In this paper we will find optimal lower bound for the first eigenvalue of the fourth order equation with integrable potentials when the L1norm of potentials is known. We establish the minimization character-ization for the first eigenvalue of the measure differential equation, which plays an important role in the extremal problem of ordinary differential equation. The conclusion of this paper will illustrate a new and very interesting phenomenon that the minimizing measures will no longer be located at the center of the interval when the norm is large enough.
Wei ChenDepartment of Mathematics, College of Science, Hohai UniversityJifeng ChuDepartment of Mathematics, College of Science, Hohai UniversityPing YanDepartment of Mathematical Sciences, Tsinghua UniversityMeirong ZhangDepartment of Mathematical Sciences, Tsinghua University
Classical Analysis and ODEsmathscidoc:1611.05002
Communications in Contemporary Mathematics, 18, (6), 1550085, 2016
To characterize the complete structure of the Fucik spectrum of the p-Laplacian on higher
dimensional domains is a long-standing problem. In this paper, we study the p-Laplacian
with integrable potentials on an interval under the Dirichlet or the Neumann boundary
conditions. Based on the strong continuity and continuous differentiability of solutions
in potentials, we will give a comprehensive characterization of the corresponding Fucik
spectra: each of them is composed of two trivial lines and a double-sequence of differentiable,
strictly decreasing, hyperbolic-like curves; all asymptotic lines of these spectral
curves are precisely described by using eigenvalues of the p-Laplacian with potentials;
and moreover, all these spectral curves have strong continuity in potentials, i.e. as potentials
vary in the weak topology, these spectral curves are continuously dependent on
potentials in a certain sense.