A variational treatment for general elliptic equations of the flame propagation type: regularity of the free boundary |
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Authors: | Eduardo V Teixeira |
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Institution: | Rutgers University, Department of Mathematics, Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA; Departamento de Matemática, Universidade Federal do Ceará, 60455-760 Fortaleza, CE, Brasil |
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Abstract: | We develop a variational theory to study the free boundary regularity problem for elliptic operators: Lu=Dj(aij(x)Diu)+biui+c(x)u=0 in {u>0}, 〈aij(x)∇u,∇u〉=2 on ∂{u>0}. We use a singular perturbation framework to approximate this free boundary problem by regularizing ones of the form: Luε=βε(uε), where βε is a suitable approximation of Dirac delta function δ0. 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. |
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Keywords: | Free boundary problems Singular perturbation Complete elliptic operator Regularity theory |
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