Abstract
The authors investigate the problem of identifying the domainG of a harmonic functionu such that Cauchy data are given on a known portion of the boundary ofG, while a zero Dirichlet condition is specified on the remaining portion of the boundary, which is to be found. Under certain conditions on the domainG, it is shown that the problem reduces to identifying the coefficients of an elliptic equation which, in turn, is converted into the problem of minimizing a functional. Under certain conditions onG, it is shown that the solution, if it exists, is unique. An application is pointed out for the problem of designing a vessel shape that realizes a given plasma shape.
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References
Cartan, H.:Calcul Differentiel, Hermann, Paris, 1966.
Courant, R. and Hilbert, D.:Methods of Mathematical Physics, vol. 1, Interscience, New York, 1953.
Cimati, G.: On a problem of the theory of lubrication governed by a variational inequality,Appl. Math. Optim. 3 (1977), 227–242.
Demidov, A. S.: The form of a steady plasma subject to the skin effect in a tokamak with noncircular cross-section,Nuclear. Fusion 15 (1975), 765–768.
Demidov, A. S.: Sur la perturbation singulière dans un problème a frontière libre, inLect. Notes in Math., Springer-Verlag, New York, 1977, pp. 123–130.
Demidov, A. S.: Equilibrium form of a steady plasma,Phys. Fliuds 21(6) (1978), 902–904.
Elliott, C. M.: On a variational inequality formulation of an electrochemical machining moving boundary problem and its applications by finite elements,J. Inst. Math. Appl. 25 (1980), 121–131.
Katano, Y., Kawamura, T. and Katami, H.: Numerical study of drop formation by a capillary jet using a general coordinate system,Theoret. Appl. Mech. 34 (1986), 3–14.
Kawarada, H., Sawaguri, T. and Imai, H.: An approximate resolution of a free boundary problem appearing in the equilibrium plasma by means of confirmal mapping,Japan J. Appl. Math. 6(3) (1989), 331–340.
Lattès, R. and Lions, J. L.:Méthode de Quasiréversibilité et Applications, Dunod, Paris, 1967, pp. 240–241.
Pironneau, O.:Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York, 1983.
Rodrigues, J. F.:Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, 1987.
Smoller, J.:Shock, Waves and Reaction Diffusion Equations, Springer-Verlag, New York, 1983.
Sasamoto, A., Imai, H. and Kawarada, H.: A practical method for an ill-conditioned optimal shape design of a vessel in which plasma is confined, in Yamagutiet al. (eds),Inverse Problems in Engineering Sciences, Springer-Verlag, New York, 1991, pp. 120–125.
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This work was completed with a financial support from the National Basic Research in the Natural Sciences.
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Ang, D.D., Vy, L.K. Domain identification for harmonic functions. Acta Appl Math 38, 217–238 (1995). https://doi.org/10.1007/BF00996147
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DOI: https://doi.org/10.1007/BF00996147