Prescribing curvatures on three dimensional Riemannian manifolds with boundaries

Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $\partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so that under these new metrics, $K$ is the scalar curvature and $c$ is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on $K$,$c$ and $(M,g)$ we show that all the solutions of the equation can only blow up at finite points over each compact subset of $\bar M$, some of them may appear on $\partial M$. We describe the asymptotic behavior of the blowup solutions around each blowup point and derive an energy estimate as a consequence.


Introduction
In geometric analysis two well known problems are closely related, the Nirenberg problem (or the Kazdan-Warner problem) and the Yamabe problem. The Nirenberg problem asks what function K(x) on S n is the scalar curvature of a metric g on S n conformal to the standard metric g 0 . From another point of view, on any given compact Riemannian manifold (M, g) without boundary, the Yamabe problem, which was solved through the works of Yamabe [36], Trudinger [35], Aubin [3] and Schoen [32], concerns whether it is possible to deform g conformally to get a new metric with constant scalar curvature. Similar questions can also be asked on general Riemannian manifolds with boundaries: Let (M, g) be a Riemannian manifold with boundary, let K ∈ C 1 (M ) and c ∈ C 1 (∂M ) be C 1 functions defined on M and ∂M , respectively, then is it possible to deform g conformally to another metric g 1 so that K is the scalar curvature and c is the boundary mean curvature under g 1 ? By writing g 1 as g 1 = u 4 n−2 g, K and c are related to the scalar curvature R g and the mean curvature h g under metric g by (1.1) K = − 4(n−1) n−2 u − n+2 n−2 (∆ g u − n−2 4(n−1) R g u). c = 2 n−2 u − n n−2 (∂ νg u + n−2 2 h g u).
where ν g is the unit normal vector pointing to the outside of ∂M . ∆ g is the Laplace-Beltrami operator, which can be written as ∆ g = 1 √ g ∂ i ( √ gg ij ∂ j ) in local coordinates. If K and c are constants and (M, g) is compact, the existence of a metric with constant scalar curvature and boundary mean curvature is always referred to as the boundary Yamabe problem (BYP). Many cases of the BYP have been solved by Escobar [17,18,19], Han-Li [22] and Marques [28]. But unlike the completely solved Yamabe problem, some cases of the BYP are still open. In general, the boundary terms in (1.1) make the nature of equation (1.1) very different from its counterpart without the boundary condition. Some difficulties created by the boundary terms are still not completely understood. In this article, we focus on three dimensional Riemannian manifolds with boundaries and consider the corresponding prescribing curvature equations defined on these manifolds.
Let (M, g) be a smooth three dimensional complete Riemannian manifold with boundary ∂M . Suppose K(x) > 0 and c(x) are C 1 functions defined on M and ∂M , respectively. If g 1 = u 4 g (u > 0 smooth) is a metric conformal to g that takes K as the scalar curvature and c as the mean curvature on ∂M , one can write the equation as The purpose of this article is to understand the bubbling phenomena of (1.2) under natural assumptions on K and c. Without the boundary condition, the bubbling phenomenon of equation (1.2) and its high dimensional variants have been studied extensively under various assumptions. The reader is referred to the following incomplete list and the references therein [4,5,6,7,11,12,13,15,16,20,21,23,24,25,26,27,28,31,32,37]. However, much less work can be found to address the case with boundary conditions. In this article we describe the blowup phenomenon for (1.2) under weak assumptions on K and c.
To state our main result, we first remark that M may not be compact, so our result concerns the blowup phenomenon for every compact subset of M that shares with M a part of its boundary. Namely, let V be a subset of M such thatV is compact and let Γ = ∂V ∩ ∂M be the part of the boundary that V shares with M , then our main result can be stated as follows: Theorem. Let (M, g), R g , h g , V , Γ, K, c be described as above. Let u > 0 be a classical solution of (1.2). Suppose Γ is umbilic and there exists ǫ 0 > 0 such that for every point p ∈ Γ, g is conformal to the Euclidean metric δ in B(p, ǫ 0 ) ∩ M . Assume in addition that for some Λ > 0, Then for some C(M, g, Λ, V, ǫ 0 ) > 0 and integer m(M, g, Λ, V, ǫ 0 ) > 0, there exist local maximum points of u, denoted as S := {P 1 , ..., P m }, such that As a consequence, we have the following energy estimate: If (M, g) is compact, Theorem concludes that if the metric is locally conformally flat near ∂M , which is umbilic, then there are only finite blowup point inM and the solutions of (1.2) can be estimated by (1.3).
The results of Theorem and the Corollary 1.1 are closely related to a priori estimates of the solutions. When (M, g) is compact with a boundary and K and c are constants, various compactness results and related discussions can be found in [2], [20], [22], [30] and the reference therein. The C 1 assumptions on K and c in Theorem should be sharp. The reason is even for the interior equations the C 1 assumption on K is necessary to have a description of the blowup phenomenon by (1.3). So Theorem implies that for this three dimensional situation, the C 1 assumption on c does not lead to more restrictions on K. For dimensions 4 and higher, we expect the situation to be much more subtle, first the assumption on K will be more delicate and should be made in neighborhoods of its critical points, then the sign of c at the blowup point will make a more significant difference. Also, the flatness assumptions in the neighborhood of the critical points of K and c might be related. As the dimension grows higher, the relations will become more subtle, new phenomenon will come out.
The major step in the proof of is to establish a Harnack type inequality near Γ. This is done by the well known method of moving planes. Unlike most of the previous works on this type of Harnack inequality established only for the interior equations, the equation in Theorem has new features that were not dealt with before. Generally speaking, the method of moving planes requires a delicate construction of some test functions. Now this becomes more difficult because of two reasons, first, the symmetry of the domain is destroyed, second, when blowup happens on the boundary or close to the boundary, the function c creates more error terms for the test functions to control. In this article we found a way to handle all the difficulties for dimension three, our approach is motivated by Caffarelli's magnificent ideas in [9][10] on free boundary problems.
In the next few sections, we mainly focus on the proof of the Harnack type inequality on upper half Euclidean balls and we shall mention how to obtain the interior estimate away from the boundary Γ.

Harnack type inequalities on the boundary
The proof of Theorem can be divided into two parts. First we consider the region close to Γ: N (Γ, ǫ 0 ). i.e. the ǫ 0 neighborhood of Γ. In this region we shall establish a Harnack type inequality. Then outside this region, we use the techniques developed in [25] to get the same type of Harnack inequality. Note that for this interior part we don't need to assume the metric to be locally conformally flat.
Since (M, g) is locally conformally flat near Γ, at each point p ∈ Γ we can find a positive smooth function φ such that δ = φ 4 g where δ is the Euclidean metric. The umbilicity of Γ implies that ∂M near p is either a piece of sphere or a hyperplane in the new metric. Since both neighborhoods are conformally equivalent through the inversion map f (x) = x|x| −2 , we just assume that near p the boundary is a hyperplane, then the equation in the neighborhood of p can be written as where ǫ 1 > 0 is a positive constant that only depends on ǫ 0 , M, g, Γ. B + ǫ 1 is the upper half ball of radius ǫ 1 , ∂ ′ B + ǫ 1 = ∂B + ǫ 1 ∩ ∂R 3 + . The computation above is based on the following conformal covariant properties of the two operators: So the whole equation is reduced to the Euclidean case. From now on in this section we just consider the following case (by abusing the notations we still use K to denote a positive C 1 function on a upper half disk B + 3R and c to denote a C 1 function defined ∂ ′ B + 3R , the lower part of ∂B + 3R ): The main result of this section is ) be a positive function that solves (2.1). Assume for some Λ > 0, K and c satisfy respectively, then for some C(Λ) > 0 This Harnack type inequality reveals important information about the interaction of bubbles (large local maximum points) of u which can be seen in the following Consequently u satisfies In the proof of Proposition we shall omit a selection process of Schoen since it is well known to experts. We only include it in the appendix. At the end of this section we indicate the outline of the proof of Corollary 2.1 based on Proposition 2.1. The proof can be found in [24] with obvious changes, but we indicate the key points for the convenience of the readers.
2.1. The Proof of Proposition 2.1. We only need to consider the case R = 1. The general case can be reduced to the case R = 1 by considering the function R 1 2 u(R·). The proof is by a contradiction. Suppose (2.4) does not hold, there exists a sequence u i (x i ) such that Clearly the above inequality implies u i (x i ) → ∞. By a standard selection process of Schoen [31] and the classification theorems of Caffarelli-Gidas-Spruck [8] and Li-Zhu [27], one can consider x i as a local maximum of u i , moreover the following sequence of functions converge in C 2 over any finite domain in the following two cases: In this casev i converge uniformly to U which satisfies Since the selection process and the application of the classification theorems are standard. We put the details in the Appendix. Similar techniques can also be found in [11], [24], [37], etc.
The proof of Proposition 2.1 that follows can be divided into three steps. First we rule out case one. i.e. We shall show that the blowup points can not be far away from the boundary. In the second step we prove the case of lim i→∞ c(x ′ i ) ≤ 0. Then in the final step we prove the case of lim i→∞ c(x ′ i ) > 0.

2.1.1.
Step one. In this subsection we derive a contradiction to With no loss of generality we assume lim i→∞ K i (x i ) = 3, in this case the U in (2.9) is of the form so the rescaled domain for v i will have a part of the boundary (the upper part) whose distance to 0 is comparable to M 2 i . By (2.7) we know on this part of the boundary Here we use y λ = λ 2 y/|y| 2 . In this step we assume λ ∈ [ 1 2 , 2]. Let Σ λ = Ω i \B λ . Note that for simplicity we shall omit i in some notations. Let w λ = v i − v λ i and we consider the equation for w λ : (2.14) where b λ and ξ i are obtained from mean value theorem: Since v i converges in C 2 norm over U over any fixed finite domain, v λ i is close to U λ , the Kelvin transformation of U . By direct computation U > U λ for |y| > λ and λ ∈ (0, 1). On the other hand U < U λ for λ > 1 and |y| > λ. So the strategy of the proof is to find a test function h λ (i omitted in this notation) so that the moving sphere method works for w λ + h λ , and the h λ is just a perturbation of w λ , which means h λ (y) = •(1)|y| −1 in Σ λ . Then it is possible to move the spheres from a position less than 1 to a position larger than 1 keeping w λ + h λ > 0 in Σ λ . But this is a contradiction since w λ + h λ converges to U − U λ in finite domains and U < U λ for λ > 1 and |y| > λ.
The test function in this section is h λ is a radial function, the function h λ (r) satisfies .16) is the same as the one in (2.15). Now we consider the equation for w λ + h λ , from (2.14) we have Then we observe that the maximum principle in the moving sphere process only needs to be applied over O λ because outside O λ , w λ + h λ > 0. Now we claim that which is the form for the application of the moving sphere method. To see (2.19), we first observe that The second term and the third term on the right are negligible comparing to the first term on the right. The reason is On the other hand, by assumption T i → ∞, we see the first term dominates the other two terms.
Next we see that the assumption T i → ∞ makes |∂ 3 (v λ i )| dominate Λ(v λ i ) 3 as the later is of the order O(|y| −3 ). By the definition of h λ and the definition of the domain Ω i , we see easily that (2.19) is proved. The process of making the moving sphere process start is standard, even though the boundary condition makes it different from the interior case.
Lemma 2.1. For any fixed λ 0 ∈ ( 1 2 , 1) and all large i, . By the convergence of v i to U and the fact h λ = •(1) over finite domains we can check easily that for any fixed R 1 >> 1, Since R 1 is chosen sufficiently large, one can find σ(λ 0 ) > 0 such that The definitions of h λ 0 and Ω i imply |h λ 0 (y)| ≤ σ 5 |y| −1 for |y| > R 1 . So to finish the proof of this Lemma it is enough to show On |y| = R 1 and ∂ ′′ Ω i we certainly have v i > (1 − 2σ)|y| −1 . To apply the maximum principle to the super harmonic function v i − (1 − 2σ)|y| −1 , we need Once this is proved, we have U > U λ 1 for some |y| > λ 1 , which is a contradiction to the fact that λ 1 > 1. To see why (2.23) holds, first, (2.17) and (2.20) means the maximum principle holds for w λ + h λ , second, (2.12) means there is no touch on ∂ ′′ Ω i .
Step one is established.

2.1.2.
Step two. In this step, we deal with the case lim i→∞ c In this case the limit ofv i is U , which satisfies By Li-Zhu's classification theorem [27], Let v λ i (y) = ( λ |y| )v i (y λ ). In this step we let and we require λ ∈ [λ 0 , λ 1 ] and the moving sphere method will be applied Also by direct computation one can verify that For this reason we require λ ∈ [λ 0 , λ 1 ] in this step. The equation for where ξ 1 and ξ 2 are obtained by mean value theorem: and Q 1 and Q 2 are error terms to be controlled by test functions: For Q 1 and Q 2 we use Note that in Q 2 it is important to have the 1 − λ |y| term, even though we don't need this term to appear in the estimate of Q 1 .
The construction of the test function here consists of two parts, h 1 and h 2 . We first define By maximum principle, h 1 (r) < 0 for r > λ. h 1 (|y|) is the first part of the test function h λ . To define the second part of the test function, we let φ : [1, ∞) → [0, ∞) be a smooth non-negative function that satisfies Let φ λ (·) = φ(λ·) and we define (2.29) h 2 (y) = −C 4 (Λ)M −2 i y 3 φ λ (|y|) where C 4 (Λ) is chosen so large that Note that on ∂ ′ Σ λ := ∂Σ λ ∩ ∂R 3 + , (2.31) For the application of the moving sphere method we show that the right hand sides of the above are non-positive. Namely we shall show (2.32) } is the only place where the maximum principle needs to hold, because by the definition of h λ , h λ (y) = •(1)|y| −1 in Σ λ , which means To see (2.32), first by (2.27) and (2.28) one sees that if C 3 is large enough For h 2 we have So by choosing C 3 larger if necessary we have To see (2.33), first by the definition of h 2 we have The second inequality above is because inŌ λ , ξ 2 ∼ |y| −2 . We also use h λ (λ) = 0 in the above. Then (2.33) follows immediately. Next we show that the moving sphere process can get started, namely we have Proof of Lemma 2.2: The proof is similar to the one in step one. For |y| ≤ R 1 (R 1 >> 1), w λ 0 + h λ 0 > 0 is guaranteed by the expressions of U , U λ 0 and the convergence of v i to U over finite domains. Also h λ 0 = •(1)|y| −1 means h λ 0 is just a perturbation over finite domains. Moreover on |y| = R 1 , there is ǫ 0 > 0 such that Since v i (y)−(1−ǫ 0 )(1+c 2 0 ) − 1 2 |y−e 3 | −1 is super-harmonic function in Σ λ \B R 1 and is positive on |y| = R 1 and |y| = ǫ i M 2 i . The only place to consider is

Lemma 2.2 is established.
Once we have Lemma 2.2, the moving sphere process can start at λ = λ 0 . Note that the equation for w λ + h λ becomes This means the maximum principle always holds for w λ + h λ as long as it is positive on the boundary ∂ ′′ Ω i , which is certainly the case because v i >> v λ i on ∂ ′ Ω i and h λ is a perturbation. So the spheres can be moved to λ 1 = 2(1 + c 2 0 ) − 1 2 to get a contradiction.

2.1.3.
Step three. In this subsection we deal with the case lim i→∞ c i (x ′ i ) = c 0 > 0.v i converges uniformly to U in all finite subsets of {y ∈ R 3 ; Let v λ i be the Kelvin transformation of v i . In this case we let Let w λ = v i − v λ i , then the equation for w λ is still described by (2.26) with ξ 1 , ξ 2 , Q 1 ,Q 2 defined as in step two and the estimates for Q 1 and Q 2 are still (2.27). Since lim i→∞ c i (x ′ i ) = c 0 > 0, the construction of the test function h λ is this case is much more delicate.
The construction of h λ consists of two parts. First we shall construct h 1 to control the region close to ∂B λ , then this function is connected smoothly to 0. Clearly h 1 creates new difficulties at the regions where it is connected to 0. Next we use h 2 to control the region far enough to ∂B λ . h 2 is 0 in regions close to ∂B λ and becomes negative as y is far away from ∂B λ . A parameter of h 2 is chosen to be large so that h 2 not only controls the difficulties from Q 1 and Q 2 , but also those from h 1 . One delicate thing is that h 2 does not create new difficulties. By choosing all the parameters carefully we shall obtain the following properties for h λ : where A > 5λ 1 is to be determined. Let Ω 1 ⊂ R 3 + be the region where φ is positive. Let Γ 1 be the 0 level surface of φ. Direct computation shows that where C 6 is a large constant to be determined. From here we see h 2 is C 1 across Γ 1 . By direct computation From here we observe that h 2 is weakly super-harmonic in Σ λ since ∇h 2 = 0 on Γ 1 . On By the definition of h 2 , if A(c 0 ) is large, we have Here A is fixed. We observe that if |y| > 16A, φ(y) > 1 2 . So by choosing C 6 large enough we have (2.39) By (2.38) we also know So the most important feature of h 2 is it does not create new difficulties, even though it does not control all the error terms in the whole Σ λ . Now we define h 1 to be (2.41) where C 5 and N are large positive constants to be determined. A is the one in the definition of h 2 , which has been determined. Also the "smooth connection" in the definition means |∇h 1 (y)|, By the definition of h 1 , Here we require C 5 to satisfy On the boundary we have So by choosing N (c 0 ) large and C 5 larger if necessary we have A are all determined, we finally determine C 6 . One last requirement for C 6 is to control the bad part of h 1 in (B 17A \ B 16A ) ∩ Σ λ . By choosing C 6 larger if necessary we see from (2.38) and (2.39) that the errors caused by bending h 1 to 0 can be controlled by h 2 . Therefore the first two equations of (2.35) are established. Here we also use the fact that h λ ≤ 0 in Σ λ . By the definitions of h 1 and h 2 , the other two equations of (2.35) are also satisfied. We are left with how to let the moving sphere process start. This part is similar to step two. We leave the details to the interested readers. Proposition 2.1 is established. Proposition 2.1 is the major step in deriving the energy estimate in Corollary 2.1. Based on the previous works of Y. Y. Li [23], Han-Li [22] and Li-Zhang [24], it is standard to derive Corollary 2.1. Therefore we only mention the major steps and the main idea in this argument: First one uses Schoen's selection process to find all large maximums of u in, say, B + 2R . Around each of these local maximums there is a small neighborhood in which u looks like a standard bubble which has most of its energy in it. The distance between these local maximums is the crucial information to find and this is the place where the Harnack inequality in Proposition 2.1 is used. The essential difference between this locally defined equation and those globally defined equations (such as the ones in [23]) is that one can not find two bubbles closest to each other. For each bubble, there certainly exists a bubble closest to it, but one certainly can not assume the first bubble is the closest one to the second bubble. This difficulty, which comes from the local nature of the equation, requires a different approach than those in [23]. The way to overcome this difficulty is to rescale the equation so that after the scaling, the equation is centered at the first bubble and the distance between the first bubble and the second bubble is one. Then the Harnack inequality in Proposition 2.1 applied to the local region implies that the first bubble and the second bubble must have comparable magnitudes. Then it is possible to show that two bubbles can not tend to the same blowup point because otherwise a harmonic function with a positive second order term can be found. This second term will lead to a contradiction in the Pohozaev Identity. Note that the important second order term can only be proved to be positive if the adjacent bubbles have comparable magnitudes, which is the key information revealed by the Harnack inequality. Once we have known that all bubbles are far apart, it is possible to use standard elliptic estimates to show that the behavior of u near each large local maximum is like a harmonic function with fast decay. So (2.5) as well as (2.6) can be obtained.

2.2.
Harnack inequality in the interior of M . Now we consider V \ N (Γ, ǫ 0 ). Let d be the distance between V \ N (Γ, ǫ 0 ) and ∂M . Over this interior region we don't need to assume M to be locally conformally flat. We shall establish the following inequality: The proof of Proposition 2.2 follows from the argument in [25]. Only small modification is needed to adjust to the current situation. The outline is as follows: Suppose (2.45) does not hold, then we can select x i as a sequence of local maximums of u i . Then the equation can be written in a conformal normal coordinates centered at x i . After rescaling, u i becomes v i , a sequence that converges in C 2 loc (R 3 ) to a standard bubble whose maximum is 1. Then consider the Kelvin transformation of v i (y): v λ i (y). By comparing the equation for v i and v λ i we see that the only new term in this context is the term: 5 in Σ λ where Σ λ becomes a symmetric domain defined appropriately. The above term is of the harmless order O(M −2 i |y| −4 ). By using the same test function we used in [25], a contradiction can be obtained correspondingly. We leave the details to the interested readers. Proposition 2.2 is proved. Proposition 2.1 and Proposition 2.2 lead to Theorem by the arguments in [24] and [26].

Appendix
In this section we provide some details of the Schoen's selection process for the convenience of readers. We only provide details for case one and case two. The details for case three are similar.
3.1. A Calculus Lemma. First we use a Calculus lemma to simplify some computations. This Calculus lemma was first used in [24]. 3.2. The selection process for case one in the proof of Proposition 2.1: In this subsection we explain why x i can be considered as a local maximum of u i , assuming u 2 i (x i )x i3 → ∞. We shall apply Lemma 3.1 for T > 1. i.e. the selection is over the whole B 1 . Let r i = |x i3 |/2, so u i (x i )r 1 2 i → ∞. Apply Lemma 3.1 to the function u i (x i + r i ·) over B 1 , then the conclusion can be translated as follows: there is a i ∈ B(x i , r i ) such that where σ i := 1 2 (r i − |a i − x i |) and for |y| ≤ u i (a i ) 2 σ i → ∞ Clearlyṽ i satisfies ∆ṽ i (y) + K i (u i (a i ) −2 y + a i )ṽ i (y) 5 = 0 |y| ≤ u i (a i ) 2 σ i So by standard elliptic theory we know there is a subsequence ofṽ i (y) (still denoted byṽ i (y)) that converges uniformly toŨ 0 (y) on all compact subsets of R 3 .Ũ 0 (y) satisfies (3.1) ∆Ũ 0 (y) +K 0Ũ0 (y) 5 = 0 y ∈ R 3 whereK 0 = lim i→∞ K i (a i ). With no loss of generality we assumeK 0 = 3, so by the well known classification theorem of Caffarelli-Gidas-Spruck [8], v i (y) = ( µ 1 + µ 2 |y − z| 2 ) 1 2 for some z ∈ R 3 and µ ≥ 1. SoŨ 0 (y) has an absolute maximum at z. Consequentlyṽ i (y) has a local maximum at z i close to z when i is large. Letx i = u i (a i ) −2 z i + a i , then {x i } are local maximum points of u i , also it is easy to verify that i → ∞ we knowv i is uniformly bounded on all compact subsets of {y 3 ≥ − lim i→∞Ti }. With no loss of generality we assume that K i (x i ) → 3, then by elliptic estimates,v i (y −T i e 3 ) converges in C 2 norm on all compact subsets ofR 3 + toŪ , which solves ∆Ū + 3Ū 5 = 0 R 3 where c = lim i→∞ c i (x ′ i ). By Li-Zhu's classification result [27], we know thatŪ has a unique maximum point z 0 in R 3 + . By the C 2 convergence ofv i (· −T i e 3 ) toŪ , we can find {y i } i=1,2,.. as local maximum points of v i (· −T i e 3 ) that approach z 0 as i → ∞. Then by the definition ofv i , we know that u i (x i ) −2 (y i −T i e 3 ) +x i are local maximum points of u i . So we redefine x i as u i (x i ) −2 (y i −T i e 3 ) +x i . So thev i defined in (2.8) converges to the function U 1 in (2.10).