75 research outputs found
Finding gaps in a spectrum
We propose a method for finding gaps in the spectrum of a differential
operator. When applied to the one-dimensional Hamiltonian of the quartic
oscillator, a simple algebraic algorithm is proposed that, step by step,
separates with a remarkable precision all the energies even for a double-well
configuration in a tunnelling regime. Our strategy may be refined and
generalised to a large class of 1d-problems
A new dynamical approach of Emden-Fowler equations and systems
We give a new approach on general systems of the form (G){[c]{c}%
-\Delta_{p}u=\operatorname{div}(|\nabla u| ^{p-2}\nabla u)=\epsilon_{1}|x|
^{a}u^{s}v^{\delta}, -\Delta_{q}v=\operatorname{div}(|\nabla v|^{q-2}\nabla
u)=\epsilon_{2}|x|^{b}u^{\mu}v^{m}, where are
real parameters, and In
the radial case we reduce the problem to a quadratic system of order 4, of
Kolmogorov type. Then we obtain new local and global existence or nonexistence
results. In the case we also describe the
behaviour of the ground states in two cases where the system is variational. We
give an important result on existence of ground states for a nonvariational
system with and In the nonradial case we solve a conjecture of
nonexistence of ground states for the system with and and
Comment: 43 page
A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle
We prove the following conjecture recently formulated by Jakobson,
Nadirashvili and Polterovich \cite{JNP}: on the Klein bottle , the
metric of revolution , , is the
\emph{unique} extremal metric of the first eigenvalue of the Laplacian viewed
as a functional on the space of all Riemannian metrics of given area. The proof
leads us to study a Hamiltonian dynamical system which turns out to be
completely integrable by quadratures
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