The one phase free boundary problem for the -Laplacian with non-constant Bernoulli boundary condition

Authors:
Antoine Henrot and Henrik Shahgholian

Journal:
Trans. Amer. Math. Soc. **354** (2002), 2399-2416

MSC (1991):
Primary 35R35, 35J70, 76S05

Published electronically:
February 14, 2002

MathSciNet review:
1885658

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Our objective, here, is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a Bernoulli-type boundary gradient condition and with the -Laplacian as the governing operator. The main theorems of this paper assert that the exterior and the interior free boundary problem with a Bernoulli law, i.e. with a prescribed pressure on the ``free'' streamline of the flow, have convex solutions provided the initial domains are convex. The continuous function is subject to certain convexity properties. In our earlier results we have considered the case of constant . In the lines of the proof of the main results we also prove the semi-continuity (up to the boundary) of the gradient of the -capacitary potentials in convex rings, with boundaries.

**[A]**Andrew Acker,*On the existence of convex classical solutions for multilayer free boundary problems with general nonlinear joining conditions*, Trans. Amer. Math. Soc.**350**(1998), no. 8, 2981–3020. MR**1422592**, 10.1090/S0002-9947-98-01943-6**[AM]**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****[AC]**H. W. Alt and L. A. Caffarelli,*Existence and regularity for a minimum problem with free boundary*, J. Reine Angew. Math.**325**(1981), 105–144. MR**618549****[BCN]**H. Berestycki, L. A. Caffarelli, and L. Nirenberg,*Uniform estimates for regularization of free boundary problems*, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 567–619. MR**1044809****[C1]**Luis A. Caffarelli,*A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are 𝐶^{1,𝛼}*, Rev. Mat. Iberoamericana**3**(1987), no. 2, 139–162. MR**990856**, 10.4171/RMI/47**[C2]**Luis A. Caffarelli,*A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz*, Comm. Pure Appl. Math.**42**(1989), no. 1, 55–78. MR**973745**, 10.1002/cpa.3160420105**[C3]**Luis A. Caffarelli,*A Harnack inequality approach to the regularity of free boundaries. III. Existence theory, compactness, and dependence on 𝑋*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**15**(1988), no. 4, 583–602 (1989). MR**1029856****[CLW]**L. A. Caffarelli, C. Lederman, and N. Wolanski,*Uniform estimates and limits for a two phase parabolic singular perturbation problem*, Indiana Univ. Math. J.**46**(1997), no. 2, 453–489. MR**1481599**, 10.1512/iumj.1997.46.1470**[CS]**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**[DH]**P. Daskalopoulos and R. Hamilton,*Regularity of the free boundary for the porous medium equation*, J. Amer. Math. Soc.**11**(1998), no. 4, 899–965. MR**1623198**, 10.1090/S0894-0347-98-00277-X**[Di]**Emmanuele DiBenedetto,*Degenerate parabolic equations*, Universitext, Springer-Verlag, New York, 1993. MR**1230384****[DPS]**D. DANIELLI, A. PETROSYAN, H. SHAHGHOLIAN, A singular perturbation problem for the -Laplacian, with Bernoulli boundary condition. Submitted.**[Do]**Manfred Dobrowolski,*On quasilinear elliptic equations in domains with conical boundary points*, J. Reine Angew. Math.**394**(1989), 186–195. MR**977441**, 10.1515/crll.1989.394.186**[FR]**M. Flucher and M. Rumpf,*Bernoulli’s free-boundary problem, qualitative theory and numerical approximation*, J. Reine Angew. Math.**486**(1997), 165–204. MR**1450755****[GNN]**B. Gidas, Wei Ming Ni, and L. Nirenberg,*Symmetry of positive solutions of nonlinear elliptic equations in 𝑅ⁿ*, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR**634248****[GT]**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****[GS]**Björn Gustafsson and Henrik Shahgholian,*Existence and geometric properties of solutions of a free boundary problem in potential theory*, J. Reine Angew. Math.**473**(1996), 137–179. MR**1390686****[H]**Richard S. Hamilton,*The inverse function theorem of Nash and Moser*, Bull. Amer. Math. Soc. (N.S.)**7**(1982), no. 1, 65–222. MR**656198**, 10.1090/S0273-0979-1982-15004-2**[HKM]**Juha Heinonen, Tero Kilpeläinen, and Olli Martio,*Nonlinear potential theory of degenerate elliptic equations*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR**1207810****[He]**Antoine Henrot,*Continuity with respect to the domain for the Laplacian: a survey*, Control Cybernet.**23**(1994), no. 3, 427–443. Shape design and optimization. MR**1303362****[HS1]**Antoine Henrot and Henrik Shahgholian,*Convexity of free boundaries with Bernoulli type boundary condition*, Nonlinear Anal.**28**(1997), no. 5, 815–823. MR**1422187**, 10.1016/0362-546X(95)00192-X**[HS2]**Antoine Henrot and Henrik Shahgholian,*Existence of classical solutions to a free boundary problem for the 𝑝-Laplace operator. I. The exterior convex case*, J. Reine Angew. Math.**521**(2000), 85–97. MR**1752296**, 10.1515/crll.2000.031**[HS3]**Antoine Henrot and Henrik Shahgholian,*Existence of classical solutions to a free boundary problem for the 𝑝-Laplace operator. II. The interior convex case*, Indiana Univ. Math. J.**49**(2000), no. 1, 311–323. MR**1777029**, 10.1512/iumj.2000.49.1711**[K]**I. N. Krol′,*The solutions of the equation 𝐷_{𝑥ᵢ}(𝐷𝑢^{𝑝-2}𝐷_{𝑥ᵢ}𝑢)=0 with a singularity at a boundary point*, Trudy Mat. Inst. Steklov.**125**(1973), 127–139, 233 (Russian). Boundary value problems of mathematical physics, 8. MR**0344670****[Lan]**N. S. Landkof,*Foundations of modern potential theory*, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR**0350027****[Lav]**M. LAVRENTIEV, Variational Methods, Noordhoff, Groningen (1963).**[L]**John L. Lewis,*Capacitary functions in convex rings*, Arch. Rational Mech. Anal.**66**(1977), no. 3, 201–224. MR**0477094****[LS]**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**[MPS]**J. MANFREDI, A. PETROSYAN, H. SHAHGHOLIAN, A free boundary problem for the -Laplacian. Calc. Var. (2001).**[PP1]**G. A. Philippin and L. E. Payne,*On the conformal capacity problem*, Symposia Mathematica, Vol. XXX (Cortona, 1988) Sympos. Math., XXX, Academic Press, London, 1989, pp. 119–136. MR**1062609****[PP2]**L. E. Payne and G. A. Philippin,*On gradient maximum principles for quasilinear elliptic equations*, Nonlinear Anal.**23**(1994), no. 3, 387–398. MR**1291578**, 10.1016/0362-546X(94)90178-3**[S]**James Serrin,*A symmetry problem in potential theory*, Arch. Rational Mech. Anal.**43**(1971), 304–318. MR**0333220****[T]**Peter Tolksdorf,*On the Dirichlet problem for quasilinear equations in domains with conical boundary points*, Comm. Partial Differential Equations**8**(1983), no. 7, 773–817. MR**700735**, 10.1080/03605308308820285**[V]**Andrew L. Vogel,*Symmetry and regularity for general regions having a solution to certain overdetermined boundary value problems*, Atti Sem. Mat. Fis. Univ. Modena**40**(1992), no. 2, 443–484. MR**1200301**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
35R35,
35J70,
76S05

Retrieve articles in all journals with MSC (1991): 35R35, 35J70, 76S05

Additional Information

**Antoine Henrot**

Affiliation:
Ecole des Mines and Institut Elie Cartan, UMR CNRS 7502 and INRIA BP 239, 54506 Vandoeuvre-les-Nancy Cedex, France

Email:
henrot@iecn.u-nancy.fr

**Henrik Shahgholian**

Affiliation:
Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden

Email:
henriks@math.kth.se

DOI:
https://doi.org/10.1090/S0002-9947-02-02892-1

Keywords:
Free boundary,
convexity,
non-linear joining conditions

Received by editor(s):
July 14, 2000

Received by editor(s) in revised form:
August 16, 2001

Published electronically:
February 14, 2002

Additional Notes:
The first author thanks Göran Gustafsson Foundation for several visiting appointments to RIT in Stockholm

The second author was partially supported by the Swedish Natural Science Research Council and STINT. He also thanks Institute Elie Cartan for their hospitality. Both authors thank A. Petrosyan for some crucial remarks

Article copyright:
© Copyright 2002
American Mathematical Society