Variational problems with two phases and their free boundary. (English) Zbl 0844.35137

The authors study the free boundary \(\Gamma=\partial\{u>0\}=\partial\{u<0\}\) of solutions \(u\) of the variational problem \(\int_\Omega|\nabla v|^2+q(x)\cdot\lambda^2(v) dx\to \min\), where \(\Omega\subset{\mathbb{R}}^n\) is open, \(0<q<\infty\) and \(\lambda(v)=\lambda_1^2\) if \(v>0\), and \(\lambda(v)=\lambda_2^2\) if \(v<0\), for some \(\lambda_1\neq\lambda_2\). The main result is that \(\Gamma\) is \(C^1\)-smooth if \(n=2\). As in the paper by the first two authors [J. Reine Angew. Math. 325, 105-144 (1981; Zbl 0449.35105)], where a similar free boundary problem is studied, they show that \(\nabla u\in L^\infty_{\text{loc}}(\Omega)\) and that the \((n-1)\)-dimensional Hausdorff measure of \(\Gamma\) is locally bounded, for arbitrary \(n\geq 2\). In contrast to the above-mentioned work, additional difficulties arise because the solutions may change sign. In order to overcome these difficulties, the authors prove a monotonicity formula which was probably inspired by a result from geometric measure theory.


35R35 Free boundary problems for PDEs
49J40 Variational inequalities
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)


Zbl 0449.35105
Full Text: DOI


[1] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105 – 144. · Zbl 0449.35105
[2] H. W. Alt, L. A. Caffarelli, and A. Friedman, Axially symmetric jet flows, Arch. Rational Mech. Anal. 81 (1983), no. 2, 97 – 149. · Zbl 0515.76017
[3] Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman, Asymmetric jet flows, Comm. Pure Appl. Math. 35 (1982), no. 1, 29 – 68. · Zbl 0515.76018
[4] Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman, Jet flows with gravity, J. Reine Angew. Math. 331 (1982), 58 – 103. · Zbl 0561.76022
[5] Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman, Jets with two fluids. I. One free boundary, Indiana Univ. Math. J. 33 (1984), no. 2, 213 – 247. · Zbl 0551.76013
[6] Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman, Jets with two fluids. II. Two free boundaries, Indiana Univ. Math. J. 33 (1984), no. 3, 367 – 391. · Zbl 0588.76017
[7] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801
[8] S. Friedland and W. K. Hayman, Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions, Comment. Math. Helv. 51 (1976), no. 2, 133 – 161. · Zbl 0339.31003
[9] Avner Friedman, Variational principles and free-boundary problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. A Wiley-Interscience Publication. · Zbl 0564.49002
[10] Enrico Giusti, Minimal surfaces and functions of bounded variation, Department of Pure Mathematics, Australian National University, Canberra, 1977. With notes by Graham H. Williams; Notes on Pure Mathematics, 10. · Zbl 0402.49033
[11] Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. · Zbl 0142.38701
[12] Emanuel Sperner Jr., Zur Symmetrisierung von Funktionen auf Sphären, Math. Z. 134 (1973), 317 – 327 (German). · Zbl 0283.26015
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