Symmetry and Overdetermined problems
|
Organizer(s): |
Name:
|
Affiliation:
|
Country:
|
Jyotshana Prajapat
|
University of Mumbai
|
India
|
|
|
|
|
|
|
|
|
|
Introduction:
| | Symmetry and Overdetermined problems
Many a times, it is useful to get qualitative information of a solution of a given PDE before even trying to find the exact solution. If we can show that a solution of PDE in $\R^n$ has radial symmetry, then the PDE reduces to an ODE. Such simplification helps in obtaining exact solutions, non existence of solutions, study of singular solutions, asymptotic behaviour of solutions and so on.
Proving symmetry of solutions has successfully led to resolving Yamabe type problems on manifolds, sub-Riemannian manifolds, Chern-Simon theories, integral equation to mention a few.
Overdetermined problems are PDEs defined on bounded domains (unbounded domains) where the boundary conditions (asymptotic behaviour ) are such that they force symmetry of the solutions and/or the domain where the problems are considered. The results in this category can be broadly summarized as
Serrin type problems on manifolds, Heisenberg group, sub-Riemannian manifolds, isoperimetric manifolds;
Allen-Cahn type equations and its variants for more general operators and geometries.
In this session I hope to bring together experts who have contributed to this field creating an opportunity for sharing recent progress, discussion on open problems and further collaborations.
|
|
|
|
| |
