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

Henrot, Antoine; Shahgholian, Henrik

doi:10.1090/S0002-9947-02-02892-1

The one phase free boundary problem for the $p$-Laplacian with non-constant Bernoulli boundary condition
HTML articles powered by AMS MathViewer

by Antoine Henrot and Henrik Shahgholian PDF

Trans. Amer. Math. Soc. 354 (2002), 2399-2416 Request permission

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 $p$-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 $a(x)$ on the “free” streamline of the flow, have convex solutions provided the initial domains are convex. The continuous function $a(x)$ is subject to certain convexity properties. In our earlier results we have considered the case of constant $a(x)$. 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 $p$-capacitary potentials in convex rings, with $C^1$ boundaries.

References

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, DOI 10.1090/S0002-9947-98-01943-6
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.}, review= MR 1334863,
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
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
Luis A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are $C^{1,\alpha }$, Rev. Mat. Iberoamericana 3 (1987), no. 2, 139–162. MR 990856, DOI 10.4171/RMI/47
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, DOI 10.1002/cpa.3160420105
J. Horn, Über eine hypergeometrische Funktion zweier Veränderlichen, Monatsh. Math. Phys. 47 (1939), 359–379 (German). MR 91, DOI 10.1007/BF01695508
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, DOI 10.1512/iumj.1997.46.1470
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, DOI 10.1080/03605308208820254
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, DOI 10.1090/S0894-0347-98-00277-X
Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384, DOI 10.1007/978-1-4612-0895-2
D. Danielli, A. Petrosyan, H. Shahgholian, A singular perturbation problem for the $p$-Laplacian, with Bernoulli boundary condition. Submitted.
Manfred Dobrowolski, On quasilinear elliptic equations in domains with conical boundary points, J. Reine Angew. Math. 394 (1989), 186–195. MR 977441, DOI 10.1515/crll.1989.394.186
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
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\textbf {R}^{n}$, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR 634248
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, DOI 10.1007/978-3-642-61798-0
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
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, DOI 10.1090/S0273-0979-1982-15004-2
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
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
Antoine Henrot and Henrik Shahgholian, Convexity of free boundaries with Bernoulli type boundary condition, Nonlinear Anal. 28 (1997), no. 5, 815–823. MR 1422187, DOI 10.1016/0362-546X(95)00192-X
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, DOI 10.1515/crll.2000.031
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, DOI 10.1512/iumj.2000.49.1711
I. N. Krol′, The solutions of the equation $D_{x_{i}}(Du^{p-2}D_{x_{i}}u)=0$ 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
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027
M. Lavrentiev, Variational Methods, Noordhoff, Groningen (1963).
John L. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal. 66 (1977), no. 3, 201–224. MR 477094, DOI 10.1007/BF00250671
Peter Laurence and Edward Stredulinsky, Existence of regular solutions with convex levels for semilinear elliptic equations with nonmonotone $L^1$ nonlinearities. I. An approximating free boundary problem, Indiana Univ. Math. J. 39 (1990), no. 4, 1081–1114. MR 1087186, DOI 10.1512/iumj.1990.39.39051
J. Manfredi, A. Petrosyan, H. Shahgholian, A free boundary problem for the $\infty$-Laplacian. Calc. Var. (2001).
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
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, DOI 10.1016/0362-546X(94)90178-3
James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318. MR 333220, DOI 10.1007/BF00250468
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, DOI 10.1080/03605308308820285
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