Abstract 4. Nonlinear Cauchy-Kowalewski theorem in extrafunctions

Date: 2015-10-07; view: 542.

Abstract 3. A proof of the Gibbs-Thomson formula in the droplet formation regime (fragment 1)

Marek Biskup, Lincoln Chayes and Roman Kotecky

Department of Mathematics, UCLA, Los Angeles, California, USA; Center for Theoretical Study, Charles University, Prague, Czech Republic

We study equilibrium droplets in two-phase systems at parameter values corresponding to phase coexistence. Specifically, we give a self-contained microscopic derivation of the Gibbs-Thomson formula for the deviation of the pressure and the density away from their equilibrium values which, according to the interpretation of the classical thermodynamics, appears due to the presence of a curved interface. The general ‑ albeit heuristic ‑ reasoning is corroborated by a rigorous proof in the case of the two-dimensional Ising lattice gas.

Mark Burgin

Department of Mathematics

University of California, Los Angeles

Many important differential equations do not have solutions in the set of differentiable functions. In the development of the theory of PDE, this lead to the introduction of weak solutions and then distributions. However, this does not completely solve the problem, since many PDE (some of them are linear and very simple) do not have solutions in the set of distributions.

In this preprint, we show that an extension of the theory of extrafunctions allows one to find generalized solutions to a much larger set of nonlinear PDE than those which are solvable in distributions. In a sense, the approach given here follows the traditional method of solving PDE by of series of functions. This succeeds because in spaces of extrafunctions all series of ordinary functions are convergent.

There are other approaches, such as theories of generalized functions of Colombeau and Egorov, which also allow one to extend the scope of solvable nonlinear PDE. However, the function spaces in which these generalized solutions are constructed do not even have a T₀ topology. In particular, in these spaces a limit of a sequence is not unique.

In contrast to this, the spaces of extrafunctions have a Hausdorff topology. Moreover, these spaces are maximal with respect to this property. This makes extrafunctions universal for solving PDE under appropriate topological conditions.

Key words:nonlinear differential equation, extended extrafunction, extended hypernumber, sequential solution, distribution