Free Boundary Problems
Free boundary problems arise naturally in a number of physical phenomena. One of the typical examples of this sort is the phase transition, e.g. thawing of ice (as solid phase) and subsequent vaporisation of the water (liquid phase).
This problem, in its simplest mathematical form called Stefan problem, has attracted a considerable attention and there is vast literature discussing various aspects of this problem.
Typically the mathematical model consists of finding a solution to overdetermined boundary value problem for given second order PDE (e.g. Laplacian) portraying a conservation law... [More].
Monge-Ampere equation
This is one of my favorite equations: for given ψ find u such that
det D2u= ψ.
Before trying to solve this equation one has to specify the class of
admissible functions. For instance if we wish to consider the elliptic
Monge-Ampere then the class of admissible functions consists of
convex (or concave e.g. n=2) functions.
More general equations of this form may be considered as well
det (D2u(x)-σ(x, u, Du))=ψ(x).
Here σ is a given matrix depending on low order terms. This equation has a number of important applications in differential geometry, optimal mass transfers, geometric optics... [More]
Nonlinear Elasticity
Suppose
that u:Ω→ Rn is a deformation of bulk Ω⊂ Rn (e.g.
one can take Ω to be B1 the unit ball centered at the origin)
and let Ω* be the deformed domain u(Ω)=Ω*. Consider the volume (or area if n=2) preserving deformations i.e.
det ∇u=1.
We wish to minimize the stored energy ∫Ω|∇u|2 over the class of volume preserving deformations... [More]