| Contents |
| Let $\Omega(t)$, \ $t \in [0,T],$ be an unknown domain in $
\mathbb{R}^n$, $n \ge 2$, with a boundary \
$\partial\Omega(t)=:\gamma(t)$, at the initial moment the domain
$\Omega(0)$ and it's boundary $\partial\Omega(0) = \gamma(0)$ are
known.
There are studied multidimensional one-phase free boundary
problems for the heat equation with unknowns $u(x,t)$ defined in
$\Omega(t)$,
and free boundary $\gamma(t)$,\ $t \in [0,T]$. On
$\gamma(t)$, $t\in (0,t)$, we have the conditions
$$u = 0, \ |\nabla u| = \varphi(x,t), \ \varphi(x,t)\ge d_0 = {\mbox {const}} >0, $$
in the first problem, and
$$u = 0, \ |\nabla u| = -V_N + \varphi(x,t) $$ in the second one,
where $V_N$ is the velocity of the free boundary on the direction
of a vector $N(\xi), \ \xi\in \gamma(0)$. In particular, the
cases, when $\varphi(x,t) = c_0$, \ $c_0=$const, are considered.
The existence, uniqueness of the solutions of these problems are
proved in the H\"{o}lder spaces locally in time, the estimates of
the solutions are derived. |
|