Kawohl, Bernhard

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeBernhard KawohlOn maximum principles and Liouville theorems for quasilinear ellipticequations and systemsCommentationes Mathematicae Universitatis Carolinae, Vol. 24 (1983), No. 4, 647--655Persistent URL: http://dml.cz/dmlcz/106262Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1983Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATiONES MATHEMATICAE UNIVERSITATIS CAROLINAE 24,4 (1983) ON MAXIMUM PRINCIPLES AND LIOUVILLE THEOREMS FOR QUASILINEAR ELLIPTIC EQUATIONS AND SYSTEMS Bernhard KAWOHL Abstract: In the study of elliptic systems of partial dif-fer enlTaTequations it is customary to prove maximum principles for the modulus I u \ of a^jrector valued solution u and Liouville theorems for the vector u. This note contains maximum principles for u and a Liouville theorem for \T$\. Key words: Quasilinear elliptic equation, maximum principle. Classification: 35B50, 35J45, 35J55 The classical maximum principle states the following: If u: £l — > 1R is a classical solution of an elliptic second or-der differential equation, then u attains its maximum on the boundary 8.CL • The strong version of the maximum principle even reads: If u attains its maximum in the interior of -ft , then u is constant. The classical Liouville Theorem says: If u: lKn—> "R is a bounded solution of an elliptic second order differential e-quation, then u is constant. The validity of these theorems is well known for linear and quasilinear differential equations 14,5,6,8,111. For elliptic systems the natural question arises which of the two quantities lul or u serves as an appropriate generali-zation of u. To the author's knowledge, up to now classical ma-ximum principles have been examined for lul [5,141 and Liouville 647 theorems for the rector function tf 16,11,12]. This note contains, among other results: a) a strong maximum principle for \u\f b) a maximum principle for the components of u, and c) a Liouville theorem for I til under weaker assumptions than the ones which guarantee such a theorem for u. Our investigations were motivated by the following examp-les* Example 1 [113. Let x € K n, u1(x) * sin x^, u2(x) « » cos x-,. The vector u(x) » (u (x),u (x)) is a bounded, real analytic solution of the system -Au1 . H Wu\2, ± . lf2$ on K n # max U , IVull x Example 2 [63: Let x c IR11 and u (x) • — i - for i =- 1,... Ixl ...,nj n>3» Then the vector u(x) is a bounded weak solution of the system -Au 1 » -=£—-* IVul2, i m l,...,n, on TRn. 1 + u r Example 3 [6]: Let x € R n and u^x) » x± (1+ Jxl2) 2 , i * l,...,n; n>J. Then the vector u(x) is a real analytic boun-ded solution of a system of type -Au 1 m u1 g(x, 7u), i « l,...,n, on JRnf with lul |g(x,u)i & 17 ul2. We use the following assumptions and notations. Let jQ. c ]Rn, nr2, be a (not necessarily bounded, but open) domain with Lipschitz continuous boundary t)il • We consider a system of uniformly strongly elliptic quasilinear differential equati-ons in XL ~X , (A . l i <x f t t f Dtt) ) i4 )_ - fk(x,u,Du), k - 1,...,H. *;j .-<l 13 *i Xj 648 In ei^rt.hand we 0hall write i t ae (1) -a ir (A(xfufittTViik) - fk(xfufDu). -We .suppose that system (1) has a weak hounded solution, i . e . , a Tector function u « (u 1 , . . . , **) with component© uke H^CCt ) n n L ^ d l ) , 0uoh that the following rtlation holdo for any teat Tootor $c tCjta)}1 . £ A(xfufBtt) Vttk V $ k dx - /& fk(x fu f.0u)$k(x)dk f X -* x f . . . f 0 . for the question of existence and regularity we refer to f l f 2 , 5 »1.U. The L^dD-nora of u i s denoted by M. the coefficients Ai;.(xfufp) are assumed to he symmetric, i%e., Aij(xfttfp) m Aj1(xftt,p) for i ,3 - l f . . . f n and x fu fp c XL t B i |(0) t R1*. and to he hounded, .furthermore, we require the e l l ipt ic!ty con-dition that there exist X9 <u> e E+ 0ueh that * Ifl2 ^ ^ A i ; j ( x f t t f p ) f i fj .£ <A,1£? for x fu fp f £ s SI f B„(0)f ft**, Ra . finally the functions Ai^(xfufp) and r(x,u tp) are assumed to he measurable in x € XI and continuous in u6By(0) and p c R for i,j » l,...,n and k « lf...fI. from time to time we shall make one or more of the follow-ing assumptions: (Al) The right hand side f - (f1,...,^) has "quadratic growth in p"f i.e. there oxiste a posltiTe number a suoh that lf(xfufp)l £ alpl2 fpjr xfufp €Xlf B,j(0)f W^ (A2) There exists a real number X* & X 0uoh that u f(xfu,p) * X* i pi2 t£T xfufp c XX f B||(0)f It1*. 649 -(A3) The coefficients A.ytefU-.p) are Lipschita oontinuoua in their arguments. Hotice that the common assumption ft* < X , which is known to he optimal in a different context, is weakened in (A2). Let us also point out that we do not require any smallness conditi-on e.g., of type a If < A • Maximum principles. Our first result shall be a maximum principle for the modulus of u. Therefore we apply the princi-pal part of system (1) to the function lul and obtain (2) -div(A(xfutDtt)?(!ul2))4 £ (2ukfk(xtutDu) - 2 A v Wc l 1 ) '2(0*-A,) I 7 u r ^ 0 t if (A2) holds. - { 2 sign u Du D(u2)t if N-l and (Al) holds. In both cases lul solves an elliptic differential inequality and therefore the following theorems are a consequence of ma-ximum principles for differential inequalities. Theorem 1. Let u he a hounded weak solution of system (1). Suppose that XL is hounded and that (A2) holds. Then the fol-lowing maximum principle holds; i) sup I u 1 .£ sup I u 11 where sup means the essential suprenorm. Xi ail Furthermore, if tt€ C2(lL)n C°(JI) and if (A3) holds, then the strong maximum principle holds, i.e. ii) lul is constant provided |u| attains its maximum in XL. Theorem 2. Suppose N»lt assumption (A3) holds and u€C 2(XL)nC°(S) is a solution of the differential inequality. ldiv(A(x,utDu) D u U a |Dul2 Then the strong maximum principle holds, i.e. lu| is eons taut - 650 -if lul attains its maximum in H . Proof: Theorem li) follows from Stampacchia's maximum principle [9, p. 393* Theorem lii) and Theorem 2 are a conse-quence of Hopf's strong maximum principle [4» p« 34]* For Theo-rem 2 one has to interpret the right hand side of (2) as a li-near term in D(lu| ) with locally bounded vtotor-coeffioient a sign u Du. Remark: Theorem 1 applies in particular to the examples above. Furthermore, Theorem 1 implies that the homogeneous Di-riohlet problem for system (1) has only the trivial solution ^ 5 0 inCH^2( . .Q)nLC 0(i l» i r . Let us now try to prove maximum principles for the compo-se nents u of u. This will not be possible without suitable as-sumptions on the structure of the right hand sides of system (!)• Since we shall concentrate on an arbitrary but fixed com-ponent u in the sequelf let ke^l,...tH} be fixed. As the si-tuation demands we shall require one of the following assump-tions* (A4) There exists a number a* € R + such that Ifk(xfufp)Ua'lpkl2 forx fu fp€il f BM(0)f R1^. Hotice that the requirement W(A4) holds for every k«l,...fHw is considerably stronger than ( A l ) . This is why Theorem 3 D contains an apparently stronger result. Other suitable assum-ptions are sign-conditions on f : (A5) sign u k sign fk^ 0 for xfufp « JCt f BM(0)f R1^, or (A6) sign u sign r ^ O for xfufp c .& f Bjj(0)f R . 651 Assumption (A6) is easily verified for examples 1 md 2* How we k can formulate the following results concerning the component u -. Theorem 3. i) Let u be a hounded weak solution of system (1) on a hounded doMain SI . Pndtr assuMPtiong (A3) and (A4) the component u ajK V k tains its maximal Modulus on d H » i»t» sup U l « sup lu I • If furthermore u€ C2(XL)nC°(5)» thtn the strong Maximum prin-ciple holds for ukt u* is constant in H f if luki attains its maximum in & . ii) Part i) remains valid if (A4) ifl replaced by (A5). iii) Let ukg C (H)rkC°(5) bt a nonnegative component of a so-lution to system (1) on a bounded domain XL and suppose that (A3) k and the sign-condition (A6) holds. Then tC attains its jiin^Mua k k on dJQ. and the strong minimum principle holds for u * »£. is constant in £ f if u^ attains its Minimum in il . k Proof* Part i) follows from the observation that u is a solution of the differential inequality Idiv(A(x,utDu) Vu ) | £ .-sal Vu | , and from Theorem 2. In case ii) wt have -div(A(xtuf0u)7((uk)2)_-. 0f i.t. lukl attains its maximum on dil. k Por iii) ont has to ust the nonnegativity of u and assumption (A6) to conclude -div(A(xfuf.Ou) Vuk)|2*0f whtnot the dtsirtd rt-sult follows. In view of tht examples abovef assumption (A6) appears to be sensible. If assumption (A6) holds for all components k * « lf...fN wt have the following corollary. Corollary 4t Let ucC2(II)n C ° ( 5 . ) bt a solution of system (1) on a bounded domain H • Suppose (A3) holds ag well as the sign condition (A6) for every k«lt...tK; and that none of tht 652 -oqwppnentaf n ^ M changes sign in .CI. * Then for each k«lf«.» k - M " IV j •--:.«" V IS i M ! attalna itg minimum on the boundary. ^iJ&f^At1^1!-^ irtitklPium principle holdgs tu ! is constant in £L9 if Jul attains its minimum in II . v E^aarkg} The nonnegatiTity of u , which wag required in Theorem 3iii) * can eoaetimee he Terified. As it wag kindly pointed out to the author hy Professor J. Frehse, systems of type (1) occur in stochastic impulse control and there the sign of u i known to be positive. In general, however, it is hard to verify the assumption of Corollary 4 that no component of u changes sign. In spite of this apparent disadvantage, Co-rollary 4 hag an interesting converse: If a nonconstant compo-k nent u of a solution u to system (1) attains its minimum in SI , then u has to change sign in SL # This is indeed the ca-se for the examples above. LiouTille theorems. Such theorems have been derived for solutions to nonlinear and quasilinear elliptic systems £3f6t 7989109119129133 and their importance lies in their close re-lation to the regularity question for solutions to gygtem (1). The following theorem is a special case of a result by M. Meier lllland can be interpreted aa being analogous to Theorem 2. Theorem 5 til]: Let w be a weak bounded solution of the differential inequality (N«l) (3) -diT(A(x,w9Vw) V w ) * a IVwl2 in, R 2. Then w ia constant. As an immediate consequence we obtain a Liouville theorem for the modulus of the solution u of system (1), which paral-653 lels Theorem 1. All one has to do is to look at (2) again. Corollary 6s Let u be a weak hounded solution of system o (1) in R and suppose that (A2) holds. Then |u| is constant. Remarks: This is the oase for example 1. Llouville theorems for the rector function u are stated elsewhere (3t6t8,10,llt12, 13J, e.g. under assumption (Al) in Jtn with n > 2 tl-U. Acknowledgement; It is my pleasure to thank Prof. 0. John for helpful disousslons. R e f e r e n c e s LI] J. PREHSE: On the regularity of solutions to elliptic dif-ferential inequalities, Preprint, Bonn 1981. [21 If. GIAQUTHTA: Multiple integrals in the calculus of varia-tions and nonlinear elliptic systems, Preprint, Bonn, 1981. 131 M. GIAQUINTA, J. NESAS: On the regularity of weak solutions to nonlinear elliptic systems via Llouville #s type property, Comment. Math. Univ. Carolinae 20(1979)§ 111-121. [41 D. GILBARG, U.S. TRUDINGER: Elliptic partial differential equations of second order, Springer, Berlin etc., 1977. 151 8* HILDEBRANDT: Nonlinear elliptic systems and harmonic mappings, Preprint, Bonn 1980. [61 S. HILDEBRANDT, K.O. WIDMAN: Satze voa Llouvilleechen Typ fur quasilineare elliptische Gleichungen und Sys-tems, Nachr. Akad. Wiss. Gottingen, II. Math. Phys. EL. 4U979), 41-59. Ill 0. JOHHJ The interior regularity and the Llouville proper-ty for the quasilinear parabolic systems, Comment. Math. Univ. Carolinae 23(1982), 685-690. (81 B. KAWOHL: On Llouville-theorem, continuity and Holder-654 continuity of weak solutions to some quasilinear elliptic systems, Comment. Math. Univ. Carolinae 21(1980), 679-697. [9] D. ICINDERLEHRER, G. STAMPACCHIAs An introduction to Varia­tional Inequalities and their Applications, Aca­demic Press, New York, etc., 1980. [101 P.L. LIONS, J. NES.AS, I. NETUKA: A Liouville theorem for nonlinear elliptic systems with isotropic nonli­near! ties, Comment. Math. Univ. Carolina* 23 * (1982), 645-655. [Ill M. MEIER: Liouville theorems for nonlinear elliptic equa­tions and systems, Manuscripta Math. 29(1979)# 207-228. [121 M. MEIER: Liouville theorems for nondiagonal elliptic sys­tems in arbitrary dimensions, Math. Z. 176(1981), 123-133. [13] M. MEIER: Liouville theorems, partial regularity and Hol­der continuity of weak solutions to quasilinear elliptic systems. To appear. H4J H. WIEGNER: Das ExL stents und Regular!tatsproblem Dei Sys-temen nichtlinearer elliptisoher Differential-gleichungen. Habilitations-schrift, Bochum 1977. Institut für angewandte Mathematik I, Universität Erlangen-Surnberg, Martenstr. 3, 18520 Erlangen, Bundesrepublik Deutschland (Oblátům 10.10. 1§83) 655 

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On maximum principles and Liouville theorems for quasilinear elliptic equations and systems

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On maximum principles and Liouville theorems for quasilinear elliptic equations and systems

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