tailieunhanh - Báo cáo hóa học: " The first nontrivial curve in the fučĺk spectrum of the dirichlet laplacian on the ball consists of nonradial eigenvalues"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : The first nontrivial curve in the fučĺk spectrum of the dirichlet laplacian on the ball consists of nonradial eigenvalues | Benedikt et al. Boundary Value Problems 2011 2011 27 http content 2011 1 27 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access The first nontrivial curve in the fucik spectrum of the dirichlet laplacian on the ball consists of nonradial eigenvalues Jiri Benedikt 1 Pavel Drábek2 and Petr Girg1 Correspondence benedikt@kma. department of Mathematics Faculty of Applied Sciences University of West Bohemia Univerzitni 22 306 14 Plzen Czech Republic Full list of author information is available at the end of the article Springer Abstract It is well-known that the second eigenvalue 12 of the Dirichlet Laplacian on the ball is not radial. Recently Bartsch Weth and Willem proved that the same conclusion holds true for the so-called nontrivial sign changing Fucik eigenvalues on the first curve of the Fucik spectrum which are close to the point l2 l2 . We show that the same conclusion is true in dimensions 2 and 3 without the last restriction. Keywords Fucik spectrum The first curve of the Fucík spectrum Radial and nonradial eigenfunctions 1. Introduction Let o c RN be a bounded domain N 2. The Fucik spectrum of -A on W0 2 f is defined as a set s of those 1 1- e R2 such that the Dirichlet problem f Au k u k u in 1 u 0 on do. 1 has a nontrivial solution u e W0 2 . In particular if 11 12 . are the eigenvalues of the Dirichlet Laplacian on o counted with multiplicity then clearly s contains each pair 1k 1k k e N and the two lines 11 X R and R X 11 . Following 1 p. 15 we call the elements of s 11 X R u R X 11 nontrivial Fucik eigenvalues. It was proved in 2 that there exists a first curve C of nontrivial Fucik eigenvalues in the sense that defining h 11 R by n z. inf p k1 k p is a nontrivial Fucik eigenvalue we have that 11 h 1 m for every 1 11 and the curve Cầẩ X n k k e ki tx consists of nontrivial Fucik eigenvalues. Moreover it was proved in 2 that C is a continuous and strictly decreasing curve which contains the point 12