behavior of the family of double sine integrals of the form ∫ ∞ 0 ∫ ∞ 0 f ( x,y ) sin ux sin vy dx dy ,      where      ( u,v ) ∈ ℝ 2 +  := ℝ +  × ℝ + ,  ℝ +  := (0, ∞), and f is a monotonically nonincreasing function. We give necessary and sufficient conditions for the uniform convergence of the ‘remainder’ integrals ∫ b 1 a 1 ∫ b 2 a 2 to zero in ( u,v ) ∈ ℝ 2 + as max{ a 1 , a 2 } → ∞ , where b j  >  a j  ≥ 0 , j  = 1,2 (called uniform convergence in the regular sense). This implies the uniform existence of the finite limits of the partial integrals ∫ b 1 0 ∫ b 2 0 in ( u,v ) ∈ ℝ 2 + as min{ b 1 , b 2 } → ∞ (called uniform convergence in Pringsheim´s sense). Our basic tool is the second mean value theorem for certain double integrals over a rectangle.