Caratheodory theory for the Bernoulli problem

Acker, A. (1980) Concerning Danjuk’s existence theorem for free-boundary-value problems with a Bernoulli condition. Proc. Amer. Math. Soc. 80, 451–454.Search in Google Scholar

Aguilera, N., Alt, H. W. and Cafarelli, L. A. (1986) An optimization problem with volume constraint. SIAM J. Control Optim. 24, 2, 191–198.Search in Google Scholar

Akhiezer, N.I. (1990) Elements of the Theory of Elliptic Functions. Translations of Mathematical Monographs 79. AMS, Providence, Rhode Island.Search in Google Scholar

Alt, H. W. and Cafarelli, L. A. (1981) Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144.Search in Google Scholar

Beurling, A. (1957) On free-boundary problems for the Laplace equation. Sem. on Analytic Functions 1, Inst. for Advanced Study, Princeton, 248–263.Search in Google Scholar

Bucur, D. and Trebesch, I. (1998) Shape optimization problems governed by nonlinear state equations. Proc. Roy. Soc. Edinburgh 128A 945-963.Search in Google Scholar

Cafarelli, L. A. and Spruck, J. (1982) Convexity properties of solutions to some classical variational problems. Comm. Partial Differential Equations ., 11, 1337–1379.Search in Google Scholar

Crank, J. (1984) Free and Moving Boundary Problems. Oxford University Press, Clarendon, Oxford.Search in Google Scholar

Daniljuk, I. I. (1972) On integral functionals with a variable domain of integration. Proc. Steklov Inst. Math. 118 and AMS (1976). Search in Google Scholar

Dieudonné, J. (1960) Foundation of Modern Analysis. Academic Press, New York and London.Search in Google Scholar

Fasano, A. (1992) Some free boundary problems with industrial applications. In: G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich and S. Ulbrich, eds., Shape Optimization and Free Boundaries, NATO ASI Series (C: Mathematical and Physical Sciences), 380, 113–142. Springer, DordrechtSearch in Google Scholar

Flucher, M. and Rumpf, M. (1997) Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 486, 165–204.Search in Google Scholar

Golusin, G. M. (1969) Geometric Theory of Functions of a Complex Variable. Translations of Mathematical Monographs. AMS.Search in Google Scholar

Hamilton, R. S. (1982) The inverse function theorems of Nash and Moser. Bull. Amer. Math. Soc. ., 65–222.Search in Google Scholar

Haslinger, J., Kozubek, T., Kunisch, K. and Peichl, G. (2004) An embedding domain approach for a class of 2-d shape optimization problems: Mathematical analysis. J. Math. Anal. Appl. 290, 665–685.Search in Google Scholar

Henrot, A. and Onodera, M. (2021) Hyperbolic solutions to Bernoulli’s free boundary problem. Arch. Rat. Mech. Anal. 240, 761–784Search in Google Scholar

Landkof, N.S. (1972) Foundations of Modern Potential Theory, Springer-Verlag, Berlin.Search in Google Scholar

Šveràk, V. (1993) On optimal shape design. J. Math. Pures Appl., 72, 537–551Search in Google Scholar

Whittaker, E.T. and Watson, G.N. (1996) A Course of Modern Analysis. 4th edition. Cambridge University PressSearch in Google Scholar