Inequalities and Applications

INEQUALITIES AND

APPLICATIONS

ii WORLD SCIENTIFIC SERIES IN APPLICABLE ANALYSIS Editor Ravi P. Agarwal (National University of Singapore) Advisory Board J. M. Ball (Heriot-Watt Univ.) Claude Brezinski (Paris Drouot, BP 18 75433 - Paris Cedex 09) Shui-Nee Chow (Georgia Inst. Tech.) R. P. Gilbert (Univ. Delaware) P. J. van der Houwen (Stichting Math. Centrum) Junji Kato (Tohoku Univ.) V. Lakshmikantham (Florida Inst Tech.) Masatake Mori (Univ. Tokyo) M. Z. Nashed (Univ. Delaware) Wolfgang Walter (Univ. Karlsruhe)

Vol. 1: Recent Trends in Differential Equations ed. R. P. Agarwal Vol. 2: Contributions in Numerical Mathematics ed. R. P. Agarwal Vol. 3: Inequalities and Applications ed. R. P. Agarwal Forthcoming Vol. 4: Dynamical Systems and Applications ed. R. P. Agarwal Vol. 5: Recent Trends in Optimization Theory and Applications ed. R. P. Agarwal

World Scientific Series inApplicable Analysis Volume 3 Editor

R. P.Agarwal Department of Mathematics National University of Singapore

INEQUALITIES AND

APPLICATIONS

l i f e World Scientific ■ T

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

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V

Contributors A.Acker (U.S.A.) J.D.Aczel (Canada) A.Alvino (Italy) K.A.Ames (U.S.A.) Y.Avishai (Israel) C.Bandle (Switzerland) B.M.Brown (U.K.) R.C.Brown (U.S.A.) D.Brydak (Poland) P.S.Bullen (Canada) K.Deimling (Germany) J.Diaz (U.S.A.) A.Elbert (Hungary) P.W.Eloe (U.S.A.) L.H.Erbe (Canada) H.Esser (Germany) M.Essen (Sweden) W.D.Evans (U.K.) W.N.Everitt (U.K.) V.Ferone (Italy) A.M.Fink (U.S.A.) R.Ger (Poland) R.Girgensohn (Canada) P.Goetgheluck (France) W.Haussmann (Germany) S.Heikkila (Finland) J.Henderson (U.S.A.) G.Herzog (Germany) D.B.Hinton (U.S.A.) T.Horiuchi (Japan) S.Hu (U.S.A.) RK hi (C \ V C K' b Vll K ) „ ' t / ! , , V m '' x N.Kirchhoff (Germany) SwKKigbhflyh(yCA) 1 H.W.Knobloch (Germany) S"£?^vi ' N H.Komg (Germany) A.Kufner ^ \ W M KKwong (U S A.) A.Laforgia (Italy)

V.Lakshmikantham (U.S.A.) S.Leela (U.S.A.) ILLemmert (Germany) E.R.Love (Australia) G.Liittgens (Germany) S.Malek (U.S.A.) RManasevich (Chile) J.Mawhin (Belgium) R.Medina (Chile) M.Migda (Poland) RJ.Nessel (Germany) Z.Pales (Hungary) N.S.Papageorgiou (U.S.A.) L.E.Payne (U.S.A.) J.Pecaric (Croatia) L.E.Persson (Sweden) A.Peterson (U.S.A.) M.Pinto (Chile) M.Plum (Germany) J.Popenda (Poland) G.Porru (Italy) R.M.Redheffer (U.S.A.) A.A.Sagle (U.S.A.) S.Saitoh (Japan) 5 c L • W e /x £■£"P"^"^;) ? . F ; S h e a (U.S.A.) A S _ »™n (France) S.S'vasundaram (U.S.A.) *fgh ^ ^ ^ ^ S - S . t a n t ^ (Uf-A-) G.lalenti (Italy) G.Trombetti (Italy) S.Varosanec (Croatia) , A, s A / A q v . R V o ^ n Germaly) R WaQg ( u ^A } V.Weckesser (Germany) F.Zanolin (Italy) K.Zeller (Germany) A#Zett, ( u s A }

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vii

PREFACE

World Scientific Series in Applicable Analysis (WSSIAA) aims at report­ ing new developments of a high mathematical standard and of current interest. Each volume in the series shall be devoted to mathematical analysis that has been applied, or potentially applicable to the solutions of scientific, engineer­ ing, and social problems. This volume of WSSIAA contains 47 research articles on Inequalities by leading mathematicians from all over the world and a trib­ ute by R.M.Redheffer to Wolfgang Walter on his 66th birthday to whom this volume has been dedicated.

R.P.Agarwal

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IX

CONTENTS

Preface Wolfgang Walter R. M. Redheffer

vii 1

On free boundary problems for quasi-linear elliptic PDE's: Uniqueness and monotone ordering of convex solutions A. Acker

17

On the determination of price and quantity indices and their monotonicity J. D. Aczel and R. Girgensohn

33

An inverse isoperimetric comparison result A. Alvino, V. Ferone, B. Kawohl and G. Trombetli Stabilizing the backward heat equation against errors in the initial time geometry K. A. Ames and L. E. Payne Canonical products of genus two Y. Avishai and R. M. Redheffer Asymptotic behaviour and convexity of large solutions to nonlinear equations C. Bandle and G. Porru

39

47 53

59

Two integral inequalities B. M. Brown, W. D. Evans, W. N. Everiti and V. G. Kirby

73

An interpolation inequality and applications R. C. Brown and D. B. Hinion

87

On the best constant for the inequality /o 0 °y , 2 Lu,

u„ > / ( « ) .

Intuitively, if the first two inequalities hold and / is not too large, one might expect global existence of u, while if the last two inequalities hold and / is not too small, one might expect u to become infinite at a finite time. Assuming f(s)f'(s) > 0 for large s, it is shown that this behavior actually holds, and that the two cases correspond respectively to divergence or convergence of

r°°

dz

I /(*)/'(*)' In a parabolic problem initiated by Karawada, singular behavior of u is caused, not by inflow of heat from the boundary as above, but by a source function in the differential equation. Karawada's problem is formulated in [55] as follows: For a > 0, let Ta be the largest value for which the problem ut = U u +

,

u < 1;

0 < t < T„,

|x| < a

has a solution with null initial and boundary conditions. When a is small one might expect the zero boundary values to overcome the heat input due to the source, so the solution exists globally and T„ = oo. But if a is large, max* u(x, t) might be expected to become infinite as t approaches a finite value Ta. In [50] and jointly with Acker in [51, 55], this so-called quenching problem is attacked by differential inequalities and generalized to equations ut = Lu+f(u, ux) where L is an elliptic operator. The bounds for Ta improve the original result of Karawada and the methods have led to further work by others. Parabolic-functional equations In [54, 56, 63, 67] the methods of [39] are extended to systems of parabolic equations containing functionals; the results include equations with retarded or deviating arguments as a special case. When the region is un­ bounded, the functionals are controlled by combining analysis of NagumoWestphal type with an iterative procedure based upon two-sided rather than one-sided estimates. In the purely parabolic case, it is found that there is a connection between the effective dimension in the sense of Meyers and Serrin and the maximum principle.

4 When there is no x dependence the problem reduces to a functional differential equation of Cauchy-Kowalevsky type. An existence proof for a broad class of such equations is given in [82]; this follows and extends a novel approach to the classical CK theorem given in [81]. Another type of functional differential equation is treated in joint work with Blaz [47]. Strongly coupled parabolic equations Nickel was the first to use inequalities of Bernstein and Kolmogorov in the study of strong coupling. In [52] a classical inequality of Bernstein that forms the basis of Nickel's work is extended to entire functions of several complex variables and used to construct a Banach space in which D> is a bounded operator; here j is a multi-index. The result is an existenceuniqueness theorem for the strongly coupled equation ut = Y, M^D'uix, lil 0

for functions u : R —» R. The object is to give sharp conditions for all solutions to be bounded, for existence of a solution with u(0) > 0, and for existence or nonexistence of a barrier. The results correct a classical theorem of Hille and give new criteria for solvability of a certain parabolic-diffusion Cauchy problem. Similar techniques are used in [104] to obtain minimum principles for ODE's like those above. Suspension bridges. Referring to the storm of February 9, 1938, R.G.Cone, chief engineer of the Golden Gate Bridge, is quoted in [98] as follows: "The force of the wind was so strong that it was impossible to stand erect on the sidewalk or roadway of the bridge .. .The truss would be quiet for a second, then in the distance one could see a running wave of several nodes approaching... I wanted a witness to substantiate what I had seen, since the oscillations and deflections of the bridge were so pronounced that they would seem unbelievable.11

9 In joint work with McKenna, References [93, 98] give a complete and satis­ fying explanation, based on the normalized equation w« + U z n i + fcmax(u,0) = 1. Existence of oscillations depends on the nonlinearity, which is effective only if u changes sign. This means that the supporting cables are sometimes slack and explains why the phenomenon requires a very high wind. Pareto 's law. In 1957 the functional-differential equation f,(t, x) + (l3 + 7 ) / ( t , x) + 0xfz(t,

x) = 7 n 2 / ( « , nx),

/ ( 0 , x) = g(x)

was deduced by Wold and Whittle in connection with Pareto's law in eco­ nomics. Under mild regularity conditions the equation is solved in [73], and the solution is used to establish a general existence-uniqueness theorem of interest in economics. The subgradient. Let / : R™ —> (—oo, oo]) be a lower semicontinuous function from Rn to the extended reals. At a point where f(x) < oo the subgradient f'(x) is the set of all vectors p € R" such that H m i n f / ( y ) - f l ' ) - , fz o y - J ; ) > - o o , v—i

|y — x|

where (p, q) denotes the inner product. Since its introduction by Francis Clarke, the subgradient has played a decisive role in optimization theory and in the theory of elasticity solutions. The main theorem of [105] is as follows: Suppose | / ' ( x ) | < K at all points of the sphere |x — c| < 4r where f'(x) exists. Suppose also that /(x<j) < oo at some point of this sphere. Then / G Lip K for | i — c| < r. Answering a question of Clarke, Walter's student Weckesser has extended the result to Hilbert and certain Banach spaces. Editorships and books Walter is editor or coeditor of six books, associate editor of three jour­ nals, and author of the seven books described below. Those labeled (abefg) are published by Springer Verlag, those labeled (cd) by Verlag Bibliographisches Institut, Mannheim. (ab) Differential and integral inequalities. The 1964 edition (a) is the first treatise on this subject. It contains, among other things, a theorem on up­ per and lower solutions for nonlinear parabolic systems that constitutes a parabolic analog of the Max Muller theorem dating from 1921. (Special cases were rediscovered much later by others.) Among many novelties are contributions to the theory of hyperbolic equations based on [7-13], a simple approach to the strong maximum principle that has often been overlooked

10

[87], and results from [15-17, 19]. The revised and enlarged English trans­ lation (b) contains additional topics from [23-34], including the line method as discussed above. A high degree of originality is found throughout, yet the books give a comprehensive, historically accurate presentation, not just of Walter's work, but of the entire subject; in fact, (b) has bibliography of some 500 papers. After thirty years, these books are still leaders in the field. (c) Distribution theory. Here we find a concise account of distribution theory, as initiated by Sobolev in 1936 and developed by Schwartz. An attractive feature is an exceptionally elegant presentation of what might be called the auxiliary apparatus of the subject: the density of test functions in the class Cjj, the i? n version of the Urysohn lemma, partition of unity, and so on. Also noteworthy is the inclusion of such topics as tempered distributions, a form of the Paley-Wiener theorem in n variables, inequalities of Poincare and Sobolev, and the Weyl lemma. As is typical of Walter's books, this one exhibits an astonishing economy of style without loss of clarity. (d) Potential theory. This down-to-earth presentation of potential theory refutes the common view that works addressed to engineers cannot contain good mathematics. Applications are treated fully, with emphasis on the many uses of single and double layers. Yet the existence of a solution to the Dirichlet problem is demonstrated in no less than three ways: by passage to the limit from the corresponding difference equations, by integral equations based on Green's formula, and by the inequality method of Perron. (e) Differential equations. Now in its fifth edition, Walter's ODE text goes beyond what one expects in an introductory course, and also beyond what is in most ODE books of twice its size. A unifying principle is the systematic use of the Banach contraction principle, together with an adroit choice of metric, to obtain the central existence theorems, including those for power-series and generalized power-series solutions in the matrix case. The latter are based on a suitable Banach space of holomorphic functions; compare [102]. Besides the usual topics there is a detailed account of the dynamics of the phase plane, Floquet theory, Sturm-Liouville theory, Liapunov methods, and much more. Peano's existence theorem, the Brouwer fixed-point theorem, and the spectral theorem for bounded self-adjoint operators are all proved in full. Despite, or perhaps because of, its unusual scope and coverage, this book has long been the most popular undergraduate text on ODE's in Germany. If a forthcoming English translation attains comparable popularity, it could inaugurate a major improvement in US undergraduate education. (fg) Analysis. Walter's two-volume text on analysis is distinguished by histor­ ical scholarship and originality of approach. With no attempt to do justice in a brief account, three highlights are mentioned here: the use of Sard's lemma in connection with the change of variables in multiple integrals, a concise de­ velopment of the Lebesgue integral, and an extension of Chernoff's proof that gives uniform (as well as ordinary) convergence of Fourier series.

11 List of Publications 1. Uber ganze Losungen der Differentialgleichung Au = / ( u ) . Jahresber. DMV 57 (1955), 94-102. 2. Mittelwertsatze und ihre Verwendung zur Losung von Randwertaufgaben. Jahresber. DMV 59 (1957), Abt. 1, 93-131. 3. Ganze Losungen der Differentialgleichung A'u = / ( u ) . Math. Z. 67 (1957), 32-37. 4. Verallgemeinerte Laplace-Operatoren und Potentiate. Math. Z. 67 (1957), 38-48. 5. Uber die Euler-Poisson-Darboux-Gleichung. Math. Z. 67 (1957), 361-376. 6. Zur Existenz ganzer Losungen der Differentialgleichung A p u = e". Arch. Math. 9 (1958), 308-312. 7. Eindeutiglceitssatze fur gewohnliche, parabolische und hyperbolische DifFerentialgleichungen. Math. Z. 74 (1960), 191-208. 8. On uniqueness theorems for ordinary differential equations and for partial differential equa­ tions of hyperbolic type (together with J.B. Diaz). Trans. Amer. Math. Soc. 96 (1960), 90-100. 9. Uber die Differentialgleichung u^v = / ( z , y , u , « , , » , ) . I. Eindeutiglceitssatze fiir das charakteristische Anfangswertproblem. Math. Z. 71 (1959), 308-324. 10. Uber die Differentialgleichung u IV = f(z,y,u,ux,iiy). II. Existenzsatze fur das charakteristische Anfangswertproblem. Math. Z. 71 (1959), 436-453. 11. Uber die Differentialgleichung u IV = f(x,y,u,ux,Uy). wertaufgabe. Math. Z. 73 (1960), 268- 279.

III. Die nichtcharakteristische Anfangs-

12. On the existence theorem of Caratheodory for ordinary and hyperbolic equations. Technical Note BN-172, AFOSR (University of Maryland) 1959. 13. Fehlerabschatzungen bei hyperbolischen Differentialgleichungen. Arch. Rat. Mech. Anal. 7 (1961), 249-272. 14. Uber die Tschebyscheff-Approximation differenzierbarer Funktionen. Z. Angew. Math. Mech. 41 (1961), T65-T66. 15. Fehlerabschatzungen und Eindeutiglceitssatze fiir gewohnliche und partielle Differentialglei­ chungen. Z. Angew. Math. Mech. 42 (1962), T49-T62. 16. Bemerkungen zu verschiedenen Eindeutigkeitskriterien fur gewohnliche Differentialgleichun­ gen. Math. Z. 84 (1964), 222-227. 17. Uber sukzessive Approximation bei Volterra-Integralgleichungen in mehreren Veranderlichen. Ann. Acad. Sci. Fenn. Ser. A I No. 345 (1965), 1-32.

12 18. Musik - naturwissenschaftlich betrachtet. n+m "Naturwissenschaft und Medizin" 2 (1965), 14-26. (C.F. Boehringer, Mannheim) 19. On nonlinear Volterra integral equations in several variables. J. Math. Mech. 16 (1967), 967-985. 20. Bemerkungen zu Iterationsverfahren bei linearen Gleichungssystemen. Numer. Math. 10 (1967), 80-85. 21. Das wissenschaftliche Werk von Erich Kamke. Jahresber. DMV 69 (1968), 193-205. 22. On the non-existence of maximal solutions for hyperbolic differential equations. Ann. Polon. Math. 19 (1967), 307-311. 23. Warmeleitung in Systemen mit mehreren Komponenten. International Series of Numerical Mathematics, Vol. 9 "Numerische Mathematik, Differentialgleichungen, Approximationstheorie", p. 177-185. Birkhauser Verlag, Basel 1968. 24. Die Linienmethode bei nichtlinearen parabolischen Differentialgleichungen. Numer. Math. 12 (1968), 307-321. 25. Ein Existenzbeweis fur nichtlineare parabolische Differentialgleichungen aufgrund der Linien­ methode. Math. Z. 107 (1968), 173-188. 26. Gewohnliche Differential-Ungleichungen im Banachraum. Arch. Math. 20 (1969), 36-47. 27. Approximation fur das Cauchy-Problem bei parabolischen Differentialgleichungen mit der Linienmethode. International Series of Numerical Mathematics, Vol. 10 "Abstract Spaces and Approximation", p. 139-145. Birkhauser Verlag, Basel 1969. 28. Existenzsatze im Grofien fur das Cauchyproblem bei nichtlinearen parabolischen Differential­ gleichungen mit der Linienmethode. Math. Ann. 183 (1969), 254-274. 29. Uber die Eindeutigkeitsbedingung von Krasnosel'skii und Krein bei hyperbolischen Differen­ tialgleichungen. Ann. Polon. Math. 22 (1970), 117-124. 30. Infinite systems of differential inequalities defined recursively (together with A. Lasota and Aaron Strauss). Math. Research Center, Technical Summary Report # 1021. J. Diff. Eqs. 9 (1971), 93-107. 31. Existence and convergence theorems for the boundary layer equations based on the line me­ thod. Math. Research Center, Technical Summary Report # 1025. Arch. Rat. Mech. Anal. 39 (1970), 169-188. Russian translation in: Matematica, Moskva 15 (1971), 46-65. 32. Ordinary differential inequalities and quasimonotonicity in ordered Banach spaces. Math. Research Center, Technical Summary Report # 1027. Published under the title: Ordinary differential inequalities in ordered Banach spaces. J. Diff. Eqs. 9 (1971), 253-261. 33. There is an elementary proof of Peano's existence theorem. Math. Research Center, Technical Summary Report # 1037. Amer. Math. Monthly 78 (1971), 170-173. 34. On the asymptotic behavior of solutions of the Prandtl boundary layer equations. Math. Research Center, Technical Summary Report # 1056. Indiana Univ. Math. J. (formerly J. Math. Mech.) 20 (1971), 829-841.

13 35. A comparison theorem and an inequality of Redheffer. Applicable Analysis 2 (1971), 111-113. 36. Some new aspects of the line method for parabolic differential equations. Conference on the Theory of Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, Vol. 280, 181-189. Springer Verlag 1972. 37. Flow-invariant sets and differential inequalities in normed spaces (together with R. Redheffer). Applicable Analysis 5 (1975), 149-162. 38. A note on contraction. SIAM Review 18 (1976), 107-111. 39. The total variation of solutions of parabolic differential equations and a maximum principle in unbounded domains (together with R. Redheffer). Math. Ann. 209 (1974), 57-67. 40. On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition. SIAM J. Math. Anal. 6 (1975), 85-90. 41. Remark on a paper by Aczel and Ostrowski. J. Australian Math. Soc. 22 (Series A) (1976), 165-166. 42. On a functional equation of Bellman in the theory of dynamic programming. Aequationes Math. 14 (1976), 435-444. 43. The line method for parabolic differential equations. Problems in boundary layer theory and existence of periodic solutions. Lecture Notes in Mathematics, Vol. 430 "Constructive and Computational Methods for Differential and Integral Equations", p. 395-413. Springer Verlag 1974. 44. A differential inequality for the distance function in normed linear spaces (together with R. RedhefTer). Math. Ann. 211 (1974), 299-314. 45. Periodic solutions to nonlinear parabolic equations (together with R. Gaines). Rocky Moun­ tain J. of Math. 7 (1977), 297-312. 46. A counterexample in connection with Egorov's theorem. Amer. Math. Monthly 84 (1977), 118-119. 47. tjber Funktional-Differentialgleichungen mit voreilendem Argument (together with J. Blai). Monatsh. Math. 82 (1976), 1-16. 48. Bemerkungen zum Kontraktionsprinzip. Elemente d. Math. 31 (1976), 90-91. 49. Konstruktive Existenzsatze in der mathematischen Grenzschichttheorie auf Grund der Linienmethode. Z. Angew. Math. Mech. 56 (1976), T36-T44. 50. Parabolic differential equations with a singular nonlinear term. Funkcialaj Ekvacioj 19 (1976), 271-277. 51. The quenching problem for nonlinear parabolic differential equations (together with A. Acker). Lecture Notes in Mathematics, Vol. 564 "Ordinary and Partial Differential Equations", pp. 1-12. Springer Verlag 1976. 52. Existence theorems for strongly coupled systems of partial differential equations over Bernstein classes (together with R. Redheffer). Bull. Amer. Math. Soc. 82 (1976), 899-902.

14 53. Counterexamples for parabolic differential equations (together with R. Redheffer). Math. Z. 153 (1977), 229-236. 54. Das Mazimumprinzip in unbeschrankten Gebieten fur parabolische Differentialungleichungen mit Funktionalen (together with R. Redheffer). Math. Ann. 226 (1977), 155-170. 55. On the global existence of solutions of parabolic differential equations with a singular nonlinear term (together with A. Acker). Nonlinear Analysis TMA 2 (1978), 499-504. 56. Uniqueness, stability and error estimation for parabolic functional-differential equations (to­ gether with R. Redheffer). 70-anniversary issue of Academician I.N. Vekua, Akad. Nauk SSSR, 494-513, Moskau 1978. 57. Invariant sets for systems of partial differential equations. I. Parabolic equations (together with R. Redheffer). Arch. Rat. Mech. Anal. 67 (1977), 41-52. 58. Existence of solutions of a nonlinear boundary value problem via the method of lines (together with K. Schmitt and R.C. Thompson). Nonlinear Analysis 2 (1978), 519-535. 59. Periodic solutions of ordinary differential equations with growth restrictions (together with J. Mawhin). Rapport no. 109, Jan. 1978, Institut de Math, pure et appl., Univ. Catholique de Louvain. Proc. Royal Soc. Edinburgh 82A (1978), 95-106. 60. On pointwise estimates for metaharmonic functions (together with Philip W. Schaefer). J. Math. Appl. 69 (1979), 171-179. 61. Entire solutions of A p u = /(r,u) (together with H. Rhee). Proc. Royal Soc. Edinburgh 82A (1979), 189-192. 62. Inequalities involving derivatives (together with R. Redheffer). Pacific J. Math. 85 (1979), 165-178. 63. Comparison theorems for parabolic functional inequalities (together with R. Redheffer). Pa­ cific J. Math. 85 (1979), 447-470. 64. Invariant sets for systems of partial differential equations. II. First order and elliptic equations (together with R. Redheffer). Arch. Rat. Mech. Anal. 73 (1980), 19-29. 65. Existence theorems for a two-point boundary value problem in Banach space. Math. Ann. 244 (1979), 55-64. 66. Counterexample for uniqueness in strongly coupled parabolic systems (together with R. Red­ heffer). Math. Z. 171 (1980), 83-90. 67. Stability of the null solution of parabolic functional inequalities (together with R. Redheffer). Trans. Amer. Math. Soc. 262 (1980), 285-302. 68. Fixpunkte der Poincare-Abbildung. Arch. Math. 33 (1979), 80-84. 69. Bedeutende Mathematiker. Deutsche Bearbeitung der Tafel "Men of Modern Mathematics'' von Charles und Ray Eames und Ray Redheffer (together with A. Stefan). IBM Deutschland 1978. 70. On generalized Wendroff type inequalities and their applications (together with B.K. Bondge and B.G. Pachpatte). Nonlinear Analysis 4 (1980), 491-495.

15 71. Remarks on a paper by F. Browder about contraction. Nonlinear Analysis 5 (1981), 21-25. 72. On Max Muller's existence-comparison theorem for infinite systems of ordinary differential equations. Ann. Polon. Math. 42 (1983), 395-401. 73. Uber ein Modell zur Pareto-Verteilung. Applicable Analysis 11 (1981), 233-239. 74. Entire solutions of the differential equation Au = /(u). J. Australian Math. Soc. (Series A) 30 (1981), 366-368. 75. Old and new approaches to Euler's trigonometric expansions. Amer. Math. Monthly 89 (1982), 225-230. 76. A comparison theorem for difference inequalities (together with R. RedhefFer). J. Diff. Eqs. 44 (1982), 111-117. 77. Comportement des solutions d'in£quations diffeVentielles deg£ne>ees du second ordre, et ap­ plications aux diffusion (together with G. Lumer and R. RedhefFer). C.R. Acad. Sci. Paris, Serie I, 294 (1982), 617-620. 78. Solution of the stability problem for a class of generalized Volterra prey-predator systems (together with R. Redheffer). J. Diff. Eqs. 52 (1984), 245-263. 79. On parabolic systems of the Volterra predator-prey type (together with R. Redheffer). Non­ linear Analysis 7 (1983), 333-347. 80. Einiuhrung zur Reprintausgabe von Eulers 'Einleitung in die Analysis des Unendlichen' (Introductio in analysin infinitorum). Springer Verlag 1983. 81. An elementary proof of the Cauchy-Kowalevsky theorem. Amer. Math. Monthly 92 (1985), 115-126. 82. Functional-differential equations of the Cauchy-Kowalevsky type. Aequationes Math. 28 (1985), 102-113. 83. On the multiplicity of the solution set of some nonlinear boundary value problems (together with P.J. McKenna). Nonlinear Analysis 8 (1984), 893-907. 84. On the Dirichlet problem for elliptic systems (together with P.J. McKenna). Applicable Analysis 21 (1986), 207-224. 85. A condition for the continuity of additive operators (together with P. Volkmann). Ann. Diff. Eqs 3 (1) (1987), 63-66. 86. On the multiplicity of the solution set of some nonlinear boundary value problems. II (together with P.J. McKenna). Nonlinear Analysis 10 (1986), 805-812. 87. On the strong maximum principle for parabolic differential equations. Proc. Edinbourgh Math. Soc. 29 (1986), 93-96. 88. Multiplicity results for asymptotically homogeneous semilinear boundary value problems (to­ gether with P.J. McKenna and R. Redlinger). Annali di Mat. Pura Appl. (IV) 143 (1986), 247-257. 89. Parabolic differential equations and inequalities with several time variables. Math. Z. 191 (1986), 319-323.

16 90. Remarks on ordinary differential equations in ordered Banach spaces (together with R. Red­ heffer). Monatsh. Math. 102 (1986), 237-249. 91. A theorem of LaSalle-Liapunov type for parabolic systems (together with R. Redheffer and R. Redlinger). SUM J. Math. Analysis 19 (1988), 121-132. 92. Finite difference approximations to the Dirichlet problem for elliptic systems (together with Ch.U. Huy and P.J. McKenna). Numer. Math. 49 (1986), 227-237. 93. Nonlinear oscillations in a suspension bridge (together with P.J. McKenna). Arch. Rat. Mech. Anal. 98 (1987), 163-177. 94. Estimates for solutions of degenerate second-order differential equations and inequalities with applications to diffusion (together with G. Lumer and R. Redheffer). Nonlinear Analysis 10 (1988), 1105-1121. 95. Weakly coupled systems of parabolic differential equations: Asymptotic behavior via inequa­ lity methods (together with R. Redlinger). Univ. Karlsruhe, Fak. f. Math., Bericht Nr. 30 (Mai 1988). 96. Differential Inequalities. Lecture Notes in Pure and Applied Math., vol. 129, "Inequalities: 50 Years on from Hardy, Littlewood and P61ya", ed. by W.N. Everitt. Marcel Dekker 1990, pp. 249-283. 97. A theorem on elliptic differential inequalities with an application to gradient bounds. Math. Z. 200 (1989), 293-299. 98. Travelling waves in a suspension bridge (together with P.J. McKenna). SIAM J. Appl. Math. 50 (1990), 703-715. 99. Convergence of the line method approximation for a parabolic free boundary problem (to­ gether with R.C. Thompson). Diff. and Int. Eqs. 3 (1990), 335-351. 100. A qualitative study on a parabolic free boundary problem (together with R.C. Thompson). Math. Z. 208 (1991), 127-138. 101. An existence theorem for a parabolic free boundary problem (together with R.C. Thompson). Diff. and Int. Eqs. 5 (1992), 43-54. 102. A useful Banach algebra. Elemente der Math. 47 (1992), 27-32. 103. The minimum principle for elliptic systems. Applicable Analysis 47 (1992), 1-6. 104. Minimum principles for weak solutions of second order ordinary differential equations. General Inequalities 6. Birkhauser Verlag, 1992, pp. 369-376. 105. The subgradient in E." (together with R. Redheffer). Nonlinear Analysis 20 (1993), 1345-1348. 106. An inequality of convolution type (together with V. Weckesser). Aequationes mathematicae 46 (1993), 212-219. 107. A general symmetry principle and some applications (together with J. Mawhin). J. Math. Anal. Appl. (accepted).

WSSIAA 3 (1994) pp. 17-32 © World Scientific Publishing Company

17

ON FREE BOUNDARY PROBLEMS FOR QUASI-LINEAR ELLIPTIC PDE'S: UNIQUENESS AND MONOTONE ORDERING OF CONVEX SOLUTIONS ANDREW ACKER Department of Mathematics and Statistics The Wichita State University, Wichita, Kansas 67!60-00SS, U.S.A. ABSTRACT We study free-boundary problems involving generalized capacitary potentials in annular domains. Given A > 0 and the convex exterior boundary component, we seek a convex interior boundary component I \ such that the capacitary potential has a normal derivative of A on T^. In this context, we study the existence of monotonically and continuously-varying families of solution surfaces, parametrized by A. These solution families are the basis for a (non-global) uniqueness theory. 1. Free-Boundary Problem. 1.1. Problem. In R , n > 2, let be given a bounded, simply-connected domain G whose boundary T* = dG is a CT' -surface for some constant O 0 , we use

N(S;e) to denote the e-neighborhood of S relative to G. Given surfaces Tl,T2 6 X, we define A(T1,T2) = A(Dt, D2) (=Hadamard distance) to be the minimum value of e > 0 such that D2 C N(D^,e) and £>, C N(D2;e). Given a positive open interval I, we call r A , A e / , a (parametrized) family of solutions of Problem 1.1 if for each A g 7, the surface Tx g X is a classical solution at A. The solution family Tx, A g 7, is called elliptically ordered if r a < T g for any a, f) g / s u c h that a0 as a—* A relative to I. 2.2. Theorem. Let Tx,\£l,

be an elliptically ordered, continuously varying

family of solutions of Problem 1.1. For fi € I, let T 6 X denote any solution of Prob­ lem 1.1 at n such that Ta0). We call Y locally bounded in (7 3,8 if: (t) For any A>0 and solution TA 6 Y of Problem 1.1 (at A), TA is a C^-surface such that Ux 6 C3,9^

A ).

(ri) For any A>0 and solution TA € Y of Problem 1.1 (at

A), there exist values 60 = tfo(rA)>0 and B0 = B0(rx)

such that if T a 6 Y,

\a — A| < S0, and A(r A , r a ) < S0, then the Cr' -norm of Ua in fi a is bounded by B0. 2.7. Definition. Given a solution-pair (r A , Ux) of Problem 1.1 at A>0, such that TA € X n C3,e and Ux € C3'8^

A ),

we use Vx G (^'"(H A) to denote the solution

of the boundary value problem: 2xV:=V-(A(\VUx\*)VV-r2A'(\VUx\2)VUx(VUx-VVJ)

= 0 in

fiA,

V(T*) = 1, K(r A ) = 0. We remark that Eqs. 4 and 5 state formal properties of the function

(i) (5) V A :=

limits-*>({U\,f ~ ^!x)/^)> where UXj solves the boundary value problem: U(T ) = 1 + S, U{TX) = 0, an( U;x) = Q in £1A. 2.8. Theorem. Assume in Problem 1.1 that G is convex, A (s) > 0, and A(s) lies between positive constants (uniformly over 0 < a0). Assume that the domains G and Dx are both convex, or else that n = 2 and Dx is simply connected. Let Af(A)=max{4(s):0 < s< A 2 }. Then Eq. 6 is satisfied if J(\VUx(x)\)>M(X)K*(x)

on T*.

(7)

2.12. Remarks. Under the assumptions of Theorem 2.11, Eq. 7 is satisfied if \VUx(x)\>QK*{x)

on T*, where Q=sup{{A(s)/A(t)):0<s,tl.

In the case

where A (s) < 0 for 0 < « 0, A(s) lies between positive constants (for 0 < s 0 in N{il;e(a)),

provided that 0 < | Q | < * 1 : = 1 / ^ ) ,

(14)

where £(a) = m i n { e 0 , | a | , / l , " H ' ) } .

Clearly, the definition of Ua implies that \VUa-VU\

< C,|o|; \Ua-U\

< 2\a\

both uniformly in ,W(ft;£0) for \a\<SJ = S2(W;E0),

(15a,b)

where C5 is a constant

depending on EQ, 62, W, and s u p { | W | } . It follows from Eqs. 10b and 15b that U(x) Ua(x) > 0 in iV(n;e 1 )\JV(r;4|a|/A) for \a\ A/4 in N{T;£l); \Ua\ > (A/4)min{e,/2,dist(p,r a )} in Ar(fi a ;e,/2), (18a,b) where Eq. 18 a follows from Eqs. 10 a and 15 a, and where Eq. 18 b follows by integrating |V{/ a | on curves of steepest ascent of Ua ending on r a . Now Eqs. 15 b and 18 b imply that r C JV(r o ;8|a|/A) for | a | < 6 4 . In view of Eq. 17 a, it follows that A ( r , r a ) < 8|a|/A for \a\ < 5 4 , completing the proof of part (d). (We omit the analogous proof of part (e).) Since Ua (with | a | < 5 4 ) is a C ' -function having no critical points in iV(r;e,), it follows from Eq. 16 that fl„ has the same connectivity as fi, and that Ta is a disjoint union of simple, closed CT' -curves, with exactly one lying in (and encircling the interior complement of) the (4|or|/A)-neighborhood of each component of T. At this point, the smooth dependence of Ta on a (part (c)) is obvious.

Also,

if

|a|0. By combining Eqs. 20 and 21, we conclude that I V ^ f c ) ! = A + [\VV(x)\-(A(X2)/B'(X2))K(x)]a

+ 0(a2)

(22)

as a—>0, where the estimate holds uniformly over all x g T. Eqs. 9a,b follow from Eq. 22, and (Ta,Ua)

is a lower (upper) solution for a > 0 (ar 0 such that Eq. 8 is satisfied on Tx, so that a family ( I \ 0 , U\,a), \a\ A such that A(IV, Tx) < (J 0 /3 A). Then IV > Tx. More generally: (/i-A)/2C, < g < 5 < 2(/i-A)/C0,

(23)

where |g|,|a| 0 and a solution IV € X of Problem 1.1 (at K) such that Eq. 8 holds. Let Tx, A 6 J, denote a parametrized family of solutions

25 which is uniformly bounded in CT' and is such that A(r A ,r K )-+0 as A—>/c. Then: (a) There exist values 60,e0 > 0 and B such that for any A € J0:= {A 6 J: IA - K\ < S0}, the solutions Ux (of Eqs. 2,3) and Vx (of Eqs. 4,5) have C3'*continuations to N(ilK;e0) such that the CT' -norms of Ux and Vx relative to M^«;£o)

are

bounded by B.

(b) For all A € J0 such that |A — K\ is sufficiently small, TA satisfies Eq. 8 with C0 replaced by C 0 /2 and Ct replaced by 2CX. (c) There exists a value S0>0 such that Theorem 3.2 holds for all A € J0 with |A — K\ sufficiently small, where £0(r.x) > 6 0 > 0 (uniformly). Proof of part (a). For this proof only, the theorem-numbers refer to Gilbarg and Trudinger11. For each A € J0, the norm of Ux in CT' (fix) is bounded by BQ. This result provides bounds, independent of A G J0, for the coefficients of the operators Q.x in the C ' -norm relative to their respective domains il x. It follows from Theorem 6.6 and Corollary 6.7 that there is a constant Bj > B0 such that for each A 6 J0, the norm of Vx in C ' (H x) is bounded by £,. Furthermore, it follows by a careful reading of the proofs of Theorems 6.17, 6.18, 6.19 that there is a constant B2 > Bt such that the norm of Vx in CT' (H x) is bounded by B2 for all A £ J0. Therefore, there exist values e 0 > 0 and B$ > £ 2 such that for each A € J0, the corresponding solutions Ux (of Eqs. 2,3) and Vx (of Eqs. 4,5) have C ' continuations to N(ilx;2e0)

such that the CT' -norms of Ux and Vx relative to

N(flx;2e0) are bounded by B3. (Regarding these continuations, see Lemma 6.37.) Proof of part (b). Clearly max{\Ux(x)-

UK(x)\:x£ d(Slxn£lK)}

Therefore, by the comparison principle, we have s\ip{\Ux(x)—

-*0 as A-+/c.

UK(x)\:xEflx(~\flK}

—»0 as A—»/c. For |A — «| < i 0 , the functions Ux, UK,Vx,VK can all be continued to N(ilK;e0) in such a way that they are uniformly bounded in the CT' (N(ilK\£o))norm. If follows using the first and second order versions of the Taylor remainder theorem that all first and second order derivatives of Ux converge (uniformly in compact subsets of iV(fi*;e0)) to their counterparts involving UK as A—*K. This proves that \Kx(y) — KK(x)\ is small when |A — K\ and |y— x\ are both small (with x € r„, y 6 Tx). This also shows that the coefficients of the operator Q.x all converge (uniformly in compact subsets of N(ilK;eQ)) to their counterparts involving the operator 2 * as A—»/c. Therefore, ZX(VK— Vx) = (2.x — 2„)V„—vO as A—*it (uniformly in compact subsets of fiK). Using uniform bounds for the coefficients of %x and the uniform ellipticity of these operators, we conclude that Vx—► VK (uniformly in

26 compact subsets of ft„) as A—>/c. Since all the third-order derivatives of the functions Vx are uniformly bounded (in compact subsets of JV(ft„;£0)) as A—»/c, the first and second order derivatives of Vx converge uniformly (in compact subsets of N(ClK;e0)) to their counterparts involving VK as A—»/c. This completes the proof. Proof of part (c). All the estimates in the proof of Theorem 3.2 depend only on e 0 and the Cr' -norms of Ux and Vx in N(ilx;e0).

It follows that (in Theorem

3.2), the values ^ ( F A ) have a uniform positive lower bound for |A — K\ small. 3.5. Proof of Theorem 2.8. By the Heine-Borel Theorem, it suffices to prove the assertion when 7= J(/i):=(/c — h,K + h), for sufficiently small h>0. By Theorems 2.5 and 3.2, there exists a Lipschitz-continuous function (j>(h) with (0) = 0 such that if A € 1(h) and h>0 is sufficiently small, then there exists a solution pair (Tx, Ux) of Problem 1.1 at A such that A ( r A , r „ ) < 4>(h). We will study the local solution families TX,X G 1(h). By assumption, these families are uniformly bounded in (j' . Therefore, by applying Lemma 3.4(b), and decreasing h>0 if necessary, we can assume Eq. 8 holds for all A € 1(h) (with C0>Ci replaced by C 0 /2 and 20-y). Again by decreasing h>0 if necessary, and applying Lemma 3.4(c), we can assume Theorem 3.2 holds for all A 6 1(h), with constants which do not depend on A € 1(h). By decreasing h>0 again if necessary, we can assume that all pairs TX,T^ with A,/J £ 1(h) satisfy the assumptions of Corollary 3.3. Then Corollary 3.3 implies that the family I\,A € 1(h), is elliptically ordered and Lipschitz-continuously varying. 4. Indirect Verification of Elliptic Ordering (Proof of Theorem 2.11). 4.1. Lemma. Let W(x) = f(\VU(x)\2)

in a domain ft, where / g C*(R+) and U

satisfies Eqs. 2,3. Then dW(x)/du= at x e f t , where s=r2,

r=\VU(x)\,

-[2sA(s)/B'(s)]f'(s)K(x)

(25)

i/ = (VJ7(z)/r) and K(x) denotes (n— 1) times

the mean curvature at z of the level surface of U through x, oriented so that K(x) > 0 if {y e ft: U(y) < U(x)} is convex near x. Proof. This follows from Eq. 2 and the definition of mean curvature. 4.2. Lemma. Let Tx be a solution of Problem 1.1 (at A>0). Then VUX ^ 0 in ilx if G and Dx are both convex, or if n = 2 and Dx is connected. Proof. For the convex case, this follows from the convexity of the level surfac­ es of Ux (see Korevaar and Lewis'3). For n = 2 , see Alessandrini , Theorem 2.1.

27 4.J. Lemma. Given a bounded C 2 -domain ft in R n , let (p) = /(|VU(x)\ 2 ) + V(x) in ft, where /(«):R+—»R is a C -function such that f(s)>Q, 2

and where

2

U€ C (ft) and Ve C (ft)n C(H) are functions such that «Dl( U\ x) = fi( I/) 7 = 0 in ft (where X(a):[0,oo)—»R satisfies the assumptions of Problem 1.1) and VU^

0 in

ft. Also assume that n = 2, or else n > 3 and the level surfaces of U are all locally convex (i.e., { U< t} is locally convex relative to ft for 16 R). Then min{ 0. We write Eq. 2 in the form 3Jl( U; x) := £ (*y + & Vt V,) U{J = 0,

(28)

ij=l

where £/, = d U/dxiy Uij = d2 U/dxfiij, 6^ = 1, and 6{j = 0 for i ^ j . At a point a*, G ft, and in coordinates chosen such that U,{XQ) = 0, t = 1,- • •, n — 1, and Un(x^) = r > 0, we have 3Hoi7=0, where we set s = r 2 and define the operator

ang = (i + sh(8))(d2/dxi) + " E V / 0 * 2 ) -

(29)

Straight-forward calculations (based on Eq. 28) show that HJlo^ + 2rhnt*U?,»

+ 2r(h+sti)Ul*

%, V + 2 r h %E Ui 3,

it follows from the assumed convexity of the level surfaces of U that the point XQ maximizes the sealer product x'VU(x0)

relative to the surface {U(x)= U(xo)}. It

follows that U?j < Uij Ujj for any », j= 1,- • -,n — 1. Therefore,

"f:1 l$i < 'Z Uiti Uu < ( "E1 UJ = Ul, (l + sh)2 i.t=:l

i.t=1

* •.♦=!

'

(33)

28 For n = 2, Eq. 33 reduces to the inequality: U?j < U2,2(l + sh) , which follows directly from the equation: URQ U= 0. Therefore, we do not require the convexity assumption when n = 2 . It follows from Eqs. 32 and 33 that SNoW < (il + sh){(4 + 2sh)f + 4sf'} -

Asih+sh')/)^^

+ Usf'+(4-2sh)f')nt,U?n We define Z(x) = (l-exp(-a\x-x*\

(34)

)), where the point x* is fixed in the

complement of ft and a > 0 is to be determined. Direct calculations show that that for any constant B, one can choose a so large that %,Z+Bt\Zi\0), where we assume G and Dx are both convex, or else n = 2 and Dx is connected. Assume that g (X ) < / 0 ( | V ^ ( * ) | 2 ) on r , wherefta)= (l/sA(,)) 2

and 90(s)=f0(s) 2

- (2/A(s)), both for all

s > 0. Then TA satisfies Eq. 6 (i.e., A{\ ) Kx{x) < B'(X ) \WX(X)\

on Tx).

29 Proof. We have V I / ^ O in flA, by Lemma 4.2. For small e > 0 , let t(x) = C«/0(|VJ7A(x)|a)+VA(a;) in H A , where C, = (l + e)A(A 2 )/2. Observe that 0 t (x) = Ce/0(AJ) on r v For sufficiently small £ > 0 , we have t(z) > Ctf0(X2) on T*, due to the assumption. Therefore e(x) > Ctf0{)?)

throughout H A, by Lemma 4.3.

Therefore, d<j>e(z)/du > 0 on Tx. However, it follows from Lemma 4.1 that d*,{z)/to=\VVx(z)\

- (l + £ )[A(A J )/B'(A 3 ))]# A (z)

on TA. We conclude from this that (|VK A (*)|/(l + £)) > [>l(AJ)/£'(AJ)]JifA(z) on TA, where e > 0 is sufficiently small. The assertion follows from this, since |V VA(z)| > 0 on TA by the Hopf boundary point lemma. 4.5. Lemma. Given a bounded domain Q in R n , let if>(x) = g(\VU(x)\ in H, where U€C2(il),

fBl(U;x)=Q in fl (where A(s):[0,oo)-+R

)+ U(x)

satisfies the

assumptions in Problem 1.1), VUX ^ 0 in H, and g(s) g C 1 (R + ) is a function such that g'(s) > 0. Also assume that n = 2, or else that n > 3 and the level surfaces of U are all locally convex. If g'(s)=(B'{s)/sAi(s))^(s),

where 2>(a)>0 and 2>'(s) < 0

(both for all s > 0), then min{V>(z): x 6 H } = min{V>(x): * € dfy.

(39)

Proof. The following discussion is in the context of the proof of Lemma 4.3. A direct calculation shows that g(s) is the most general function with strictly positive derivative such that 2s{l + sh{3))g"{a)+{\il

+ sh{s)][2 + 3h(s)]-28[h(s)

for all a > 0 , where h(s) = 2A (s)/A(s).

+ sh'(s)})g'{s)

< 0

(40)

If the assertion is false, then for any

sufficiently small £ > 0 , the function if>t(x):=g(\W(x)\

)+ U(x) + eZ(x) achieves its

absolute minimum at a critical point z, € il. In coordinates chosen such that t/,{xe) = 0, i = l y , n — 1, and Un(z,) = r>0, we have 2rg'(s)Un,»+r+eZn

= 0and2rg>(3)Uiil,+eZi=Q,

i=l,---,n-l,

(41)

at Xt, where we have set s = r s . In view of Eq. 40, Eq. 34 (with f(s) replaced by g(s), and with W(x) = g(\VU(x)\ )), reduces to the inequality: SDlo W < 2(2sg" + {2-sh)g') at Xf. It follows from Eqs. 41 and 42 that

'£* U?%

(42)

30 (l/ejaBo^,

moZ+e(5(,)*£lZ?,


0 is sufficiently large, then 5DloZ + (1/2) %tlZ?

< 0

(44)

in fl. It follows from Eqs. 43 and 44 that for or>0 sufficiently large and e > 0 sufficiently

small (both independent of Xt), we have 3Jl o V' e 0), where we assume that G and Dx are both convex, or else that n = 2 and Dx is connected. Assume that 2CK*{x)

:= \V Ux(x)\ A(\V Ux(x)\2)

< J(\VUx(x)\)

on T , for some constant C> 0. Then for gQ(s) :=fQ{s) — (2/i4(s)), we have Cg0(\VUx(x)\2) Proof. We have VUx(x)j^0 y, e (i) = ( l - f e ) Cg0(\VUx(x)\2)+ d*t(x)/dv on T

l>Cg0(X2)oaT*.

+

in flA by Lemma 4.2. We define the function Ux{x):fi X-+R for small e > 0 . Observe that 2{l+e)CK\x)IA{\VVx(x)\2)

= |WA(x)| -

by Lemma 4.1, where v denotes the exterior normal to T

at x. For

sufficiently small e, we have by assumption that dipe(x)/dv > 0 on T , so that il>e(x) cannot achieve its global minimum on T*. Since g£(s)

:=(B'(s)/sA2(s))>0,

Lemma 4.5 implies that i>t{x) > (1+e) Cg {X ) throughout A A , where t/>e(x) = .(l + e)Cg0(X2)

on Tx. Thus C Cg0(X2) on T* for suffic­

iently small e > 0, from which the assertion follows. 4.7. Proof of Theorem 2.11. Under the assumptions of Theorem 2.11, Lemma 4.6 implies that (l/C)>

g0(\')

- g0(\VUx(x)\2)

(45)

on T*, where C= (1/2) M(X). It follows that (2/A(\VUx(x)\2))

> g0(X2) - g0(\VUx(x)\2)

(46)

on T . Observe that Eq. 46 holds automatically when 0 l and p / 2. Assume n = 2 and Dx is connected, or else that and G and Dx are both convex. Also assume the solution (Tx, Ux) satisfies one of the following two conditions: | W A | > A ( p - l ) 1 / ( 2 - ' ' on T*,

(47)

( p - l ^ - ' I V t f J ' " ^ K*(x) on T*.

(48)

Then Da C Dx, where Ta denotes any classical solution of Problem 1.1 at a < A; in

fact

DaCDxiia(x) := (1 / (2 - p))(|V{7 A |/A) 2 ~' + UX:U X-*R must achieve - ( l / ( p - lJHIVtfjJ/A) 2 " ' # * ( * )

its minimum on dQx. We have d(x) = (l/(2 —p)) on Tx, it follows from either Eq. 47 or Eq. 48 that <j>(x) > ( 1 / ( 2 - p ) ) throughout ftv Therefore, \VUX\ > A(l + ( p - 2 ) ^ ) , / ( 2 - ' '

(49)

throughout fl x. Now suppose, for the purpose of obtaining a contradiction, that Ta intersects ilx,

and let 0 < C— UX(XQ)= max{ Ux(x): i £ r a PIQ x } < 1,where Xg 6 Ta-

We define W(x) = ((Ux{x) - C ) / ( l - C)), observing that ApW(x) = 0 in QX. Now W{T*)=Ua(T*) = \,

and

W{x) = 0 < Ua{x) on

{UX=C}.

Therefore,

W< Ua

throughout { Ux > C}, by the comparison principle. Since W(xo)= (7a(jfl) = 0, we conclude that a = dUa(x0)/du>dW(x0)/du=\VUx(x0)\/(l the direction of VUX(XQ).

— C), where v points in

Therefore, |VIto,)| < a(l -

Ux(xo)).

(50)

However, one can show that ( l - < ) < ( l + ( p - 2 ) t ) 1 / ( 2 ' ' whenever p > l , p jt 2, and 0 < < < 1 . Thus, Eqs. 49 and 50 are inconsistent when a < A, so that r a n n A = 0 in that case. If a < A, then Eqs. 49 and 50 are inconsistent even when Xg £ Tx f~l Ta. 5.2. Corollary. In the context of Theorem 5.1, assume l < p < 2 . Choose A o >0

32 and a C2-domain D0 such that IVt^^Ao on T0 and Eq. 48 holds on T*. If G is convex and A> Ag, then there exists at most one convex solution I \ of Problem 1.1 (at A) such that DA D B 0 . If n = 2 and A> AQ, there exists at most one solution Tx of Problem 1.1 (at A) such that Dx is connected and DA D D 0 . Proof. Tx satisfies Eq. 48 if Dx D Z>„, since then \VUX\ > |VZ70| on T*. References 1.

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A. Acker: Uniqueness and monotonicity of solutions for the interior Bernoulli free boundary problem in the convex, n-dimensional case. Nonlinear Analysis, T.M.A. 13(1989), 1409-1425.

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A. Acker: On the existence of convex solutions of free-boundary problems involving quasi-linear partial differential equations, (manuscript)

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A. Acker: Uniqueness for a free boundary problem involving the p-Laplacian. A.M.S. Abstracts 14(1993), 93T-35-39.

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A. Acker: On the monotonic ordering of free boundaries. A.M.S. 14(1993), 93T-35-58.

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S. Agmon, A. Doughs, L. Nirenberg: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12(1959), 623-727.

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G. Alessandrini: Isoperimetric inequalities for the length of level lines of solutions of quasilinear capacity problems in the plane. J. Appl. Math. Phys. (ZAMP) 40(1989), 920-924.

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H.W. Alt, L.A. Caffarelli, A. Friedman: A free boundary problem for quasilinear elliptic equations. Ann. Sen. norm. sup. Pisa (Ser. 4) 11(1984), 1-44.

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A. Beurling: On free boundary problems for the Laplace equation. Seminars on Analytic Functions (Institute for Advanced Study, Princeton, N.J.) 1(1957), 248-263.

Abstracts

10. L.A. Caffarelli: A Harnack inequality approach to regularity of free boundaries. Part I: Lipschitz free boundaries are C1,