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**Two-phase Stefan problems in several space variables.**
*(Italian)*
Zbl 0545.35096

This is a very clear and comprehensive exposition of recent results on the two-phase multidimensional Stefan problem. Let \(\Omega \subset {\mathbb{R}}^ n\) be a bounded domain whose boundary \(\Gamma\) is supposed to be sufficiently regular. In the following \(Q=\Omega \times(0,T),\quad \Sigma =\Gamma \times(0,T)\) and N denotes the free boundary. Special reference is made to problems with a nonlinear boundary condition of the form \(k \partial \theta /\partial n+g(x,t,\theta)=0\) on \(\Sigma\), where \(\theta\) is the temperature and \(\partial /\partial n\) denotes the outward normal derivative.

First of all the problem is stated in its classical form (Problem P1), i.e. \(c \partial \theta /\partial t-div(k\quad \text{grad} \theta)=f\) on \(Q\backslash N\), \(\theta =0\) on N, where the Stefan condition is imposed: \[ \lambda \quad \cos(\nu,t)-\Sigma_ i[k\partial \theta /\partial x_ i]\quad \cos(\nu,x_ i)=0 \] (\(\lambda\) latent heat, \(\nu\) unit normal vector to N, [ ] jump accross N). Recent results are quoted about local existence of classical solutions in the case of g linear w.r.t. \(\theta\).

Section 3 presents weak formulations involving temperature and enthalpy (Problem 2) or just enthalpy (Problem 3), adding some comments about their equivalence and their meaning. The author also illustrates other points of view in interpreting weak formulations within the framework of the theories of (i) linear monotone, coercive, hemicontinuous operators, and (ii) of nonlinear semigroups. One more weak formulation (Problem 4), based upon the freezing index \(w(x,t)=\int^{t}_{0}\theta(x,\tau) d\tau,\) and leading to a variational inequality is discussed in Section 4, emphasizing that solutions of P3 can be used to construct solutions of P4. It is stressed that the formulation P4 can be greatly simplified when dealing with one-phase problems. This simplified version has been used in the literature to obtain important regularity properties of the solutions.

With reference to P3 results about existence and uniqueness are reviewed in Section 5, with a brief sketch of the regularization procedure, which is one of the main tools in proving existence. Some regularity theorems and other qualitative properties are stated and discussed in Section 6, where some fundamental open questions are outlined.

In Section 7 the general difficulties in setting up a numerical procedure are pointed out and some specific finite elements or finite difference schemes are illustrated. This section is followed by a detailed discussion on error estimation and by one more section about different approaches for numerical computation. The bibliography includes 130 items, almost all from the last decade.

First of all the problem is stated in its classical form (Problem P1), i.e. \(c \partial \theta /\partial t-div(k\quad \text{grad} \theta)=f\) on \(Q\backslash N\), \(\theta =0\) on N, where the Stefan condition is imposed: \[ \lambda \quad \cos(\nu,t)-\Sigma_ i[k\partial \theta /\partial x_ i]\quad \cos(\nu,x_ i)=0 \] (\(\lambda\) latent heat, \(\nu\) unit normal vector to N, [ ] jump accross N). Recent results are quoted about local existence of classical solutions in the case of g linear w.r.t. \(\theta\).

Section 3 presents weak formulations involving temperature and enthalpy (Problem 2) or just enthalpy (Problem 3), adding some comments about their equivalence and their meaning. The author also illustrates other points of view in interpreting weak formulations within the framework of the theories of (i) linear monotone, coercive, hemicontinuous operators, and (ii) of nonlinear semigroups. One more weak formulation (Problem 4), based upon the freezing index \(w(x,t)=\int^{t}_{0}\theta(x,\tau) d\tau,\) and leading to a variational inequality is discussed in Section 4, emphasizing that solutions of P3 can be used to construct solutions of P4. It is stressed that the formulation P4 can be greatly simplified when dealing with one-phase problems. This simplified version has been used in the literature to obtain important regularity properties of the solutions.

With reference to P3 results about existence and uniqueness are reviewed in Section 5, with a brief sketch of the regularization procedure, which is one of the main tools in proving existence. Some regularity theorems and other qualitative properties are stated and discussed in Section 6, where some fundamental open questions are outlined.

In Section 7 the general difficulties in setting up a numerical procedure are pointed out and some specific finite elements or finite difference schemes are illustrated. This section is followed by a detailed discussion on error estimation and by one more section about different approaches for numerical computation. The bibliography includes 130 items, almost all from the last decade.

Reviewer: A.Fasano

### MSC:

35R35 | Free boundary problems for PDEs |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

65Z05 | Applications to the sciences |

65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

65N99 | Numerical methods for partial differential equations, boundary value problems |

47A20 | Dilations, extensions, compressions of linear operators |

47B44 | Linear accretive operators, dissipative operators, etc. |

35B65 | Smoothness and regularity of solutions to PDEs |

49J40 | Variational inequalities |