Variational problems with two phases and their free boundaries

Variational Problems with Two Phases and Their Free Boundaries

The problem of minimizing /[Vu|2 + q2(x)\2(v)] dx in an appropriate class of functions v is considered. Here q(x) ¥= 0 and A2(t>) = X2 if v < 0, = X22 if v > 0. Any minimizer u is harmonic in {u ¥= 0} and | Vu|2 has a jump a2(x)(\]-\22) across the free boundary [u ¥= 0). Regularity and various properties are established for the minimizer u and for the free boundary. Introduction. In this paper ...

Variational Iterative Method Applied to Variational Problems with Moving Boundaries

In this paper, He’s variational iterative method has been applied to give exact solution of the Euler Lagrange equation which arises from the variational problems with moving boundaries and isoperimetric problems. In this method, general Lagrange multipliers are introduced to construct correction functional for the variational problems. The initial approximations can be freely chosen with possi...

Dimensional Estimates for Singular Sets in Geometric Variational Problems with Free Boundaries

We show that singular sets of free boundaries arising in codimension one anisotropic geometric variational problems are Hn−3-negligible, where n is the ambient space dimension. In particular our results apply to capillarity type problems, and establish everywhere regularity in the three-dimensional case.

Free boundaries in problems with hysteresis.

Here, we present a survey concerning parabolic free boundary problems involving a discontinuous hysteresis operator. Such problems describe biological and chemical processes 'with memory' in which various substances interact according to hysteresis law. Our main objective is to discuss the structure of the free boundaries and the properties of the so-called 'strong solutions' belonging to the a...

Variational Problems with Moving Boundaries Using Decomposition Method