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Общегородской семинар по математической физике им. В. И. Смирнова
16 октября 2023 г. 16:30, г. Санкт-Петербург, онлайн-конференция zoom
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Existence of planar H-loops via Hardy's inequality
R. Musina University of Udine
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Количество просмотров: |
Эта страница: | 88 | Материалы: | 14 |
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Аннотация:
Given a continuous function on $\mathbb{R}^2$, we study the existence of non-constant, $2\pi$-periodic solutions to the
$2$-dimensional Hamiltonian system
\begin{equation}
\label{eq:problem}
u''=|u'|~\!{\rm H}(u) iu'.
\end{equation}
Any non-constant solution to \eqref{eq:problem} parametrizes a closed planar curve having prescribed curvature ${\rm H}$ at each point.
Our interest in \eqref{eq:problem} is motivated also by its relations with Arnold's problem on ${\rm H}$-magnetic geodesics.
Problem \eqref{eq:problem} admits a variational formulation; under
reasonable assumptions on the prescribed (non-constant) curvature ${\rm H}$, the associated energy functional has a
nice Mountain-Pass geometry. However, due to the groups of dilations and translations
in $\mathbb{R}^2$, the Palais-Smale condition fails to hold, and in fact there could exists unbounded Palais-Smale sequences.
We will present an existence result which is strongly based on a Hardy type inequality for functions of two variables.
This is joint work with Gabriele Cora (Università di Torino, Italy).
Дополнительные материалы:
abstract_musina_stp.pdf (71.3 Kb)
Язык доклада: английский
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