Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2015, Article ID 410896, 7 pages
http://dx.doi.org/10.1155/2015/410896
Research Article

Some Inequalities for the Omori-Yau Maximum Principle

Korea Institute for Advanced Study, Hoegiro 85, Seoul 130-722, Republic of Korea

Received 22 January 2015; Revised 25 June 2015; Accepted 2 July 2015

Academic Editor: Leszek Gasinski

Copyright © 2015 Kyusik Hong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We generalize A. Borbély’s condition for the conclusion of the Omori-Yau maximum principle for the Laplace operator on a complete Riemannian manifold to a second-order linear semielliptic operator with bounded coefficients and no zeroth order term. Also, we consider a new sufficient condition for the existence of a tamed exhaustion function. From these results, we may remark that the existence of a tamed exhaustion function is more general than the hypotheses in the version of the Omori-Yau maximum principle that was given by A. Ratto, M. Rigoli, and A. G. Setti.

1. Introduction

Let be a smooth complete Riemannian manifold of dimension . For a smooth real-valued function on , a second-order linear differential operator without zeroth-order term can be written aswhere is self-adjoint with respect to , is the Hessian of in the form defined by for , and finally . In this paper, we will deal with the semielliptic case, that is, is positive semidefinite at each point, and we always assume that

Definition 1. A smooth complete Riemannian manifold is said to satisfy the Omori-Yau maximum principle for the Laplace operator (the above semielliptic operator ) if for any function which is bounded from above and for any there is a point such that , , and ().

The Omori-Yau maximum principle is a useful substitute of the usual maximum principle in noncompact settings. For the operator , Definition 1 is the well-known Omori-Yau maximum principle for the Laplacian, which was first proven by Omori [1] and Yau [2] when the Ricci curvature is bounded below. This was improved upon by Chen and Xin [3] and Ratto et al. [4] when the Ricci curvature decays were slower than a certain decreasing function tending to minus infinity. For instance, we have the following.

Theorem 2 (Ratto-Rigoli-Setti’s condition [4, Theorem  2.3]). Let be a fixed point and be the distance function from . Let one assumes that away from the cut locus of one haswhere is some constant and on satisfiesThen satisfies the Omori-Yau maximum principle for the Laplacian .

Borbély [5, Theorem] has given an elegant proof of the validity of the Omori-Yau maximum principle where the Ricci curvature condition (3) is replaced by the assumption without (4) and (5). Also, Bessa et al. [6, Theorem ] proved Borbély’s theorem [5, Theorem] for the -Laplacian for a selected smooth function on . In this paper, we first show that Borbély’s theorem [5, Theorem] is also true for our semielliptic operator by following his method in [5] (see Theorem 5).

To state other results, we need the following definitions.

Definition 3. Let be a real-valued continuous function on and let a point . (i)A function is called proper, if the set is compact for every real number .(ii)A function defined on a neighborhood of is called an upper-supporting function for at , if the conditions and hold in .

Definition 4. A proper continuous function is called a -tamed exhaustion, if the following condition holds:(1).(2)At all points it has a smooth, upper-supporting function at defined on an open neighborhood such that and .

Royden [7] showed that every complete Riemannian manifold satisfying Omori-Yau’s condition (i.e., the Ricci curvature is bounded from below) admits a -tamed exhaustion function. Inspired by Royden’s article [7], Kim and Lee [8, Theorem ] proved the Omori-Yau maximum principle for the Laplacian when there exists a -tamed exhaustion function. Moreover, they proved that every complete Riemannian manifold satisfying Ratto-Rigoli-Setti’s condition admits a -tamed exhaustion function [8]. Similar to Definition 4, we define an -tamed exhaustion function (i.e., we replace with ) [9, Definition ]. Then, using the existence of an -tamed exhaustion function, Hong and Sung [9, Theorem ] generalized the Omori-Yau maximum principle for the Laplacian to the operator . In this paper, we give a new sufficient condition for the existence of an -tamed exhaustion function (see Theorem 6). We prove this result using the ideas adapted from [8]. Note that Theorem 6, together with [9, Theorem ], implies the maximum principle of Omori and Yau for the operator . As a corollary, we prove that the existence of a -tamed exhaustion is more general than Ratto-Rigoli-Setti’s condition. Unfortunately, for the operator , the relation between Borbély’s condition (or the existence of an -tamed exhaustion) and Ratto-Rigoli-Setti’s condition remains for further study.

Now, we formulate our main results. From (1), is diagonalizable at each point on an orthonormal basis, since is symmetric. Then one can take a normal coordinate around such that at is represented as a diagonal matrix. Thus, we havefor a real-valued function on , where each is nonnegative; the entries and are bounded above as varies by (2). We introduce a locally defined differential operator for convenience as follows:Put and for . We may assume that and are the largest of and , respectively.

Then we have the following.

Theorem 5. Let be a fixed point and be the distance function from . Assume that for all where is smooth, , and on satisfies Then satisfies the Omori-Yau maximum principle for the operator .

Theorem 6. Let be a fixed point and be the distance function from . Assume that for all where is smooth, , and on satisfiesThen admits an -tamed exhaustion function.

Remark 7. By [5, Corollary] and Theorem 6, Ratto-Rigoli-Setti’s condition without implies the existence of a -tamed exhaustion function. Therefore, the existence of a -tamed exhaustion function for the conclusion of the Omori-Yau maximum principle for the Laplacian is more general than the hypothesis in Theorem 2.

There are some other sufficient conditions under which the Omori-Yau maximum principle for the Laplacian holds [1012]. Also, [13] deals with the general setting of semielliptic operators (trace type operators). Recently, Bessa and Pessoa [14, Theorem ] present a sufficient condition for the conclusion of the Omori-Yau maximum principle for a second-order linear semielliptic operator with bounded first-order coefficients and no zeroth-order term. However, they will not consider the existence of a tamed exhaustion function as sufficient conditions for the conclusion of the Omori-Yau maximum principle.

2. Proof of Theorem 5

The proof is similar to the method in [5]. Let . We may assume that at every point of ; otherwise, has its maximum at some point and that point directly satisfies the Omori-Yau maximum principle for a semielliptic operator .

Define the function asThenSince on , we have , and . Hence the function is strictly increasing, and . Since the set is compact, we have For any positive constant , we define the function as ThenBecause, for all , and . If , then we haveDefine as Then, clearly, . Furthermore, we can obtain for all ; that is, there is a point such that . Assume that to the contrary for all . Then we will show that there is a constant with such that for all . This is a contradiction to the definition of .

Let . Because , there is a sufficiently large positive number such that for . Also, because the set is compact, the statement for all implies that there is a constant with such that for . Now, let . Then, for , we have for all . Moreover, by (17) and , we have .

Next, we have to show that is smooth at . Since , it is enough to show that is smooth at . To avoid confusion, the point , in the statement of Theorem 5, is switched to . Note that is a Lipschitz function and is smooth on , where is the cut locus of . Suppose that . Then we have two possibilities (Petersen [15, Lemma ]); either there are two distinct minimizing geodesic segments joining to , or there is a geodesic segment from to along which is conjugate to . Notice thatWe consider the first case. Let and . Since and are distinct segments, we have . For or , the functions are differentiable on and they have a left-derivative at . Note that is smooth on . From the definition of , , and we obtainwhere denotes the directional derivative of at the point in the direction of . Furthermore, since has a directional derivative at in the direction of , we haveThis yieldsHence, by (21) and (23), we get the following inequality:Note that and . Recall that . Then, from (24), we can getThe inequality (25) will lead to a contradiction. Since and are different segments, by connecting from the point to the point with a geodesic segment, there is a constant with such that, for a sufficiently small , the distance . Thus there is a constant with depending only on the angle of and such thatfor a sufficiently small . Note that . By plugging (26) to (25), we have a contradiction.

From now, let us consider the second case. Since is distance minimizing between and , is smooth at for . Let . Then is also smooth for . Because is conjugate to along , by a simple calculation, we getBecause , by (23), we get ; that is, . Hence the level surface is a smooth hypersurface near . Denote by the surface parallel to and passing through the point for some . Since is smooth near , the surface is also smooth near for a sufficiently small . Therefore, by (27), for some sufficiently small , the trace of the second fundamental form of at in the direction of is greater than , where is the trace of the second fundamental form of the geodesic sphere at with respect to the normal vector . This implies that there has to be a point sufficiently close to , which lies inside ; that is,Since is parallel to , we also have a point on such that the distance . By (28), we have Since is strictly increasing, we get This is a contradiction to the fact that for all . Therefore, the function must be smooth at .

By the definition of , , , and , we haveBecause , , and , we haveHenceRecall notations (6) and (7). Sincewe haveNote that . By (31), (33), and , the first equality of (35) yieldsAlso, by (2), (31), (33), (36), , and , the second inequality of (35) yieldsIf we replace with , then the above inequality, (32), and (36) show that the point satisfies the conditions in Definition 1.

3. Proof of Theorem 6

The proof is similar to the method in [8]. Let be a fixed point and be the distance function from . Define a function by Assume that a smooth complete Riemannian manifold satisfies assumption (10). Then we will prove that is an -tamed exhaustion function. We consider two cases.

First Case. Assume that has no cut points in .

By the definition, the function is an exhaustion function for . We have to show that, for certain positive constants and , and outside a ball of a certain radius with center . Let and . Then . By a direct calculation, one getsBy (12), there is a positive constant such thatThen, for , we obtainMoreover, by (11), we haveBy plugging (41) to (39), we haveNote that . Applying (42) givesBy (2) and (44), one getsBy assumption (11), we haveBecause of the above inequality, , (41), and (42), we have for By our assumption (10), there exits such thatThus, by (45) and (48), we have If we replace with , then satisfies the additional conditions for an -tamed exhaustion function.

Second Case. Assume that the cut locus of is nonempty.

Let be a cut point of and let for . We choose a point outside of cut locus of such that and . Denote by . Take such that and does not have cut point of .

Now, we present several functions to find an upper-supporting function for .

For a neighborhood , we define a smooth map with , and it is translation sending to in a coordinate chart including both and and satisfying . Also, we define a function such that , , , and Since and , we get . Finally, for , we define a functionwhere and for . Note that we choose as close to such that . Therefore, .

Let . Then one gets . Because of the fact and the inequality (41), we getMoreover, we have two inequalities; that is, for ,Hence is an upper-supporting function for at the point .

Since , , , and , we have By our assumption (2), the above inequality implies thatNotice thatwhere . By a simple calculation, we have and henceUsing , , (52), (56), and (58), we haveLet be the distance to a closest cut point of . Because the point is a cut point of , by (41) and (42), we getBy plugging (62) to (60), our assumption (10) tells us that, for ,Therefore, by (55) and (63), we obtain, for , So satisfies the conditions for an -tamed exhaustion function.

Altogether, we can conclude that must be an -tamed exhaustion function for .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the referee for valuable comments and corrections. Also, the author thanks Professor G. P. Bessa for pointing out [6, 14].

References

  1. H. Omori, “Isometric immersions of Riemannian manifolds,” Journal of the Mathematical Society of Japan, vol. 19, pp. 205–214, 1967. View at Publisher · View at Google Scholar · View at MathSciNet
  2. S. T. Yau, “Harmonic functions on complete Riemannian manifolds,” Communications on Pure and Applied Mathematics, vol. 28, pp. 201–228, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. Q. Chen and Y. L. Xin, “A generalized maximum principle and its applications in geometry,” American Journal of Mathematics, vol. 114, no. 2, pp. 355–366, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  4. A. Ratto, M. Rigoli, and A. G. Setti, “On the Omori-Yau maximum principle and its applications to differential equations and geometry,” Journal of Functional Analysis, vol. 134, no. 2, pp. 486–510, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. A. Borbély, “A remark on Omori-Yau maximum principle,” Kuwait Journal of Science, vol. 39, no. 2, pp. 45–56, 2012. View at Google Scholar
  6. G. P. Bessa, S. Pigola, and A. G. Setti, “Spectral and stochastic properties of the f-laplacian, solutions of PDEs at infinity and geometric applications,” Revista Matemática Iberoamericana, vol. 29, no. 2, pp. 579–610, 2013. View at Google Scholar
  7. H. L. Royden, “The Ahlfors-Schwarz lemma in several complex variables,” Commentarii Mathematici Helvetici, vol. 55, no. 4, pp. 547–558, 1980. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. K.-T. Kim and H. Lee, “On the Omori-Yau almost maximum principle,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 332–340, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. K. Hong and C. Sung, “An Omori-Yau maximum principle for semi-elliptic operators and Liouville-type theorems,” Differential Geometry and its Applications, vol. 31, no. 4, pp. 533–539, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. S. Pigola, M. Rigoli, and A. G. Setti, “A remark on the maximum principle and stochastic completeness,” Proceedings of the American Mathematical Society, vol. 131, no. 4, pp. 1283–1288, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. S. Pigola, M. Rigoli, and A. G. Setti, Maximum Principles on Riemannian Manifolds and Applications, vol. 882 of Memoirs of the American Mathematical Society, American Mathematical Society, 2005.
  12. K. Takegoshi, “A volume estimate for strong subharmonicity and maximum principle on complete Riemannian manifolds,” Nagoya Mathematical Journal, vol. 151, pp. 25–36, 1998. View at Google Scholar · View at MathSciNet · View at Scopus
  13. G. Albanese, L. J. Alías, and M. Rigoli, “A general form of the weak maximum principle and some applications,” Revista Matemática Iberoamericana, vol. 29, no. 4, pp. 1437–1476, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. G. P. Bessa and L. F. Pessoa, “Maximum principle for semi-elliptic trace operators and geometric applications,” Bulletin of the Brazilian Mathematical Society, vol. 45, no. 2, pp. 243–265, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. P. Petersen, Riemannian Geometry, vol. 171 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1998. View at Publisher · View at Google Scholar · View at MathSciNet