The obstacle problem for conformal metrics on compact Riemannian manifolds

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1 Bao Xing Journal of Inequalities Applications :238 R E S E A R C H Open Access The obstacle problem for conformal metrics on compact Riemannian manifolds Sijia Bao 1* YumingXing 1 * Correspondence: Baosj11@163.com 1 Department of Mathematics, Harbin Institute of Technology, Harbin, China Abstract We prove a priori estimates up to their second order derivatives for solutions to the obstacle problem of curvature equations on Riemannian manifolds M n, g arising from conformal deformation. With the a priori estimates the existence of a C 1,1 solution to the obstacle problem with Dirichlet boundary value is obtained by approximation. Keywords: Obstacle problem; A priori estimates; Hessian equations; Viscosity solutions; Riemannian manifolds 1 Introduction Let M n, g be a compact Riemannian manifold of dimension n 3 with smooth boundary M, M := M M. In conformal geometry, it is interesting to find a complete metric g [g], the conformalclass of g, with which the manifold has prescribed curvature. In general, such conformal deformation can be interpreted by certain partial differential equations. See [8, 13, 22, 25, 26] for more details. In [8], Guan studied the existence of a complete conformal metric g of negative Ricci curvature on M satisfying f λ g 1 Ric g = ψ in M, 1.1 where Ric g is the Ricci tensor of g, λ g 1 Ric g =λ 1,...,λ n are the eigenvalues of g 1 Ric g. The transformation formula for the Ricci tensor under conformal deformation g = e 2u g is given by 1 n 2 Ric g = 1 u n 2 Ric g 2 u n 2 + u 2 g + du du, where u, 2 u, u denote the gradient, Hessian, Laplacian of u with respect to the metric g, respectively. When f is homogenous of degree one, it is easy to verify that equation 1.1 is equivalent to the following form: [ f λ g 1 2 u + u n 2 g + u 2 g du du Ric ] g = ψx n 2 n 2 e2u. 1.2 The Authors This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, reproduction in any medium, provided you give appropriate credit to the original authors the source, provide a link to the Creative Commons license, indicate if changes were made.

2 Bao XingJournal of Inequalities Applications :238 Page 2 of 12 In this paper, we study the obstacle problem of equation 1.2. More generally, let T[u]:= 2 u + sdu du + γ u t2 u 2 g + χ, where χ is a smooth 0, 2 tensor, γ > 0 is a constant, s, t R. We consider the following equation: max { u h, f λ g 1 T[u] ψ[u] } =0 inm 1.3 with the Dirichlet boundary condition u = ϕ on M, 1.4 where h C 3 M, ϕ C 4 M, h > ϕ on M, ψ[u] =ψx, u is a positive function in C 3 M R. Equationsas are the Hessian equations, which were well studied by many authors such as [2, 7, 9 12, 23, 24]. Generally, f C 2 Ɣ C 0 Ɣ is a symmetric function of λ R n, defined in an open, convex, symmetric cone Ɣ R n, with vertex at the origin, which contains the positive cone: Ɣ + n := {λ Rn : each component λ i >0} satisfies the following fundamental structure conditions: f i f λ i >0 inɣ,1 i n, 1.5 f is a concave function, 1.6 f >0 inɣ, f =0 on Ɣ. 1.7 Here, for convenience, we also assume that f is homogeneous of degree one. 1.8 We observe that by the concavity homogeneity of f, fi λ=f λ+ f i λ1 λ i f 1,...,1>0 inɣ. 1.9 Importantclasses of f are the elementary symmetric functions their quotients, i.e., f λ=σ k 1 k λ:= f λ= σk σ l 1 i 1 < <i k n 1 k l, 0 l < k n. λ i1 λ ik 1 k, 1 k n,

3 Bao XingJournal of Inequalities Applications :238 Page 3 of 12 Let F be defined by Fr=f λr for r = {r ij } S n n with λr Ɣ,whereS n n is the set of n n symmetric matrices. It is shown in [2] that1.5 impliesf is an elliptic operator 1.6ensuresthatF is concave. Afunctionu C 2 M is called admissible at x M if λg 1 T[u]x Ɣ, wecallit admissible in M when it is admissible at each x in M.Inthispaper,weprovetheexistence of an admissible viscosity solution of inC 1,1 Msee[1, 3] for the definition of viscosity solutions. Many authorshave studied various obstacle problems. In [6], Gerhardt considered a hypersurface bounded from below by an obstacle with prescribed mean curvature in R n.lee [17] considered the obstacle problem for the Monge Ampère equation i.e., f =σ n n 1 for the case that T[u]=D 2 u, ψ 1, ϕ 0, proved the C 1,1 regularity of the viscosity solution in a strictly convex domain in R n. Xiong Bao [27]extendedtheworkofLeeto a nonconvex domain in R n with general ψ ϕ under additional assumptions. Bao, Dong, Jiao treated a class of obstacle problems in [1] assuming that T[u]= 2 u + Ax, u, u, under a certain technical assumption. Because of the term γ u γ > 0, here we only need a minimal amount of assumptions. For other works, see [4, 14, 15, 18 21]. Our main result is the following theorem. Theorem 1.1 Assume that either the following condition or lim ψx, z=+, x M, 1.10 z + 2s nt 1+nγ <2λ hold, where λ 1 isthefirsteigenvalueoftheproblem u + λtr χ + u =0 u =0 on M, on M 1.12 λ 1 =+ if tr χ 0. Then there exists a viscosity solution u C 1,1 M to , if there exists a subsolution u C 0 M C 1 M δ for some δ >0such that f λg 1 T[u] ψ[u], in M, u = ϕ, on M, u h, in M, 1.13 where M δ = {x M : distx, M δ}. Moreover, we have that u C 3,α E for any α 0, 1, f λg 1 T[u] = ψ[u] in E, where E := {x M : ux<hx}. Remark , as well as 1.11, is used in Lemma 3.2 to derive an upper bound for u. Assumption 1.13 is just applied to derive a lower bound for u on M ν u on M, where ν is the interior unit normal to M.

4 Bao XingJournal of Inequalities Applications :238 Page 4 of 12 Remark 1.3 We can construct some subsolutions of 1.2 satisfying1.13 asin[15] following ideas from [2][7]since u 2 g du du is positive definite that we can obtain a priori upper bound of any admissible function Lemma 3.2 under additional conditions that there exists a sufficiently large number R > 0 such that at each point x M, κ 1,...,κ n 1, R Ɣ, 1.14 where κ 1,...,κ n 1 are the principal curvatures of M with respect to the interior normal, that for every C > 0 every compact set K in Ɣ there is a number R = RC, Ksuch that f Rλ C for all λ K We use a penalization technique to prove the existence of viscosity solutions to We shall consider the following singular perturbation problem: f λg 1 T[u] = ψ[u]+β ε u h inm, 1.16 u = ϕ on M, where the penalty function β ε C 2 Rsatisfies β ε, β ε, β ε 0 onr, β εz = 0, whenever z 0; β ε z as ε 0 +,wheneverz >0. An example given in [27]is 0, z 0, β ε z= z 3 /ε, z >0, for ε 0, 1. Observe that u is also a subsolution to Let U = { u ε u ε C 4 M is an admissible solution of 1.16withu ε u on M }. We aim to derive the uniform bound u ε C 2 M C 1.19 for u ε U, wherec is independent of ε. Afterestablishing1.19, the equation 1.16 becomes uniformly elliptic by 1.7. By Evans Krylov [5], [16] theorem, we can derive the C 2,α estimates which may depend on ε ofu ε. Higher estimates can be derived by

5 Bao XingJournal of Inequalities Applications :238 Page 5 of 12 Schauder theory. Following the proof as in [8]or[1], we can prove there exists an admissible solution u ε to Then we can conclude by 1.19 that there exists a viscosity solution u C 1,1 Mto1.31.4, see [1, 27]. Thus, our main work is focused on the a priori estimates for admissible solutions up to their second order derivatives. In Sect. 2, we achieve the estimates for second order derivatives. Finally, we end this paper with gradient C 0 estimates in Sect Estimates for second order derivatives In this section, we prove a priori estimates of second order derivatives for admissible solutions. Fromnow on, we drop the subscript ε when there is no possible confusion. Theorem 2.1 Assume that f satisfies u C 4 M is an admissible solution to Then sup 2 u C 1+sup 2 u, 2.1 M M where C depends on u C 1 M other known data. Proof Set Wx= max ξ T x M, ξ =1 ξξ u + s ξ u 2 e φ, x M, where φ isafunctiontobedetermined.assumethatw is achieved at an interior point x 0 M a unit direction ξ T x0 M. Choose a smooth orthonormal local frame e 1,...,e n about x 0 such that ξ = e 1, i e j x 0 =0thatT ij x 0 is diagonal. We write G = 11 u + s 1 u 2. Assume Gx 0 >0otherwisewearedone. At the point x 0,wherethefunctionlog G + φ defined near x 0 attains its maximum, we have i G G + iφ =0, i =1,...,n, 2.2 ii G G i G 2 + iiφ G By 2.3wehave F ii ii G + G ii φ G i φ G + G φ G φ Since γ >0,weobtain F ii ii G + γ G + G ii φ + γ G φ G i φ 2 γ G φ

6 Bao XingJournal of Inequalities Applications :238 Page 6 of 12 By calculation, we get i G = i11 u +2s 1 u i1 u, 2.7 ii G = ii11 u +2s i1 u u ii1 u. 2.8 Recall the formula for interchanging order of covariant derivatives ijk v kij v = R l kij lv, 2.9 ijkl v klij v = R m ljk imv + i R m ljk mv + R m lik jmv + R m jik lmv + R m jil kmv + k R m jil mv It follows from 2.10 ii G 11ii u +2s i1 u u 1ii u C1 + G, 2.11 ii G + γ G 11ii u +2s i1 u u 1ii u + γ 11 u +2sγ i1 u u 1 u C1 + G Differentiating equation 1.16onceatx 0, we obtain for 1 k n, k F = F ii k T ii = ψ xk + ψ z k u + k β ε u h It is easy to see that F ii 1 ii u + γ u=f ii 1 T ii [u] s i u 2 + t 2 u 2 χ ii 1 F 2sF ii i u 1i u + t k u 1k u i F ii i F ii 2.14 that F ii 11 ii u + γ u=f ii 11 T ii [u] s i u 2 + t 2 u 2 χ ii F ii 11 T ii [u] 2sF ii i u 11i u + 1i u 2 + t k u 11k u + 1k u 2 F ii C k i i F ii. 2.15

7 Bao XingJournal of Inequalities Applications :238 Page 7 of 12 With 2.9wesee 2s i u 11i u 2s i u i G 2s 1 u i1 u+c 4s 2 i u 1 u i1 u + C 1+G φ, 2.16 similarly t k u 11k u 2st k u 1 u k1 u C 1+G φ With 2.12, , the concavity of F,wederive F ii ii G + γ G 11 F +2s 1 u 1 F +2sγ + t k 1k u 2 F ii C G + G φ + j,k jk u F +2s 1 u 1 F C G 2 + G φ. By 1.9β ε > 0 it follows from that Let F ii ii φ i φ 2 + γ φ φ 2 F ii C G + φ F ii + φ := ηw= 1 w 1/2, w = u 2, 2a 2 where a > sup M w is a constant to be determined. We have C G 1 β ε u h η < 2, η = η3 4a, η = 3η 2 η ii φ i φ 2 = η ii w + η η 2 i w 2 η ii w Next, by 2.14 F ii ii w + γ w = F ii l il u 2 + γ k,l kl u 2 + F ii l u iil u + γ l u F ii l u lii u + γ k lkk u + γ G 2 CG F ii Cβ ε u h+ γ G 2 CG F ii. 2.21

8 Bao XingJournal of Inequalities Applications :238 Page 8 of 12 Combining 2.19, 2.20, 2.21, φ Cη G,wehave η γ G 2 CG F ii C G + η G C F ii + G 1+Cη β ε u h We could assume that G 2C.Whena >2C, the coefficient of β ε u h is negative. Then we can derive G 4aC γ. To derive the boundary estimates for 2 u,wenotethattrsdu du t 2 u 2 g + χ C on M, wherec is independent of ε, thoughitmaydependon u C 1 M. Asin[1, 4], let H be the solution to 1 + nγ H + C =0 inm, H = ϕ on M. Then we have u H in M by the maximum principle β ε u h 0inM δ = {x M : distx, M δ},whereδ is sufficiently small. Thus, f λg 1 T[u] = ψ[u] inm δ, 2.23 u = ϕ on M. By the same arguments of Sect. 4 in [8], we obtain that sup 2 u C, 2.24 M where C depends on u C 1 M other known data. Combining , we therefore get the full estimates for second order derivatives. 3 Gradient estimates, maximum principle, existence For the gradient estimates, we have the following theorem. Theorem 3.1 Assume that hold. Let u C 3 M be an admissible solution to Then sup u C M 1+sup M where C depends on u C 0 M other known data. u, 3.1 Proof Suppose that we φ,wherew = u 2 φ = φu tobedeterminedsatisfyingthat 2 φ u > 0, achieves a maximum at an interior point x 0 M. As before, we choose a smooth orthonormal local frame e 1,...,e n about x 0 such that ei e j =0atx 0 {T ij x 0 } is diagonal. Differentiating we φ at x 0 twice, we have i w + w i φ =0 3.2

9 Bao XingJournal of Inequalities Applications :238 Page 9 of 12 ii w w i φ 2 + w ii φ Differentiating w,wesee i w = k k u ik u, ii w = k ik u 2 + k k u iik u. Using 3.2 it followsfrom 3.3that F ii δ kl ku l u ik u il u + F ii k u iik u wf ii i φ 2 ii φ w 2 i,k,l δ kl ku l u 2w ik u il u + k u k u w 2 φ 2 + w φ Note that the first term in is nonnegative. Multiply γ F ii to 3.5add what we got to 3.4. Thus, by 2.9weobtain F ii k u kii u + γ k u w 2 Fii i φ 2 + γ φ 2 + wf ii ii φ + γ φ C u 2 F ii. 3.6 Now we compute the first term in 3.6. Firstly, we have i φ = φ i u, ii φ = φ ii u + φ i u 2. Using 3.2, we easily get that F ii k u kii u + γ k u = F ii k u k T ii s i u 2 + t 2 u 2 χ ii = k u k ψ + β ε +wφ F ii 2s i u 2 t u 2 F ii k u k χ ii. 3.7 By the homogeneity of F,we also get F ii ii φ + γ φ = φ F ii i u 2 + γ u 2 + φ F ii T ii s i u 2 + t 2 u 2 χ ii = φ F ii i u 2 + γ u 2 + φ F sf ii i u 2 + t 2 Fii u 2 F ii χ ii. 3.8

10 Bao XingJournal of Inequalities Applications :238 Page 10 of 12 According to , it followsfrom 3.6 Let γ u 2 φ 1 φ 2 t 2 2γ φ F ii + φ 1 φ 2 + sφ F ii i u 2 2 φ ψ + β ε F ii χ ii + C F ii ku k ψ + β ε w φ β ε + ku k u hβ ε + C F ii + C. 3.9 w φu=v a, v =1 u + sup u. M We have φ u=av a 1, φ u= a +1φ, v φ 1 2 φ 2 = φ a +1 v av a φ a 2v 2v >0 since v a 1. When ux 0 is sufficiently large, we see k u k u h>0.hencewehave that the first term on the right-h side of 3.9 isnegativeasβ ε, β ε >0.From whena is sufficiently large, we then obtain that φ aγ u 2 C, v from which we conclude that 3.1holds. In order to prove 1.19, it remains to bound sup M u + sup M u.wequotetwolemmas in [8], the ingredients of whose proofs are the maximum principle. Lemma 3.2 If either 1.10 or 1.11 holds, then any admissible solution u of 1.16 admits the a priori bound sup u c M Lemma 3.3 If u is admissible such that tr T[u] 0 u C 0 M μ, then sup ν u c 1 μ, 3.12 M where ν is the interior unit normal to M. Now with the above two lemmas the fact ν u ν u on M when u U, wethen have the following.

11 Bao XingJournal of Inequalities Applications :238 Page 11 of 12 Theorem 3.4 Suppose that , either 1.10 or 1.11 hold. Then, for u U, 1.19 holds. Therefore, the uniform estimates 1.19 ensure that there exist a subsequence {u εk } of {u ε } a function u C 1,1 Msuchthatu εk u in M as ε k 0. It is easy to verify that u satisfies u C 3,α E for any α 0, 1. Consequently, Theorem 1.1 is established. Funding The research was supported by the National Natural Science Foundation of China No Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps institutional affiliations. Received: 16 April 2018 Accepted: 25 August 2018 References 1. Bao, G.-J., Dong, W.-S., Jiao, H.-M.: Regularity for an obstacle problem of Hessian equations on Riemannian manifolds. J. Differ. Equ. 258, Caffarelli, L.A., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations III, functions of the eigenvalues of the Hessian. Acta Math. 155, Crall, M., Ishii, H., Lions, P.: User s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, Dong, W.-S., Wang, T.-T., Bao, G.-J.: A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Commun. Pure Appl. Anal. 15, Evans, L.C.: Classical solutions of fully nonlinear, convex, second order elliptic equations. Commun. Pure Appl. Math. 35, Gerhardt, C.: Hypersurfaces of prescribed mean curvature over obstacles. Math. Z. 133, Guan, B.: The Dirichlet problem for Hessian equations on Riemannian manifolds. Calc. Var. Partial Differ. Equ. 8, Guan, B.: Complete conformal metrics of negative Ricci curvature on compact manifolds with boundary. Int. Math. Res. Not. 2008, rnn Addendum, IMRN , , rnp Guan, B.: Second order estimates regularity for fully nonlinear elliptic equations on Riemannian manifolds. Duke Math. J. 163, Guan, B.: The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds. arxiv: Guan, B., Jiao, H.-M.: Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds. Calc. Var. Partial Differ. Equ. 54, Guan, B., Jiao, H.-M.: The Dirichlet problem for Hessian type fully nonlinear elliptic equations on Riemannian manifolds. Discrete Contin. Dyn. Syst. 36, Guan, P.-F., Wang, G.-F.: Local estimates for a class of fully nonlinear equations arising from conformal geometry. Int. Math. Res. Not. 26, Jiao, H.-M.: C 1,1 regularity for an obstacle problem of Hessian equations on Riemannian manifolds. Proc. Am. Math. Soc. 144, Jiao, H.-M., Wang, Y.: The obstacle problem for Hessian equations on Riemannian manifolds. Nonlinear Anal. TMA 95, Krylov, N.V.: Boundedly inhomogeneous elliptic parabolic equations in a domain. Izvestia Math. Ser. 47, Lee, K.: The obstacle problem for Monge Ampère equation. Commun. Partial Differ. Equ. 26, Liu, J.-K., Zhou, B.: An obstacle problem for a class of Monge Ampère type functionals. J. Differ. Equ. 254, Oberman, A.: The convex envelope is the solution of a nonlinear obstacle problem. Proc. Am. Math. Soc. 135, Oberman, A., Silvestre, L.: The Dirichlet problem for the convex envelope. Trans. Am. Math. Soc. 363, Savin, O.: The obstacle problem for Monge Ampere equation. Calc. Var. Partial Differ. Equ. 22, Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, Trudinger, N.S.: On the Dirichlet problem for Hessian equations. Acta Math. 175, Urbas, J.: Hessian equations on compact Riemannian manifolds. In: Nonlinear Problems in Mathematical Physics Related Topics, II, pp Kluwer/Plenum, New York 2002

12 Bao XingJournal of Inequalities Applications :238 Page 12 of Viaclovsky, J.A.: Conformal geometry, contact geometry, the calculus of variations. Duke Math. J. 101, Viaclovsky, J.A.: Conformal Geometry Fully Nonlinear Equations. Nankai Tracts in Mathematics, vol. 11, pp World Scientific, Hackensack Xiong, J.-G., Bao, J.-G.: The obstacle problem for Monge Ampère type equations in non-convex domains. Commun. Pure Appl. Anal. 10,