| Abstract: |
| We study two--phase multidimensional Stefan problem with two
small parameters $\varepsilon > 0, \ \kappa > 0$ at the highest
derivatives in the condition on the free boundary. This problem
is considered in the H\{o}lder space.
The small parameters in the problems have very deep physical
cense. They characterize the properties of the medium, in which
the processes go. And it is very important to know the behavior of
the solutions of the problems, when the properties of the medium
are changes, i.e. small parameters tend to zero.
The aim of the work is to obtain from the solution of the fully
perturbed nonlinear problem (problem A) with $\varepsilon
> 0, \ \kappa > 0$ the
solutions of the partially and fully unperturbed B, C, D problems
with $ \varepsilon
> 0, \ \kappa = 0$; \ $\kappa > 0$, \ $ \varepsilon = 0; \ \kappa
= 0$, \ $ \varepsilon = 0$ respectively do not solving them.
There are shown that the solution of the problem A converges as
$\kappa \to 0, \ \varepsilon > 0$; \ $\kappa > 0, \ \varepsilon
\to 0$; $\kappa \to 0, \ \varepsilon = 0$ to the unique solution
of the corresponding partially and fully unperturbed problem. We
prove that the limit solutions have the maximal regularity, that
is the solutions of the problems B, C, D are obtained without loss
of the smoothness of the given functions.The coercive estimates of
all problems are established in the H\{o}lder space.
From here it follows that the boundary layers do not appear,
although the small parameters are at the principal terms in the
condition on a free boundary. |
|