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Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations. (English) Zbl 0593.35047
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.
Reviewer: D.Tiba

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B65 Smoothness and regularity of solutions to PDEs 35R35 Free boundary problems for PDEs
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##### References:
 [1] Aronszajn, N, A unique continuation for solutions of elliptic partial differential equations or inequalities of second order, J. math. pures appl, 36, 235-249, (1957) · Zbl 0084.30402 [2] Caffarelli, L.A; Friedman, A, The free boundary in the Thomas-Fermi atomic model, J. differential equations, 32, 335-356, (1979) · Zbl 0408.35083 [3] Caffarelli, L.A; Friedman, A, The shape of axisymmetric rotating fluids, J. funct. anal, 35, 109-142, (1980) · Zbl 0439.35068 [4] Cordes, H.O, Über die bestimmtheit der Lösungen elliptischer differentialgleichungen durch anfangsvorgaben, Nachr. akad. wiss. Göttingen iia math. phys. kl, 239-258, (1956) · Zbl 0074.08002 [5] \scE. DiBenedetto and A. Friedman, Conduction-convection problems with change of phase, J. Differential Equations, to appear. · Zbl 0593.35085 [6] Erdély, A, () [7] Friedman, A, Variational principles and free boundary problems, (1982), Wiley New York · Zbl 0564.49002
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