A Remark on -harmonic Functions on Riemannian Manifolds
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1 Electronic Journal of ifferential Equations Vol. 1995(1995), No. 07, pp Published June 15, ISSN URL: or ftp (login: ftp) or A Remark on -harmonic Functions on Riemannian Manifolds Nobumitsu Nakauchi Abstract In this note we prove an equality for -harmonic functions on Riemannian manifolds. As a corollary, there is no non-constant -harmonic function on positively (or negatively) curved manifolds. 1 Introduction In [1], [2], Aronsson studied solutions of the boundary value problem for the degenerate elliptic equation i u j u i j u =0 (1) i,j in a bounded subdomain of R n with the boundary condition u = ϕ on. His motivation is to consider the absolutely minimizing Lipschitz extension problem, which means the problem of finding an extension u in W 1, () of any given Lipschitz function ϕ on satisfying the minimization property u L (U) v L (U) for any open set U and for v W 1, (U) such that v u W 1, 0 (U). The equation (1) is the Euler-Lagrange equation of the functional F (u) = u L in the following sense. A p-harmonic function u is a solution of div( u p 2 u) = 0, (2) which is the Euler-Lagrange equation of the functional F p (u) = u L p. Rewrite (2) to read 1 p 2 u 2 u + i u j u i j u = 0. i,j 1991 Mathematics Subject Classifications: 35J70, 35J20, 26B35. ey words and phrases: -harmonic function, -Laplacian. c 1995 Southwest Texas State University and University of North Texas. Submitted: February 20,
2 2 -harmonic Functions EJE 1995/07 Formally passing to the limit as p tends to infinity, the Euler-Lagrange equation (2) of the functional F p converges in some sense to the Euler-Lagrange equation (1) of the functional F. From the point of view by Aronsson, Jensen [6] obtained existence and uniqueness results. (See also Bhattacharya, ibenedetto and Manfredi [4].) He proved 1. any solution of the absolutely minimizing Lipschitz extension problem is a viscosity solution of (1), and 2. there exists a unique viscosity solution of (1). Any bounded such solution is ally Lipschitz continuous. Aronsson s pioneering papers [1], [2] investigated classical solutions. Recently Evans [5] obtained a Harnack inequality for classical solutions. The absolutely minimizing Lipschitz extension problem is considered also on subdomains of Riemannian manifolds M. Then the associated equation corresponding to (1) is g ip g jq i u j u p q u =0, (3) where g ij (resp. g ij ) is the metric of M (resp. the inverse matrix of g ij ), and denotes the Levi-Civita connection of g. (Throughout this note, we use the Einstein summation convention; if the same index appears twice, once as a superscript and once as a subscript, then the index is summed over all possible values.) In this note we are concerned with W 2,2+ε -solutions of (3) ( ε>0). We say that u is a W 2,2+ε -solution of (3) in if the following two conditions hold: 1. u is ally Lipschitz continuous, and 2. u W 2,2+ε (), and u satisfies (3) a.e., where W 2,2+ε () denotes the Sobolev space of functions whose second derivatives belong to L 2+ε (). On this general setting, the curvature of M provides an obstruction on existence of nontrivial W 2,2+ε -solutions of (3). The purpose of this note is to prove the following equality. Theorem 1 Let M be a Riemannian manifold, and let be a domain in M. Let u be a W 2,2+ε -solution of the equation (3) in.then g ip g jq g kr g ls R ijkl p u q u r u s u = 0 a.e. in, (4) where R ijkl is the Riemannian curvature tensor of M. Note that when M = R n, R ijkl 0 ; hence the equality (4) holds automatically in this case. From equality (4), we have u = 0 at any point where the curvature is positive (or negative). So we have:
3 EJE 1995/07 Nobumitsu Nakauchi 3 Corollary 1 Suppose that the sectional curvature of M is positive (or negative) in.thenanyw 2,2+ε -solution of (3) in is a constant function. We mention a related fact on harmonic functions. Let u be a harmonic function on a Riemannian manifold M. Thenu is a constant function if one of the following two conditions holds: 1. M is compact (the maximum principle). 2. M is complete and non-compact, the Ricci curvature of M is nonnegative, and u is bounded on M (Yau [7]). These results need the assumption that u is globally defined on compact or complete manifolds. On the other hand, the above equality (4) holds when an -harmonic function u is defined on a subdomain of M ; the structure of -Laplacian gives a restriction on al existence of solutions. The author thinks that our theorem holds without the assumption that solutions belong to the class W 2,2+ε (), though we use this assumption. Then Aronsson s minimization approach of the Lipschitz extention problem does not seem to work on any positively (or negatively) curved manifold. 2 A Bochner type formula In this section we prove the following formula of Bochner type. Lemma 1 Let u be a C 3 -solution of (3) on a subdomain of a Riemannian manifold M. Then the following equality holds. g ip g jq i u j u p q u u 2 2 (5) +2 g ip g jq g kr g ls R ikjl p u q u r u s u = 0 in, where u 2 = g ij i u j u and u 2 = g ij i u j u. Proof. Note g ij = g ij = 0, since is the Levi-Civita connection. Applying r to both sides of (3), we have g ip g jq i u j u r p q u + 2g ip g jq i u r j u p q u = 0. (6) We see that p q r u = p r q u (7) = r p q u g ls R prqs l u (by the Ricci formula).
4 4 -harmonic Functions EJE 1995/07 We get g ip g jq g kr i u j u p k u q r u (8) = g kr 1 2 k(g ip i u p u) 1 2 r(g jq j u q u) = 1 4 gkr k u 2 r u 2 = 1 4 u 2 2. Then we have g ip g jq i u j u p q u 2 = g ip g jq i u j u p q (g kr r u k u) = 2g ip g jq g kr i u j u p q r u k u +2 g ip g jq g kr i u j u q r u p k u = 2g ip g jq g kr i u j u r p q u k u 2 g ip g jq g kr i u j ug ls R prqs l u k u +2 g ip g jq g kr i u j u q r u p k u (by (7) ) = 4 g ip g jq g kr i u k u r j u p q u 2 g ip g jq g kr g ls R prqs i u j u k u l u +2 g ip g jq g kr i u j u q r u p k u (by (6) ) = 2 g ip g jq g kr g ls R pqrs i u j u k u l u 2 g ip g jq g kr i u j u q r u p k u (by exchange of indices ) = 2 g ip g jq g kr g ls R pqrs i u j u k u l u 1 2 u 2 2 (by (8) ). 3 Proof of Theorem 1 for C 3 -solutions Take any η C 0 (). Then from (5), we have ηg ip g jq i u j u p q u u 2 2 η (9) +2 ηg ip g jq g kr g ls R ikjl p u q u r u s u = 0. Note here g jq j u q u 2 = g jq j u q (g ip i u p u) (10) = 2g ip g jq j u i u q p u = 0.
5 EJE 1995/07 Nobumitsu Nakauchi 5 Using integration by parts, we get ηg ip g jq i u j u p q u 2 (11) = g ip g jq p η i u j u q u 2 ηg ip g jq p i u j u q u 2 ηg ip g jq i u p j u q u 2 = ηg ip g jq i u p j u q u 2 (by (10) ) = η 1 2 gjq j (g ip i u p u) q u 2 = 1 ηg jq j u 2 q u 2 2 = 1 u 2 2 η. 2 From (9) and (11), we have ηg ip g jq g kr g ls R ikjl p u q u r u s u = 0. (12) Since η is an arbitrary test function in C 0 (), we have g ip g jq g kr g ls R ikjl p u q u r u s u = 0 a.e. in. 2 4 Proof of Theorem 1 In this section we complete our proof of Theorem 1 using an approximation. For any W 2,2+ε -solution u of (3), we take an approximating sequence { u (ν) } ν =1 C 3 () such that for any compact set in, 1. ϕ (ν) :=u (ν) uapproaches zero in W 2,2+ε () asνtends to infinity, and 2. the Lipschitz constants of u (ν) (ν = 1, 2,...) are uniformly bounded on : hence u (ν) L and ϕ (ν) () L (ν = 1, 2,...) are () uniformly bounded on. Since u = u (ν) ϕ (ν) satisfies (3), we have g ip g jq i (u (ν) ϕ (ν) ) j (u (ν) ϕ (ν) ) p q (u (ν) ϕ (ν) )=0
6 6 -harmonic Functions EJE 1995/07 i.e., where g ip g jq i u (ν) j u (ν) p q u (ν) + F(ϕ (ν),u (ν) ) = 0 (13) F (ϕ (ν),u (ν) ) = g ip g jq i ϕ (ν) j u (ν) p q u (ν) g ip g jq i u (ν) j ϕ (ν) p q u (ν) g ip g jq i u (ν) j u (ν) p q ϕ (ν) + g ip g jq i ϕ (ν) j ϕ (ν) p q u (ν) +g ip g jq i u (ν) j ϕ (ν) p q ϕ (ν) + g ip g jq i ϕ (ν) j u (ν) p q ϕ (ν) g ip g jq i ϕ (ν) j ϕ (ν) p q ϕ (ν). Let ψ W 1,1 0 (). Multiply by rψ both sides of (13) and use integration by parts, then we have ψg ip g jq i u (ν) j u (ν) r p q u (ν) +2 ψg ip g jq i u (ν) r j u (ν) p q u (ν) F(ϕ (ν),u (ν) ) r ψ = 0. M Let ψ = ηg kr k u (ν) and sum them up with respect to r.thenweget ηg ip g jq g kr i u (ν) j u (ν) r p q u (ν) k u (ν) (14) +2 ηg ip g jq g kr i u (ν) k u (ν) r j u (ν) p q u (ν) F (ϕ (ν),u (ν) )g kr r (η k u (ν) ) = 0 M We see ηg ip g jq i u (ν) j u (ν) p q u (ν) 2 = ηg ip g jq i u (ν) j u (ν) p q (g kr r u k u) = 2 ηg ip g jq g kr i u (ν) j u (ν) p q r u (ν) k u (ν) +2 ηg ip g jq g kr i u (ν) j u (ν) q r u (ν) p k u (ν) = 2 ηg ip g jq g kr i u (ν) j u (ν) r p q u (ν) k u (ν) 2 ηg ip g jq g kr i u (ν) j u (ν) g ls R prqs l u (ν) k u (ν)
7 EJE 1995/07 Nobumitsu Nakauchi 7 +2 = = = ηg ip g jq g kr i u (ν) j u (ν) q r u (ν) p k u (ν) (by (7) ) ηg ip g jq g kr i u (ν) k u (ν) r j u (ν) p q u (ν) F (ϕ (ν),u (ν) )g kr k (η r u (ν) ) ηg ip g jq g kr g ls R prqs i u (ν) j u (ν) k u (ν) l u (ν) ηg ip g jq g kr i u (ν) j u (ν) q r u (ν) p k u (ν) (by (14) ) ηg ip g jq g kr g ls R pqrs i u (ν) j u (ν) k u (ν) l u (ν) ηg ip g jq g kr i u (ν) j u (ν) q r u (ν) p k u (ν) F (ϕ (ν),u (ν) )g kr k (η r u (ν) ) (by exchange of indices) ηg ip g jq g kr g ls R pqrs i u (ν) j u (ν) k u (ν) l u (ν) u (ν) F(ϕ (ν),u (ν) )g kr k (η r u (ν) ) (by (8)). Therefore we obtain an integral form of the Bochner equality for u (ν) : ηg ip g jq i u (ν) j u (ν) p q u (ν) u (ν) 2 2 η M 2 M 2 F (ϕ (ν),u (ν) )g kr k (η r u (ν) ) (15) M +2 ηg ip g jq g kr g ls R ikjl p u (ν) q u (ν) r u (ν) s u (ν) = 0. M for any η C 0 (). Note here g jq j u (ν) q u (ν) 2 = g jq j u (ν) q (g ip i u (ν) p u (ν) ) (16) = 2g ip g jq j u (ν) i u (ν) q p u (ν) = 2 F(ϕ (ν),u (ν) ) (by (13) ). Then using integration by parts, we get ηg ip g jq i u (ν) j u (ν) p q u (ν) 2 (17) = g ip g jq p η i u (ν) j u (ν) q u (ν) 2
8 8 -harmonic Functions EJE 1995/07 ηg ip g jq p i u (ν) j u (ν) q u (ν) 2 ηg ip g jq i u (ν) p j u (ν) q u (ν) 2 = 2 g ip p η i u (ν) F (ϕ (ν),u (ν) )=2 ηg ip p i u (ν) F (ϕ (ν),u (ν) ) ηg ip g jq i u (ν) p j u (ν) q u (ν) 2 (by (16) ) = 2 g ip i u (ν) p ηf(ϕ (ν),u (ν) ) + 2 η u (ν) F(ϕ (ν),u (ν) ) 1 u (ν) 2 η, 2 because g ip i u (ν) p j u (ν) = 1 2 j(g ip i u (ν) p u (ν) )= 1 2 j u (ν) 2. Then by (15) and (17), we obtain 2 ηg ip g jq g kr g ls R ijkl p u (ν) q u (ν) r u (ν) s u (ν) = 2 g ip i u (ν) p ηf(ϕ (ν),u (ν) ) (18) 2 η u (ν) F (ϕ (ν),u (ν) ) +2 g kr r (η k u (ν) ) F (ϕ (ν),u (ν) ). Since u (ν) is bounded uniformly on,weget the right hand side of (18) C F (ϕ (ν),u (ν) ) + C u (ν) F(ϕ (ν),u (ν) ) C ϕ (ν) u (ν) + C ϕ (ν) + C ϕ (ν) 2 u (ν) + C ϕ (ν) ϕ (ν) + C ϕ (ν) 2 ϕ (ν) + C ϕ (ν) u (ν) 2 + C ϕ (ν) u (ν) + C ϕ (ν) 2 u (ν) 2 + C ϕ (ν) ϕ (ν) u (ν)
9 EJE 1995/07 Nobumitsu Nakauchi 9 + C ϕ (ν) 2 ϕ (ν) u (ν) =: I 1 + I 2 + I 3 + I 4 + I 5 + I 6 + I 7 + I 8 + I 9 + I 10. Since ϕ (ν) converges to in W 2,2+ε () as ν tends to infinity, ϕ (ν) and ϕ (ν) approaches zero in L 2 (). Then { } 1/2 { } 1/2 C ϕ (ν) 2 u (ν) 2 I 1 I 2 C { ϕ (ν) 2 } 1/2 { 1 2 } 1/2 I 3 C sup { ϕ (ν) ϕ (ν) 2 } 1/2 { u (ν) 2 } 1/2 I 4 C { ϕ (ν) 2 } 1/2 { ϕ (ν) 2 } 1/2 I 5 C sup { ϕ (ν) ϕ (ν) 2 } 1/2 { ϕ (ν) 2 } 1/2 I 7 C { ϕ (ν) 2 } 1/2 { u (ν) 2 } 1/2 I 9 C sup { ϕ (ν) ϕ (ν) 2 } 1/2 { u (ν) 2 } 1 2 { } 1/2 { } 1/2 I 10 C sup ϕ (ν) 2 ϕ (ν) 2 u (ν) 2 0. Furthermore, since ϕ (ν) converges to zero in W 2,2+ε (), we have { } ε/(2+ε) { } 2/(2+ε) I 6 Csup ϕ (ν) 1 ε ϕ (ν) 2+ε u (ν) 2+ε { } ε/(2+ε) { } 2/(2+ε) I 8 C sup ϕ (ν) 2 ε ϕ (ν) 2+ε u (ν) 2+ε 0. Thus the right hand side of (18) converges to zero as ν tends to infinity. Then, letting ν go to infinity in (18), we have (12). This completes the proof. 2
10 10 -harmonic Functions EJE 1995/07 References [1] Aronsson, G., Extension of function satisfying Lipschitz conditions, Ark. Mat. 6 (1967), [2] Aronsson, G., On the partial differential equation u 2 x u xx +2u x u y u xy + u 2 y u yy = 0, Ark. Mat. 7 (1968), [3] Aronsson, G., On certain singular solutions of the partial differential equation u 2 x u xx + 2u x u y u xy + u 2 y u yy = 0, Manuscripta Math. 47 (1984), [4] Bhattacharya, T., ibenedetto, E. & Manfredi, J., Limits as p of p u p = f and related extremal problems, Rend. Sem. Mat. Univ. Pol. Torino, Fascicolo Speciale 1989 Nonlinear PE s, [5] Evans, L.C., Estimates for smooth absolutely minimizing Lipschitz extensions, Electronic J. iff. Equations 1993 (1993), No.3, 1-9. [6] Jensen, R., Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 123 (1993), [7] Yau, S.-T., Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. XXVIII (1975), Nobumitsu Nakauchi epartment of Mathematics Faculty of Science Yamaguchi University Yamaguchi 753, Japan nakauchi@ccy.yamaguchi-u.ac.jp
11 EJE 1995/07 Nobumitsu Nakauchi 11 ecember 11, 1996 Addendum In this article, Theorem 1 follows from general properties of the Riemannian curvature tensor, and Corollary 1 is incorrect. The Bochner formula does not seem to work in this situation. Lemma 1 can be used in proving the following Liouville theorem for C 3 - solutions. Theorem A. Let M be a complete noncompact Riemannian manifold of nonnegative (sectional) curvature. Let u be a bounded -harmonic function of C 3 -class on M. Thenuis a constant function. The curvature assumption in Theorem A is necessary only for applying the Hessian comparison theorem in the proof (Here we use the operator Q ij = g ip g jq p q ). Theorem A also follows from arguments in [Cheng, S.Y., Liouville theorem for harmonic maps, Proc. Symp. Pure Math. 36(1980), ]. See also the article [Hong, N.C., Liouville theorems for exponentially harmonic functions on Riemannian manifolds, Manuscripta Math. 77(1992), 41-46]. Sincerely yours, Nobumitsu Nakauchi
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