A free boundary problem for semilinear elliptic equations.

*(English)*Zbl 0598.35132The regularity of the free boundary for the following problem was investigated:
\[
\Delta u=a(x,u)\gamma u^{\gamma -1}\text{ on } \Omega \cap \{u>0\},\quad 0<\gamma <2;\quad u=| \nabla u| =0\text{ on } \Omega \cap \{u=0\}.
\]
Such an equation is used for modeling the distribution of a gas with density u(x) in reaction with a porous catalyst pellet. The authors proved that,

1) In an average sense, \(u(x)^{1/\beta}\), \(\beta =2/2-\gamma\), must grow like distant to the free boundary;

2) \(\partial \{u>0\}\) has locally finite n-1 dimensional Hausdorff measure in \(\Omega\). This allows to distinguish a subset of \(\Omega \cap \partial \{u>0\}\) where a tangent plane in measure exists, which is denoted by \(\partial_{red}\{u>0\};\)

3) A notion of ’flat’ free boundary point was introduced;

4) The \(C^{1,\alpha}\) surface regularity of the free boundary near such a point was proved. In particular, locally \(u=0\) on one side of the free boundary and \(u>0\) on the other side.

The work extends the results of L. A. Caffarelli [Acta. Math. 139, 155-184 (1978; Zbl 0386.35046); Commun. Partial Differ. Equations 5, 427- 448 (1980; Zbl 0437.35070)] in which \(\gamma =1\). However, the regularity hypothesis for a(x,z) in Caffarelli’s papers is weaker; and his hypothesis that replace the flatness condition is also weaker. The work can match the above results for \(n=2\). The notion of the flatness of a free boundary point was proceed along the line developed by the first author and L. A. Caffarelli [J. Reine Angew. Math. 325, 105-144 (1981; Zbl 0449.35105)].

1) In an average sense, \(u(x)^{1/\beta}\), \(\beta =2/2-\gamma\), must grow like distant to the free boundary;

2) \(\partial \{u>0\}\) has locally finite n-1 dimensional Hausdorff measure in \(\Omega\). This allows to distinguish a subset of \(\Omega \cap \partial \{u>0\}\) where a tangent plane in measure exists, which is denoted by \(\partial_{red}\{u>0\};\)

3) A notion of ’flat’ free boundary point was introduced;

4) The \(C^{1,\alpha}\) surface regularity of the free boundary near such a point was proved. In particular, locally \(u=0\) on one side of the free boundary and \(u>0\) on the other side.

The work extends the results of L. A. Caffarelli [Acta. Math. 139, 155-184 (1978; Zbl 0386.35046); Commun. Partial Differ. Equations 5, 427- 448 (1980; Zbl 0437.35070)] in which \(\gamma =1\). However, the regularity hypothesis for a(x,z) in Caffarelli’s papers is weaker; and his hypothesis that replace the flatness condition is also weaker. The work can match the above results for \(n=2\). The notion of the flatness of a free boundary point was proceed along the line developed by the first author and L. A. Caffarelli [J. Reine Angew. Math. 325, 105-144 (1981; Zbl 0449.35105)].

Reviewer: K.-C.Chang

##### MSC:

35R35 | Free boundary problems for PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

35J60 | Nonlinear elliptic equations |

76S05 | Flows in porous media; filtration; seepage |