On symmetrization in the single-phase Stefan problem
The investigated problem is as follows: to obtain such function \(u>0\) for which \[ u_ t-\partial_ i(a_{ij}(x,t,u)\partial_ ju=f(x,t),\quad f\geq 0\text{ on } G\subset {\mathbb{R}}^ n\times {\mathbb{R}}_+ \] with initial condition \(u|_{t=0}=u_ 0(x)\) and Stefan condition on the free boundary; \(a_{ij}\partial_ ju\partial \phi =\mu \phi_ t\), \(\mu >0\), where the unknown free boundary \(\Gamma\) is given by the equation \(\phi (t,x)=0\) and \(a_{ij}\) assumed to be elliptic. The tool of solution of this problem is the following: Instead of the original problem the solution of the symmetrized problem is investigated in which the domain \(\tilde G,\) \(\tilde a_{ij}\) and \(\tilde u_ 0\) are assumed to be symmetric. In this paper a theorem is proved which says: The norm of solution of the original problem can be estimated by the norm of solution of the symmetrized problem.
scientific article; zbMATH DE number 4179975