Abstract
This paper deals with a free boundary porblem connected with the concept “quadrature surface”. Let Ω⊂R n be a bounded domain with aC 2 boundary and μ a measure compactly supported in Ω. Then we say ∂Ω is a quadrature surface with respect to μ if the following overdetermined Cauchy problem has a solution.
Applying simple techniques, we derive basic inequalities and show uniform boundedness for the set of solutions. Distance estimates as well as uniqueness results are obtained in special cases, e.g. we show that if ∂Ω and ∂D are two quadrature surfaces for a fixed measure μ and Ω is convex, thenD⊂Ω. The main observation, however, is that if ∂Ω is a quadrature surface for μ≥0 andxε∂Ω, then the inward normal ray to ∂Ω atx intersects the convex hull of supp μ. We also study relations between quadrature surfaces and quadrature domains.D is said to be a quadrature domain with respect to a mesure μ if there is a solution to the following overdetermined Cauchy problem:
Finally, we apply our results to a problem of electrochemical machining.
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References
Acker, A., Heat flow inequalities with applications to heat flow and optimization problems,SIAM J. Math. Anal. 8 (1977), 604–618.
Alt, H. W. andCaffarelli, L. A., Existence and regularity for a minimum problem with free boundary,J. Reine Angew. Math. 325 (1981), 434–448.
Caffarelli, L. A., Compactness methods in free boundary problems,Comm. Partial Differential Equations 5 (1980), 427–448.
Gidas, G., Wei-Ming Ni andNirenberg, L., Symmetry and related properties via the maximum principle,Comm. Math. Phys. 68 (1979), 209–243.
Gustafsson, B., Applications of variational inequalities to a moving boundary problem for Hele Shaw flows,SIAM J. Math. Anal. 16 (1985), 279–300.
Gustafsson, B., On quadrature domains and an inverse problem in potential theory,J. Analyse Math. 55 (1990), 172–216.
Gustafsson, B., Application of half-order differentials on Riemann surfaces to quadrature identities for arc-length,J. Analyse Math. 49 (1988), 54–89.
Gustafsson, B. andSakai, M., Properties of some balayage operators, with applications to quadrature domains and moving boundary problems,Preprint.
Helms, L. L.,Introduction to Potential Theory, John Wiley & Sons, New York, 1969.
Hopf, H.,Differential Geometry in the Large,Lecture Notes in Math. 1000 Springer-Verlag, Berlin-Heidelberg, 1983.
Isakov, V.,Inverse Source Problems,Mathematical Surveys and Monographs 34, Amer. Math. Soc., Providence, R. I., 1990.
Kellogg, O. D.,Foundations of Potential Theory, Ungar, New York, 1970, 4th printing.
Khavinson, D. andShapiro, H. S., The Schwarz potential inR n and Cauchy's problem for the Laplace equation,Research report TRITA-MATH-1989-36, Royal Inst. of Technology, Stockholm.
Kinderlehrer, D. andNirenberg, L., Regularity in free boundary problems,Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1977), 373–391.
Lacey, A. A. andShillor, M., Electrochemical machining with a threshold current, to appear inIMA J. Appl. Math.
Matano, H., Asymptotic behavior of the free boundaries arising in one face Stefan problems in multi-dimensional spaces, inNonlinear Partial Differential Equations in Applied Science (Fujita, H., Lax, P. D., Strang, G., eds.),Lecture Notes Numer. Appl. Anal. 5, pp. 133–151, North-Holland, Amsterdam, 1983.
Protter, M. H. andWeinberger, H. F.,Maximum Principle in Differential Equations, Prentic-Hall, Englewood Cliffs, N. J., 1967.
Sakai, M.,Quadrature Domains,Lecture Notes in Math. 934, Springer-Verlag, Berlin-Heidelberg, 1982.
Sakai, M., Application of variational inequalities to the existence theorem on quadrature domains,Trans. Amer. Math. Soc. 276 (1983), 267–279.
Serrin, J., A symmetry problem in potential theory,Arch. Rational Mech. Anal. 43 (1971), 304–318.
Shahgholian, H., On quadrature domains and the Schwarz potential,Preprint (1989).
Shahgholian, H., A characterization of the sphere in terms of single-layer potentials, to appear inProc. Amer. Math. Soc.
Shapiro, H. S. andUllemar, C., Conformal mappings satisfying certain extremal properties, and associated quadrature identities,Research report TRITAMATH-1981-6, Royal Inst. of Technology, Stockholm.
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The author is grateful to Professor H. S. Shapiro for valuable suggestions. He also thanks Professor B. Gustafsson for his constructive criticism, which led to improvement of some technical details.
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Shahgholian, H. Quadrature surfaces as free boundaries. Ark. Mat. 32, 475–492 (1994). https://doi.org/10.1007/BF02559582
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DOI: https://doi.org/10.1007/BF02559582