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 | References | Similar Articles | Additional Information

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.

**1.**Andrew Acker,*Free boundary optimization—a constructive iterative method*, Z. Angew. Math. Phys.**30**(1979), no. 6, 885–900 (English, with German summary). MR**559244**, 10.1007/BF01590487**2.**Andrew Acker,*Interior free boundary problems for the Laplace equation*, Arch. Rational Mech. Anal.**75**(1980/81), no. 2, 157–168. MR**605527**, 10.1007/BF00250477**3.**A. Acker,*On the convexity of equilibrium plasma configurations*, Math. Methods Appl. Sci.**3**(1981), no. 3, 435–443. MR**657307**, 10.1002/mma.1670030130**4.**Andrew Acker,*How to approximate the solutions of certain free boundary problems for the Laplace equation by using the contraction principle*, Z. Angew. Math. Phys.**32**(1981), no. 1, 22–33 (English, with German summary). MR**615552**, 10.1007/BF00953547**5.**Andrew Acker,*On the convexity and on the successive approximation of solutions in a free-boundary problem with two fluid phases*, Comm. Partial Differential Equations**14**(1989), no. 12, 1635–1652. MR**1039913**, 10.1080/03605308908820671**6.**Andrew Acker,*On the nonconvexity of solutions in free-boundary problems arising in plasma physics and fluid dynamics*, Comm. Pure Appl. Math.**42**(1989), no. 8, 1165–1174. MR**1029123**, 10.1002/cpa.3160420808

Andrew Acker,*Addendum to: “On the nonconvexity of solutions in free-boundary problems arising in plasma physics and fluid dynamics” [Comm. Pure Appl. Math. 42 (1989), no. 8, 1165–1174; MR1029123 (91a:35167)]*, Comm. Pure Appl. Math.**44**(1991), no. 7, 869–872. MR**1115097**, 10.1002/cpa.3160440707**7.**A. Acker,*A multi-layer fluid problem*, Free boundary problems in fluid flow with applications (Montreal, PQ, 1990), Pitman Res. Notes Math. Ser., vol. 282, Longman Sci. Tech., Harlow, 1993, pp. 44–52. MR**1216377****8.**Andrew Acker,*On the multi-layer fluid problem: regularity, uniqueness, convexity, and successive approximation of solutions*, Comm. Partial Differential Equations**16**(1991), no. 4-5, 647–666. MR**1113101**, 10.1080/03605309108820772**9.**Andrew Acker,*On the existence of convex classical solutions to multilayer fluid problems in arbitrary space dimensions*, Pacific J. Math.**162**(1994), no. 2, 201–231. MR**1251898****10.**Andrew Acker,*Convex free boundaries and the operator method*, Variational and free boundary problems, IMA Vol. Math. Appl., vol. 53, Springer, New York, 1993, pp. 11–27. MR**1320771**, 10.1007/978-1-4613-8357-4_2**11.**A. Acker,*On 2-layer free-boundary problems with generalized joining conditions: convexity and successive approximation of solutions*, Comparison methods and stability theory (Waterloo, ON, 1993) Lecture Notes in Pure and Appl. Math., vol. 162, Dekker, New York, 1994, pp. 1–15. MR**1291605****12.**A. Acker and R. Meyer,*A free boundary problem for the 𝑝-Laplacian: uniqueness, convexity, and successive approximation of solutions*, Electron. J. Differential Equations (1995), No. 08, approx. 20 pp. (electronic). MR**1334863****13.**Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman,*Variational problems with two phases and their free boundaries*, Trans. Amer. Math. Soc.**282**(1984), no. 2, 431–461. MR**732100**, 10.1090/S0002-9947-1984-0732100-6**14.**Luis A. Caffarelli and Avner Friedman,*Convexity of solutions of semilinear elliptic equations*, Duke Math. J.**52**(1985), no. 2, 431–456. MR**792181**, 10.1215/S0012-7094-85-05221-4**15.**Luis A. Caffarelli and Joel Spruck,*Convexity properties of solutions to some classical variational problems*, Comm. Partial Differential Equations**7**(1982), no. 11, 1337–1379. MR**678504**, 10.1080/03605308208820254**16.**Avner Friedman,*Variational principles and free-boundary problems*, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. A Wiley-Interscience Publication. MR**679313****17.**R. M. Gabriel,*A result concerning convex level surfaces of 3-dimensional harmonic functions*, J. London Math. Soc.**32**(1957), 286–294. MR**0090662****18.**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190****19.**Nicholas J. Korevaar and John L. Lewis,*Convex solutions of certain elliptic equations have constant rank Hessians*, Arch. Rational Mech. Anal.**97**(1987), no. 1, 19–32. MR**856307**, 10.1007/BF00279844**20.**Peter Laurence and E. W. Stredulinsky,*A new approach to queer differential equations*, Comm. Pure Appl. Math.**38**(1985), no. 3, 333–355. MR**784478**, 10.1002/cpa.3160380306**21.**Peter Laurence and Edward Stredulinsky,*Existence of regular solutions with convex levels for semilinear elliptic equations with nonmonotone 𝐿¹ nonlinearities. I. An approximating free boundary problem*, Indiana Univ. Math. J.**39**(1990), no. 4, 1081–1114. MR**1087186**, 10.1512/iumj.1990.39.39051**22.**John L. Lewis,*Capacitary functions in convex rings*, Arch. Rational Mech. Anal.**66**(1977), no. 3, 201–224. MR**0477094****23.**R. Meyer,*Approximation of the solutions of free boundary problems for the -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

DOI:
https://doi.org/10.1090/S0002-9947-98-01943-6

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