A variational treatment for general elliptic equations of the flame propagation type: regularity of the free boundary

摘要

We develop a variational theory to study the free boundary regularity problem for elliptic operators: Lu = D-j(a(ij)(x)D(i)u) + b(i)u(i) + c(x)u = 0 in {u > 0}, < a(ij)(x)del u, del u > = 2 on partial derivative{u > 0}. We use a singular perturbation framework to approximate this free boundary problem by regularizing ones of the form: Lu-epsilon = beta(epsilon)(u(epsilon)), where beta(epsilon) is asuitable approximation of Dirac delta function delta(0). A useful variational characterization to solutions of the above approximating problem is established and used to obtain important geometric properties that enable regularity of the free boundary. This theory has been developed in connection to a very recent line of research as an effort to study existence and regularity theory for free boundary problems with gradient dependence upon the penalization. (C) 2007 Elsevier Masson SAS. All rights reserved.