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Boundary regularity for free boundary problems. (English) Zbl 0928.35213

The purpose of this paper is to study up-to-the-boundary regularity of solutions of elliptic free boundary problems with two phases. The solution of a typical problem can be described as a function \(u\) that is harmonic in \(\{u\neq 0\}\) and satisfies the gradient jump condition: \(| \nabla u^+|^2 -|\nabla u^-|^2=1\) on the free boundary \(\{u= 0\}\). There exist several ways to state the problem rigorously. The variational formulation is given in H. W. Alt and L. A. Caffarelli [J. Reine Angew. Math. 325, 105-144 (1981; Zbl 0449.35105)] and H. W. Alt, L. A. Cafarelli and A. Friedman [Trans. Am. Math. Soc. 282, No. 2, 431-461 (1981; Zbl 0844.35137)], and a treatment of weak solutions is given in L. A. Caffarelli [Rev. Mat. Iberom. 4, No. 2, 139-162 (1987; Zbl 0676.35085)]. Notice that these papers contain only interior regularity results.
In order to obtain the regularity of the solutions the author studies the regularization of the problem by the singular perturbation family of elliptic operators: \(Lu=\beta_\varepsilon(u)\), where \(L\) is a uniformly elliptic operator in divergence form, and where \(\beta_\varepsilon (s)={1\over \varepsilon} \beta(s/ \varepsilon)\) with \(\beta\in C^\infty (\mathbb{R})\), \(\beta\geq 0\), and \(\text{supp} \beta\subset [0,1]\). This approach was used in H. Berestycki, L. A. Cafarelli and L. Nirenberg [Uniform estimates for regularization of free boundary problems. Lect. Notes Pure Appl. Math. 122, 567-619 (1990; Zbl 0702.35252)], where the up-to-the-boundary uniform Lipschitz estimate for this family in the one-phase situation \((u\geq 0)\) was proved.
To be precise, the author obtains up-to-the-boundary gradient estimates for the solution \(u\) of \[ Lu=\beta_\varepsilon (u)\quad\text{in }\Omega,\quad u=f\text{ on }\partial \Omega, \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^n\) and \(f\in C^{2, \alpha}(\overline\Omega)\) is a given Dirichlet data. The smooth Dirichlet data is not naturally compatible with the gradient jump condition, and a logarithmic blowup of \(\nabla u\) as \(\varepsilon\to 0\) can be expected at the intersection of the boundary \(\partial\Omega\) and the free boundary. For general Dirichlet data \(f\), the uniform estimate \[ \bigl|\nabla u(x) \bigr|\leq C\bigl(1+ \bigl| \log\bigl( \text{dist} (x,\partial \Omega)\bigr) \bigr| \bigr), \] where \(C\) is independent of \(\varepsilon>0\), is obtained. This estimate is sufficient to get for \(\varepsilon\to 0\) a subsequence converging locally uniformly to a weak solution of the free boundary problem. In addition, for Dirichlet data \(f\) satisfying the condition \(\sharp\), the uniform Lipschitz estimate \(|\nabla u(x)|\leq C\) is given, where the condition \(\sharp\) is such that the tangential gradient of \(f\) has to vanish wherever \(f\) vanishes.

MSC:

35R35 Free boundary problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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References:

[1] Alt, J Reine Angew Math 325 pp 105– (1981)
[2] Alt, Trans Amer Math Soc 282 pp 431– (1984)
[3] ; ; Uniform estimates for regularization of free boundary problems. Analysis and partial differential equations, 567-619, Lecture Notes in Pure and Appl Math, 122, Dekker, New York, 1990.
[4] Caffarelli, Rev Mat Iberoamericana 3 pp 139– (1987)
[5] Caffarelli, Amer J Math 120 pp 391– (1998)
[6] ; Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences, 224. Springer-Verlag, Berlin-New York, 1983.
[7] Grter, Manuscripta Math 37 pp 303– (1982)
[8] Boundary regularity for free boundary problems. Doctoral Dissertation, University of Chicago, 1997.
[9] Harmonic analysis techniques for second order elliptic boundary value problems. CBMS Regional Conference Series in Mathematics, 83. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 1994.
[10] Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130 Springer-Verlag New York, Inc., New York 1966.
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