Project number: PN-II-ID-PCE-2011-3-0241
Symmetries in elliptic problems: Euclidean and non-Euclidean techniques
The primary goal of the present project is to study symmetry phenomena in nonlinear elliptic problems within non-euclidean frameworks.
First, we investigate elliptic PDEs which involve the Finsler-Laplace operator associated with asymmetric Minkowski norms modeling
for instance the Matsumoto mountain slope metric or various Randers-type norms coming from Mathematical Physics. Quantitative
anisotropic isoperimetric inequalities, improved anisotropic Hardy-Rellich inequalities, and the Wulff shape of minimizers for sharp optimal
constants in important integral inequalities will be considered by employing critical point theory and symmetrization arguments for
asymmetric Minkowski norms. Second, we are interested to identify those group actions which are hidden in non-euclidean phenomena by
solving elliptic PDEs via the principle of symmetric criticality.
We intend to describe the group actions with respect to Wulff shapes in elliptic problems involving asymmetric Minkowski norms,
the symmetries in elliptic problems on Riemannian manifolds via isometric actions, and the symmetry actions in higher-dimensional
Heisenberg groups via special subgroups of the unitary group by exploiting an idea from solving the Rubik-cube. We believe the proposed
approaches will generate further constructive ideas related to symmetries in other fields of nonlinear analysis (theory of bifurcations, N-body
problems, evolution equations, adiabatic perfect gas dynamics, crystal structures, etc).
[Plan of the project 2013 (RO)]