## Subsolutions and supersolutions in a free boundary problem.(English)Zbl 0809.35172

Summary: We begin by giving some results of continuity with respect to the domain for the Dirichlet problem (without any assumption of regularity on the domains). Then, following an idea of A. Beurling, a technique of subsolutions and supersolutions for the so-called quadrature surface free boundary problem is presented. This technique would apply to many free boundary problems in $$\mathbb{R}^ N$$, $$N\geq 2$$, which have overdetermined Cauchy data on the free boundary. Some applications to concrete examples are also given.

### MSC:

 35R35 Free boundary problems for PDEs 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
Full Text:

### References:

 [1] Acker, A., Heat flow inequalities with applications to heat flow optimization problem,SIAM J. Math. Anal 8 (1977), 604–618. · Zbl 0349.31003 [2] Aharonov, D. andShapiro, H. S., Domains on which analytic functions satisfy quadrature identities,J. Analyse Math. 30 (1976), 39–73. · Zbl 0337.30029 [3] Alt, H. W. andCaffarelli, L. A., Existence and regularity for a minimum problem with free boundary,J. Reine Angew. Math. 325 (1981), 105–144. · Zbl 0449.35105 [4] Beurling, A., On free boundary problems for the Laplace equation,Seminars on analytic functions I, Institute Advanced Studies Seminars (1957), Princeton, 248–263.Collected works of Arne Beurling I, pp. 250–265, Birkhäuser, Boston, 1989. [5] Carleman, T., Über ein Minimalproblem der mathematischen Physik,Math. Z. 1 (1918), 208–212. · JFM 46.0765.02 [6] Crouzeix, M. andDescloux, J., A bidimensional electromagnetic problem,SIAM J. Math. Anal. 21 (1990), 577–592. · Zbl 0709.35029 [7] Dautray, R. andLions, J. L.,Analyse mathématique et calcul numérique pour les sciences et les techniques, vol I and II, Masson, Paris, 1984. [8] Deny, J., Les potentiels d’énergie finie,Acta Math. 82 (1950), 107–183. · Zbl 0034.36201 [9] Gustafsson, B., Application of half-order differentials on Riemann surfaces to quadrature identities for arc-length,J. Analyse Math. 49 (1987), 54–89. · Zbl 0652.30029 [10] Gustafsson, B., On quadrature domains and an inverse problem in potential theory,J. Analyse Math. 55 (1990), 172–216. · Zbl 0745.31002 [11] Hedberg, L. I., Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem,Acta Math. 147 (1981), 237–264. · Zbl 0504.35018 [12] Henrot, A. andPierre, M., About existence of a free boundary in electromagnetic shaping, inRecent Advances in Nonlinear Elliptic and Parabolic Problems (Bénilan, P., Chipot, M., Evans, L. C. and Pierre M., eds.),Pitman Research Notes Series 208, pp. 283–293, Harlow, New York, 1989. [13] Henrot, A. andPierre, M., About existence of equilibria in electromagnetic casting,Quart. Appl. Math. 49 (1991), 563–575. · Zbl 0731.76092 [14] Mikhailov, V.,Equations aux dérivées partielles Mir, Moscow, 1980. [15] Osipov, Y. S. andSuetov, A. P. Existence of optimal domains for elliptic problems with Dirichlet boundary condition,Academy of Sciences, Institute of Mathematics and Mechanics, Sverdlovsk, USSR (1990). [16] Pironneau, O.,Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York, 1984. · Zbl 0534.49001 [17] Sakai, M., Application of variational inequalities to the existence theorem on quadrature domains,Trans. Amer. Math. Soc. 276 (1983), 267–279. · Zbl 0515.31001 [18] Shahgolian, H., Quadrature surfaces as free boundaries, to appear in Ark. Mat. [19] Shapiro, H. S. andUllemar, C., Conformal mappings satisfying certain extremal properties, and associated quadrature identities,Research report TRITA MATH, 1981–6, Royal Institute of Technology, Stockholm. [20] Sverak, V., On optimal shape design,Preprint, Heriot-Watt University, 1992.C. R. Acad. Sci. Paris Sér I Math. 315 (1992), 545–549. [21] Tepper, D. E., Free boundary problem,SIAM J. Math. Anal. 5 (1974), 841–846. · Zbl 0293.31004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.