Catherine Bandle

On Positive Solutions of Emden Equations in Cone-like Domains

Let x ∈ ℝ N , N ≥ 2 be a generic point, S N-1 = x ∶ |x| = 1 be the (N—l)-dimensional unit sphere, Ω ⊂ S n-1 be an arbitrary domain and denote by C the cone x ∶ x/|x| ∈ Ω. Consider the nonlinear Dirichlet problem where δ ∈ ℝ and p >1. We shall be interested in positive classical solutions. They might be discontinuous at the origin, and in this case they will be called singular solutions and will be denoted by us, in contrast to the regular solutions u ∈ C 2 (C) ∩ C 0 (C ∪ ∂C), u ≢ 0.

Applications of Rellich's perturbation theory to a classical boundary and eigenvalue problem

ZusammenfassungMit Hilfe der Rellichschen Störungstheorie wird die analytische Abhängigkeit der Lösungen eines Randwert- und eines Eigenwertproblems gezeigt, sowie des tiefsten Eigenwertes λ1(α) von einem Parameter α, der in den Randbedingungen auftritt.Als Anwendung ergeben sich Schranken für das Energieintegral und λ1(α). Ferner wird die Definition der RobinfunktionR(P, Q; α) erweitert auf negative Werte von α, insbesondere singuläre Werte −pj.

A Geometrical Isoperimetric Inequality and Applications to Problems of Mathematical Physics

The classical isoperimetric inequality states that among all closed curves of given circumference the circle encloses the largest area. This inequality has been considerably generalized by A. D. Alexandrow. He derived [1] inequalities for the case where the curve lies on an abstract surface, and obtained lower bounds for the length of the curve in terms of the area of the domain and an expression involving the curvature of the surface. In this paper we consider a curve Fo on an abstract surface whose endpoints lie on a curve F 1. With the help of Alexandrows inequality we construct lower bounds for the length of Fo. These bounds depend on the area of the domain between Fo and F1, the curvature of the surface and the geodesic curvature of F~. By use of the geometrical inequalities we derive a monotony property of the Greens function. The geometrical inequalities lead also to an estimate for the fundamental frequency of an inhomogeneous membrane with partially free boundary. The result extends the Rayleigh-Faber-Krahn inequality [12] and its generalizations obtained by Nehari [11] and the author [2, 3, 6]. At the end we indicate how to generalize the concept of Schwarz symmetrization [12] for functions which do not vanish at the whole boundary. This symmetrization combines in a certain way the ones defined in [2] and [3]. The principal results of this paper have already been announced in [4].

Some norm estimates for the solutions of a nonlinear diffusion problem

SummaryBy means of Rellichs identity bounds for the spectrum of the nonlinear problem Δv+λev=0 are derived and certain norms for the solutions are estimated.ZusammenfassungMit Hilfe einer Rellichschen Identität werden Schranken für das Spektrum des nichtlinearen Problems Δv+λev=0 angegeben. Ferner werden gewisse Normen für die Lösungen abgeschätzt.