an:03953370
Zbl 0593.35047
Caffarelli, Luis A.; Friedman, Avner
Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations
EN
J. Differ. Equations 60, 420-433 (1985).
00149366
1985
j
35J65 35B65 35R35
nonlinear elliptic problems; Hausdorff dimension; free boundary problems
Let u be a solution of \(\Delta u=f(x,u,\nabla u)\) in \(\Omega\), where \(\Omega\) is an open region in \(R^ m\). It is shown that the Hausdorff dimension of the singular subset
\[
S=\{x\in \Omega:\quad u(x)=0\quad and\quad \nabla u(x)=0\}
\]
of the zero-set \(\{u=0\}\) is at most m-2. The superlinear case \(| f(x,u,\nabla u)| \leq A| u|^{\alpha}+B| \nabla u|^{\beta},\) \(\alpha\geq 1\), \(\beta\geq 1\) and the linear case are discussed separately. The main application concerns the study of the free boundary in free boundary problems.
D.Tiba