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On the existence of convex classical solutions
for multilayer free boundary problems
with general nonlinear joining conditions

Author: Andrew Acker
Journal: Trans. Amer. Math. Soc. 350 (1998), 2981-3020
MSC (1991): Primary 35R35, 35J05, 76T05
MathSciNet review: 1422592
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Abstract: We prove the existence of convex classical solutions for a general multidimensional, multilayer free-boundary problem. The geometric context of this problem is a nested family of closed, convex surfaces. Except for the innermost and outermost surfaces, which are given, these surfaces are interpreted as unknown layer-interfaces, where the layers are the bounded annular domains between them. Each unknown interface is characterized by a quite general nonlinear equation, called a joining condition, which relates the first derivatives (along the interface) of the capacitary potentials in the two adjoining layers, as well as the spatial variables. A well-known special case of this problem involves several stationary, immiscible, two-dimensional flows of ideal fluid, related along their interfaces by Bernoulli's law.

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  • 1. A. Acker, Free-boundary optimization-a constructive, iterative method, J. Appl. Math. Phys. (ZAMP) 30 (1979), 885-900. MR 81k:31006
  • 2. -, Interior free boundary problems for the Laplace equation, Arch. Rat'l. Mech. Anal. 75 (1981), 157-168. MR 82f:35185
  • 3. -, On the convexity of equilibrium plasma configurations, Math. Meth. Appl. Sci. 3 (1981), 435-443. MR 83g:76106
  • 4. -, How to approximate the solutions for certain free boundary problems for the Laplace equation by using the contraction principle, Z. Angew. Math. Phys. (ZAMP) 32 (1981), 22-33. MR 82m:35004
  • 5. -, On the convexity and the successive approximation of solutions in a free boundary problem with two fluid phases, Comm. Part. Diff. Eqs. 14 (1989), 1635-1652. MR 91d:35235
  • 6. -, On the nonconvexity of solutions in free boundary problems arising in plasma physics and fluid dynamics, Comm. Pure Appl. Math. 42 (1989), 1165-1174 (Addendum: 44 (1991), 869-872). MR 91a:35167; MR 92h:35243
  • 7. -, A multilayer problem. In : Free Boundary Problems in Fluid Flow with Applications (Proceedings of the International Conference on Free Boundary Problems, Montreal, 1990), pp. 44-52. Pitman Res. Notes in Math. Ser., # 282. Longman Sci. Tech., Harlow, 1993. MR 94g:35222
  • 8. -, On the multilayer fluid problem: regularity, uniqueness, convexity, and successive approximation of solutions, Comm. Part. Diff. Eqs. 16 (1991), 647-666. MR 92k:35290
  • 9. -, On the existence of convex classical solutions to multilayer fluid problems in arbitrary space dimensions, Pacific J. Math. 162 (1994), 201-231. MR 94j:35196
  • 10. -, Convex free boundaries and the operator method, in Variational Problems (A. Friedman and J. Spruck, Editors), IMA Volumes in Mathematics and its Applications # 53, Springer-Verlag, 1993. MR 96a:35222
  • 11. -, On $2$-layer free boundary problems with generalized joining conditions: convexity and successive approximation of solutions. In: Comparison Methods and Stability Theory, edited by Xinzhi Liu and David Siegel. Lecture Notes in Pure and Applied Mathematics, Vol. 162, Marcel Dekker, Inc., New York, 1994. MR 95e:35229
  • 12. A. Acker and R. Meyer, A free boundary problem for the $p$-laplacian: uniqueness, convexity, and successive approximation of solutions, Electronic J. Diff. Eqs. 1995, No. 8, approx. 20 pp. MR 96c:35198
  • 13. H. W. Alt, L. A. Caffarelli and A. Friedman, Variational problems with two fluid phases and their free boundaries, Trans. Amer. Math. Soc. 282 (1984), 431-461. MR 85h:49014
  • 14. L. A. Caffarelli and A. Friedman, Convexity of solutions of semilinear elliptic equations, Duke Math. J. 52 (1985), 431-456. MR 87a:35028
  • 15. L. A. Caffarelli and J. Spruck, Convexity properties of some classical variational problems, Comm. Part. Diff. Eqs. 7 (1982), 1337-1379. MR 85f:49062
  • 16. A. Friedman, Variational principles and free boundary problems, Wiley, 1982. MR 84e:35153
  • 17. R. Gabriel, A result concerning convex level surfaces of $3$-dimensional harmonic functions, J. London Math. Soc. 32 (1957), 286-294. MR 19:848a
  • 18. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order (2nd ed.), Springer-Verlag, 1983. MR 86c:35035
  • 19. N. Korevaar and J. Lewis, Convex solutions to certain elliptic equations have constant rank Hessians, Arch. Rat'l. Mech. Anal. 97 (1987), 19-32. MR 88i:35054
  • 20. P. Laurence and E. Stredulinsky, A new approach to queer differential equations, Comm. Pure Appl. Math. 38 (1985), 333-355. MR 87f:35246
  • 21. -, Existence of regular solutions with convex level sets for semilinear elliptic equations with nonmonotone $L^1$ nonlinearities, Part I: an approximating free boundary problem, Indiana U. Math. J. 39 (1990), 1081-1114. MR 92m:35279
  • 22. J. L. Lewis, Capacitary functions in convex rings, Arch. Rat'l. Mech. Anal. 66 (1977), 201-224. MR 57:16638
  • 23. R. Meyer, Approximation of the solutions of free boundary problems for the $p$-Laplace equation, Doctoral dissertation, Wichita State University, May, 1993.

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Additional Information

Andrew Acker
Affiliation: Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260-0033

Keywords: Multilayer elliptic free boundary problem, convexity, non-linear joining conditions
Received by editor(s): August 15, 1995
Article copyright: © Copyright 1998 American Mathematical Society

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