Regularity of viscosity solutions of the $\sigma_k$-Loewner-Nirenberg problem

We study the regularity of the viscosity solution $u$ of the $\sigma_k$-Loewner-Nirenberg problem on a bounded smooth domain $\Omega \subset \mathbb{R}^n$ for $k \geq 2$. It was known that $u$ is locally Lipschitz in $\Omega$. We prove that, with $d$ being the distance function to $\partial\Omega$ and $\delta>0$ sufficiently small, $u$ is smooth in $\{0<d(x)<\delta\}$ and the first $(n-1)$ derivatives of $d^{\frac{n-2}{2}} u$ are H\"older continuous in $\{0 \leq d(x)<\delta\}$. Moreover, we identify a boundary invariant which is a polynomial of the principal curvatures of $\partial\Omega$ and its covariant derivatives and vanishes if and only if $d^{\frac{n-2}{2}} u$ is smooth in $\{0 \leq d(x)<\delta\}$. Using a relation between the Schouten tensor of the ambient manifold and the mean curvature of a submanifold and related tools from geometric measure theory, we further prove that, when $\partial\Omega$ contains more than one connected components, $u$ is not differentiable in $\Omega$.

Let Ω be a bounded domain in R n and consider the σ k -Loewner-Nirenberg problem in Ω, i.e. the problem of finding a positive solution to 2) where N k = 2 −k n k and d(x) = dist(x, ∂Ω). The geometric nature of problem (1.1)-(1.2) comes from the fact that u This problem was first studied in the classical paper of Loewner and Nirenberg [28] where, among other results, the existence of a unique smooth positive solution was proved when the boundary of Ω is smooth and compact. Since then, further studies of problem (1.3) and its generalization in manifold settings have been done by many authors; see e.g. Allen, Isenberg, Lee and Allen [1], Andersson, Chruściel and Friedrich [3], Aviles [4], Aviles and McOwen [5], Finn [9], Gover and Waldron [13], Graham [14], Han, Jiang and Shen [19], Han and Shen [20], Jiang [21], Mazzeo [30], Véron [35] and the references therein. When 2 ≤ k ≤ n, the σ k -Loewner-Nirenberg problem (1.1)-(1.2) is a fully nonlinear (non-uniformly) elliptic problem of Hessian type. Some key results for this problem and its analogues on manifolds have been obtained by Chang, Han and Yang [6], González, Li and Nguyen [12], Guan [15], Gursky, Streets and Warren [16], Gurksy and Viaclovsky [18], and Li and Nguyen [25]. For other related works, see also Li and Sheng [22], Sui [32], Wang [36] and the references therein.
When Ω is an annulus {a < |x| < b} for some positive constants a < b, it was shown in Chang, Han and Yang [6] that (1.1)-(1.2) has no rotationally symmetric C 2 solution.
In a closely related context, Li in [23] gave the definition of viscosity solutions to nonlinear Yamabe-type equations, proved comparision principles for Lipschitz viscosity solutions and in turn established Liouville-type theorems and local gradient estimates for solutions of general nonlinear Yamabe-type equations. Such comparison principles were strengthened by Li, Nguyen and Wang in [26] showing that they hold for continuous viscosity solutions and also for a larger class of equations. Based on the latter comparison principles, González, Li and Nguyen proved in [12] the uniqueness of continuous viscosity solutions to the σ k -Loewner-Nirenberg problem for general Ω with smooth boundary. The combination of this uniqueness result and the above mentioned result of Chang, Han and Yang implies that there is no C 2 solution on any annulus. González, Li and Nguyen also proved in [12] the existence of a locally Lipschitz viscosity solution to the σ k -Loewner-Nirenberg problem for general Ω with smooth boundary. Furthermore, by the same work, the solution u satisfies (1.4) It was then shown in Li and Nguyen [25] that in the case of an annulus Ω = {a < |x| < b}, (1.1)-(1.2) admits no C 1 solution: Speficially, the unique viscosity solution of (1.1)-(1.2) is not C 1 across the sphere S * = {|x| = √ ab}, is smooth in Ω \ S * , and is C 1, 1 k but not C 1,γ for any γ > 1 k in each of {a < |x| ≤ √ ab} and { √ ab ≤ |x| < b}. More generally, it is worthwhile to understand for what domain Ω problem (1.1)-(1.2) admits a C 2 solution and for what domain Ω it admits no C 2 solution. For this purpose, several specific questions were raised in [25]. Our results in the sequel are efforts in addressing this issue, and in particular affirmatively confirm [25, Conjecture 1.1] and give a complete answer to [25,Question 1.3].
Our first theorem concerns the non-existence of C 1 solutions for k ≥ 2 when ∂Ω is disconnected. We would like to point out that the situation is very different when k = 1: The solution of the Loewner-Nirenberg problem is smooth in Ω. Theorem 1.1. Let Ω be a bounded domain in R n , n ≥ 3, with smooth and disconnected boundary ∂Ω. Suppose that 2 ≤ k ≤ n. Then problem (1.1)-(1.2) has no positive solution in C 1 (Ω). In particular, the unique locally Lipschitz viscosity solution of (1.1)-(1.2) does not belong to C 1 (Ω).
Some ideas supporting Theorem 1.1 were described in [25]. Roughly speaking, they are as follows. On one hand, by [25,Theorem 1.1], the complete Riemannian manifold (Ω, u 4 n−2g ), if smooth, cannot admit a minimal immersion. On the other hand, in view of the fact that u behaves like d − n−2 2 near ∂Ω, the hypersurface Σ δ = {x ∈ Ω : d(x) = δ} is expected to be mean convex with respect to u 4 n−2g for small δ > 0 and so there should be a minimal hypersurface in Ω confined by Σ δ . In fact, this line of argument gives the following result for the differential inclusion λ(−A u ) ∈Γ 2 : Let Ω be a bounded domain in R n , n ≥ 3, with smooth and disconnected boundary ∂Ω. There exists no positive function u ∈ C 1 (Ω) such that λ(−A u ) ∈Γ 2 in Ω in the viscosity sense and ∂Ω has non-negative mean curvature H ∂Ω ≥ 0 with respect to g = g u and the normal pointing towards Ω.
It follows from the proof that in Theorem 1.2 it is enough to assume that ∂Ω is C 1,α ∩ W 2,p regular for some α ∈ (0, 1] and p ≥ 2. In the proof of Theorem 1.2, the assumption that u is differentiable is used in the application of Corollary 3.9 and in asserting that the hypersurface ∂E \ S in Proposition 3.7 indeed has zero mean curvature. The rest of the proof uses only that u is Lipschitz continuous. Nevertheless, the assumption u ∈ C 1 (Ω) cannot be relaxed to u ∈ C 0,1 (Ω), even when one imposes the stronger assumption that λ(−A u ) ∈ Γ 2 in Ω and H ∂Ω > 0. Indeed, it can be easily deduced from [25,Theorem 1.2] that, for any 0 < a < c < b < ∞ and with in the viscosity sense and that H ∂Ω > 0 with respect to g = g u and the normal pointing towards Ω. In the particular case of this example, if one follows the strategy of our proof, the minimal set E one would obtain in the context of Proposition 3.7 is the set {|x| < c} which has smooth boundary ∂E = {|x| = c}. However, u fails to be differentiable exactly on this set, the mean curvature of ∂E with respect to g u is undefined, and Corollary 3.9 is inapplicable there.
Our next theorem shows that, for small δ > 0, the solution u of (1.1)-(1.2) is smooth in {0 < d(x) < δ}, and confirms the aforementioned expectation that Σ δ = {x ∈ Ω : d(x) = δ} is mean convex with respect to u 4 n−2g , which is needed in the passage from Theorem 1.2 to Theorem 1.1. Let π be a smooth map defined in a neighborhood of ∂Ω such that π(x) is the point on ∂Ω which is closest to x, i.e. |x − π(x)| = d(x) near ∂Ω. We show that, near ∂Ω, ln(d u 2 n−2 ) admits an expansion of the form Here (1.5) is understood in the sense that, for every m ≥ 1 and with for j ≥ 0, s ≥ 0, γ ∈ (0, 1), where the implicit constant in (1.6) depends only on Ω, m, j, s, γ. Here ∇ T denotes the gradient along the hypersurfaces orthogonal to ∇d. It is clear that such expansion is unique.
Let Ω be a bounded domain in R n , n ≥ 3, with smooth boundary ∂Ω. Then there exists δ 1 = δ 1 (n, k, Ω) > 0 such that the viscosity solution u of (1.1)-(1.2) satisfies Moreover, near ∂Ω, d u 2 n−2 admits an expansion of the form (1.5) in the sense that (1.6) holds for every m ≥ 1 where coefficients c p,q are smooth functions on ∂Ω. Finally, if When k = 1, Theorem 1.3 was proved by Andersson, Chruściel and Friedrich [3] and Mazzeo [30]. See Han, Jiang and Shen [19] and Han and Shen [20] for results when Ω is not a smooth domain. Also in the case k = 1, it was shown by Gover and Waldron [13] and by Graham [14] via a volume renormalization procedure that the obstruction for the smoothness of du 2 n−2 (i.e. the vanishing of the coefficient c n,1 ) can be characterized using a conformally invariant energy, which can be viewed as a higher dimensional analogue of the Willmore energy. There has been an interest in understanding if analogous results can be obtained in fully nonlinear settings. For related discussions, see e.g. Fefferman [7], Fefferman and Graham [8], Gursky, Han and Stolz [17] and the references therein.
For a discussion about the proof of Theorem 1.3 and related works in the literature, see Subsection 2.1.
We conclude the introduction with a more general version of Theorem 1.1 and Theorem 1.3. Let Γ ⊂ R n be an open convex symmetric cone with vertex at the origin, be concave, homogeneous of degree one, and symmetric, (1.9) (1.10) In (1.9), we say that f is symmetric if f (λ 1 , . . . , λ n ) = f (λ σ(1) , . . . , λ σ(n) ) for any permutation σ. It follows from a more general result of [12] that the problem  The rest of the paper is organized as follows. We start with the proof of Theorem 1.3 and the first statement of Theorem 1.6 in Section 2. We then give the proof of Theorem 1.1, Theorem 1.2 and the second statement of Theorem 1.6 in Section 3.

Acknowledgement:
The authors would like to thank Fang-Hua Lin for stimulating discussions. Part of this work was completed while J. Xiong was visiting Rutgers University, to which he is grateful for providing very stimulating research environments and supports.

and its generalization
In this section, we prove Theorem 1.3 and the first part of Theorem 1.6.

The setup and main ideas of the proof
Before discussing the strategy of the proof of Theorem 1.3 (and its generalization in Theorem 1.6), some comments are in order. In fully nonlinear settings, closest to our result above are those of Gursky, Streets and Warren [16] and Wang [36] for the equation σ k (λ(−Ricg)) = 1 on a given compact manifold M with non-empty boundary ∂M for an unknown metricg which is conformal to a given Riemannian metric g on M and whose conformal factor blows up at ∂M. It is known that this equation has better ellipticity property than the σ k -Loewner-Nirenberg equation (1.1). In [16], it was established that this problem has a unique solution which is smooth in M \ ∂M. In [36], a finite boundary expansion as in (1.5) up to term of order O(d n ln d) was established in C 0 sense. In these works as well as works cited earlier on in the case k = 1, the smoothness of the solution in Ω or M \ ∂M plays a role. In contrast, in our case, smoothness of solution is not available a priori; in fact global differentiability fails in light of Theorem 1.1. Thus, a novelty of our proof of Theorem 1.3 concerns not only the smoothness of the solution near the boundary but also the quantitative nature of estimate (1.6). In fact, even if it is given a priori that the solution is smooth near the boundary or in the whole domain, we do not know of a simpler proof. Of importance in our argument is our use of a result of Savin [31] for small perturbation solutions for elliptic equations to obtain certain needed quantitative C 2 estimate near the boundary. See below for more details. Let u = e n−2 2 w and Then we see that Thus problem (1.1)-(1.2) can be rewritten as For any function v ∈ C 2 (Ω), define Let Ω δ 0 = {x ∈ Ω : d(x) > δ 0 } with δ 0 > 0 sufficiently small such that d(x) ∈ C ∞ (Ω \ Ω δ 0 ), and π : Ω \ Ω δ 0 → ∂Ω be the orthogonal projection map, i.e. |x − π(x)| = d(x).
The idea of the proof of Theorem 1.3 is roughly as follows.
Step 2: Using a barrier argument, we obtain w − W = O(d n ) at C 0 level. This gives the first n terms in the expansion (1.5) at C 0 level. We then use a result of Savin [31] for small perturbation solutions for elliptic equations to show that w − W = O(d n ) also at derivative level -see Proposition 2.8. This gives local smoothness near the boundary of u and the C n−1,γ -regularity of d n−2 2 u up to the boundary. To explain briefly how Savin's result is applied, let us consider a point x 0 ∈ ∂Ω near which we would like to prove our estimate. Without loss of generality, we may assume that x 0 is the origin, the x n axis point toward Ω and the axes x 1 , . . . , x n−1 are along directions tangential to ∂Ω. Then the function h(x) := − ln x n satisfies G(h) = 0 in {x n > 0}. This leads us to consider the functionŵ( ClearlyĜ is smooth and, as G(h) = 0, 0 is a solution forĜ, i.e.Ĝ[0] = 0, andĜ is elliptic near 0. Note thatĜ has the following scaling property: Now, the fact that |w − W | = O(d n ) obtained in Step 1 implies that, for large r, |ŵ r | ≤ C r in a small ball centered at (0, . . . , 0, 1). We may then apply Savin's result toŵ r , yielding the C 2,α regularity ofŵ r in a smaller ball around (0, . . . , 0, 1). Returning to w, this shows the C 2,α regularity of w near ∂Ω.
Step 3: The estimates obtained in Step 2 have the consequence that, along the integral curves of ∇d, the equation G(w) = 0 can be recast as an ODE in the form from which one can deduce the existence of the coefficient function c n,0 and hence all higher coefficient functions in the expansion (1.5).
In particular, in a neighborhood of ∂Ω, G(W 0 ) = O(d), and G(W 0 ) is smooth and can be written as a convergent power series of d: p is an explicitly computable polynomial of the principal curvatures κ 1 , . . . , κ n−1 .
It will be convenient to use the following notations.
We denote by W the set of functions f defined in a neighborbood of ∂Ω for which there exist a sequence {N p } of non-negative integers and a sequence {a p,q } p≥0,0≤q≤Np of smooth functions on ∂Ω such that, for every m ≥ 0, γ ∈ (0, 1), s, j ≥ 0, where ∇ T denotes the gradient along the hypersurfaces orthogonal to ∇d. Functions in W are sometimes known as polyhomogeneous functions. For ℓ ≥ 0, we denote by W ℓ the set of functions f ∈ W satisfying f = o(d ℓ ) near ∂Ω, i.e. the coefficients a p,q as above vanish for all 0 ≤ p ≤ ℓ.
By Lemma 2.1, G(W 0 ) ∈ W 0 . The coefficient functions c 1,0 , . . . , c n−1,0 are constructed inductively via the next lemma. This lemma does not determine the coefficients functions c n,q for all q. However, if those coefficient functions are known, it can be used to determine all the coefficient functions c p,q with p ≥ n + 1.

Lemma 2.3.
Let Ω be a bounded domain in R n , n ≥ 3, with smooth boundary ∂Ω. Assume for some m ≥ 1 that one has found a sequence {N p } 1≤p≤m−1 of non-negative integers and a sequence {c p,q } 1≤p≤m−1,0≤q≤Np of smooth functions on ∂Ω such that the function If m = n, then there exist a minimal N m ≥ 0 and a unique sequence {c m,q } 0≤q≤Nm such that the function The following remarks are clear from the proof of the lemma.
and their derivatives up to second order.
Proof. For convenience, we also write c p,q = 0 for q > N p . Letĉ p,q = c p,q • π. Pick an arbitrary pointx ∈ Ω \ Ω δ 0 and, after a translation and rotation of the coordinate axes, writē x = (x ′ ,x n ) = (0 ′ ,x n ),x n > 0 and π(x) = 0 ∈ ∂Ω as in the proof of Lemma 2.1. For N m , c m,1 , . . . , c m,Nm to be chosen, we compute atx, and In the sequel, we shall use err m to denote some function of the form ∞ j are polynomials of the principal curvatures κ 1 , . . . , κ n−1 , the functionsĉ p,q with 1 ≤ p ≤ m, 0 ≤ q ≤ N p and their derivatives up to the second order.
We have, atx, . Thus a simple induction using (2.7) gives that To proceed, we consider separately the cases Then there existÑ m ≥ 0 and a sequence {a m−1,m,q } 0≤q≤Ñm with a m−1,m,Ñm ≡ 0 such that, for every j ≥ m and γ ∈ (0, 1), When N m takes the minimal valuesÑ m , since m = n and c m,Nm+1 = c m,Nm+2 = 0 (by our convention), we may solve (2.11) successively and uniquely for c m,Nm , c m,Nm−1 , . . . , and eventually c m,0 , which concludes the proof in this case. It remains to consider the case G(W m−1 ) ∈ W m . The same argument as above, but simpler, gives that G(W ) ∈ W m if and only if c m,q = 0 for all q. The conclusion then follows with N m = 0 and c m,0 ≡ 0.
While Lemma 2.3 cannot determine the coefficient functions c n,q for all q, a careful inspection of the proof gives that c n,1 can be determined this way. We have: Let Ω be a bounded domain in R n , n ≥ 3, with smooth boundary ∂Ω. Then there exist n smooth functions c 1,0 , . . . , c n−1,0 , c n,1 on ∂Ω such that, for any smooth function µ on ∂Ω, the function satisfies λ(−S(W )) ∈ Γ k near ∂Ω and G(W ) ∈ W n . Moreover, ifc 1,0 , . . . ,c n−1,0 ,c n,1 ,μ are n + 1 smooth functions on ∂Ω such that the functioñ Remark 2.6. It is seen from the proof that each c p,q in the proposition above other than c n,0 is a polynomial in terms of the principal curvatures κ 1 , . . . , κ n−1 of ∂Ω and their covariant derivatives up to order 2(p − 1). In particular, Proof. Using Lemma 2.1 and applying Lemma 2.3 successively (n − 1) times with we have λ(−S(W m )) ∈ Γ k near ∂Ω and G(W m ) ∈ W m . In addition, it is seen from the proof of Lemma 2.3 that if G(W ) ∈ W m for some 1 ≤ m ≤ n − 1, thenc 1,0 = c 1,0 , . . . ,c m,0 = c m,0 . Moreover, so far no logarithmic terms appear in the expansion of In particular, there exists a smooth function a n,0 on ∂Ω such that G(W n−1 ) − a n,0 • πd n ∈ W n .
Thus, by formula (2.10), a function of the form and µ is arbitrary.

Step 2: Approximations of w up to order O(d n )
We next show that the function W constructed in Proposition 2.5 is a good approximation for the solution of (2.1)-(2.2) up to an O(d n ) error at any derivative level. To show that the approximation is good in C 0 sense, we use the following lemma on sub-and supersolutions.
We now show that w −W = O(d n ) at all derivative level and in particular the regularity of w near the boundary.
Step 2: We prove that w is smooth in Ω \ Ω δ 1 for some δ 1 ∈ (0,δ 1 ). This will be achieved by using (2.16) to recast (1.1) near ∂Ω in a form suitable to apply a C 2,α result of Savin [31] for small perturbation solutions for elliptic equations.
Consider a point x 0 ∈ ∂Ω, which is taken without loss of generality as the origin, and set up coordinate axes so that the x n axis point toward Ω and the axes x 1 , . . . , x n−1 are along direction tangential to ∂Ω. Since ∂Ω is smooth, there exists ̺ 0 > 0 depending only on ∂Ω such that, after possibly shrinkingδ 1 , the wedges are subsets of Ω for ̺ > ̺ 0 and 0 < δ < δ 1 . It suffices to show that, for some δ 1 ∈ (0,δ 1 ) independent of x 0 , w is C 2,α regular in Ω x 0 ,2̺ 0 ,δ 1 , as smoothness follows from elliptic regularity theories.
It is important to observe that the function h(x) := − ln x n satisfies G(h) = 0 in {x n > 0} and hence in Ω x 0 ,̺ 0 ,δ 0 . Consider the function in Ω x 0 ,̺ 0 ,δ 0 in the viscosity sense.
To proceed, note that the fact that G(W ) ∈ W n in fact gives an estimate stronger than (2.18), namely where, by a slight abuse of notation, ∇ T now denotes the gradient along directions orthogonal to ∇d(x 0 + K −1 0 r 0 ·). Also, estimate (2.15) for s = 1 and j = 0 gives Therefore, the argument in Step 3 in fact gives which up on returning to w gives (2.15) for s = 1 and all j ≥ 0. We may then repeatedly differentiate (2.19) and apply the above argument to obtain (2.15) for all s ≥ 1, j ≥ 0.

2.4
Step 3: The coefficient function c n,0 and the rest of the proof The next step in the proof of Theorem 1.3 is to find W n = W +(c n,0 •π)d n where W is given by Proposition 2.8 such that w − W n = o(d n ) near ∂Ω. Once this is done, W n satisfies the condition of Lemma 2.3 with m = n + 1 (by Proposition 2.5), and we can resume the application of Lemma 2.3 to construct higher order approximations W n+1 , W n+2 , . . . of w. We introduce some notations. For small δ, set π δ = π| ∂Ω δ : ∂Ω δ → ∂Ω and let π −1 δ : ∂Ω → ∂Ω δ be its inverse. Define By estimate (2.14) in Proposition 2.8, the family {c δ } is bounded in C 0 (∂Ω) as δ → 0. However, because of the loss of d −ε in estimate (2.15) we do not know yet the boundedness of {c δ } in stronger norms. This is addressed in the next lemma.
By Proposition 2.5, we have that λ(−S(W n )) ∈ Γ k near ∂Ω and G(W n ) ∈ W n . Using Lemma 2.3, we can find a sequence {N p } p≥n+1 and coefficients {c p,q } p≥n+1,0≤q≤Np such that the functions satisfy λ(−S(W m )) ∈ Γ k near ∂Ω and G(W m ) ∈ W m . Note also that, if c n,1 vanishes in some ∂Ω∩B(x 0 , r 0 ), then it can be seen from the proof of Lemma 2.3 that, in the expansion of G(W n ), all coefficients carrying a non-trivial power of ln d actually vanish in ∂Ω ∩ B(x 0 , r 0 ), and so (2.11) implies that c n+1,1 , . . . , c n+1,N n+1 also vanish in ∂Ω ∩ B(x 0 , r 0 ).
It remains to prove (1.6). By estimate (2.21), for any smooth vector fields s ≥ 0, We may then argue as in Step 3 of the proof of Proposition 2.8 to obtain for any s, . We thus have that (1.6) holds for m = n and hence for all m ≤ n. Suppose by induction that (1.6) has been established for some m = m 0 ≥ n. Let us prove (1.6) for m = m 0 +1. As in Step 4 of the proof of Proposition 2.8, if Y 1 , . . . , Y s are vector fields which are orthogonal to and commute with ∇d, we have that We then use the argument in the proof of Lemma 2.9. In the coordinate system in which (2.5) holds, along the x n -axis, we have by (1.6) for m = m 0 that |∇ ℓ x ′ ϕ m 0 +1 (0, x n )| = O(d m 0 +γ ) for ℓ ≥ 0, and so the above PDE reduces to an ODE: Also by (1.6) for m = m 0 , we have |∂ ℓ n ϕ m 0 +1 (0, Integrating, we obtain ξ m 0 +1 = O(d m 0 +1−n+γ ), which gives ϕ m 0 +1 (0, x n ) = O(d m 0 +1+γ ). We may then argue as in Step 3 of the proof of Proposition 2.8 to obtain for any j ≥ 0 that which gives (1.6) for m = m 0 + 1. The proof is complete.
Proof of the first statement of Theorem 1.6. The proof is essentially the same as that of Theorem 1.3 except for a few changes which we list below.
1. By a rescaling, we may assume that f (1/2, . . . , 1/2) = 1. The expression of G is modified to 2. In Lemma 2.1, the conclusion is that G(W 0 ) ∈ W 0 and the polyhomogeneous expansion of G(W 0 ) has no logarithmic term, i.e. G(W 0 ) = ∞ p=0 (a p,0 • π)d p in the sense of Definition 2.2 (instead of in the sense of a convergent power series).

In
Step 2 of the proof of Proposition 2.8, the expression ofĜ is modified tô
Note that, since u ∈ C 1 (Ω), if Σ is a smooth surface in Ω, then the mean curvature H Σ of Σ (in a specified normal direction) with respect to g is well-defined. To dispel confusion, in our notation, the mean curvature is the trace of the second fundamental form. Moreover, if we denote byH Σ the mean curvature of Σ with respect to the Euclidean metricg, then where ν is theg-unit normal to Σ along the specified normal direction of Σ.
Since u is the solution of (1.1)-(1.2), by Theorem 1.3, there exists δ 0 > 0 such that u ∈ C ∞ (Ω \ Ω δ 0 ) and where the mean curvature is computed with respect to the normal pointing towards Ω δ . Using (3.2) and applying Theorem 1.2 to any Ω δ in place of Ω with some small δ > 0, the result follows immediately.
The rest of the section is devoted to the proof of Theorem 1.2. The plan is as follows. In Subsection 3.1, we set up an obstacle problem to look for a set Ω 1 ⊂ E ⊂ Ω ∪ Ω 1 which minimizes the perimeter function with respect to g. We show that either a component of ∂Ω has zero mean curvature or Σ = ∂E detaches from ∂Ω, in which case Σ is a C 1 hypersurface away from a singular set S ⊂ Σ of Hausdorff dimension at most n − 8 and Σ \ S has zero mean curvature (see Proposition 3.7). In Subsection 3.2, we then extend an argument made in [25] using an obstruction for the existence of a minimal hypersurface for metrics whose Schouten tensors belong to the negativeΓ 2 cone in a smooth context to the current situation (see Lemma 3.8) to deduce a contradiction and conclude the proof.

Existence and regularity of a minimal hypersurface
Let us recall the notion of perimeter with respect to g = g u = u 4 n−2g when u > 0 and u 2n n−2 ∈ W 1,1 (Ω ℓ ), where throughout the section, unless otherwise stated, all Sobolev spaces are defined using the measure given by the Euclidean metric. For an open set A ⊂ Ω ℓ , let X (A) be the set of smooth vector fields compactly supported in A. Recall that, for a measurable set E ⊂ Ω ℓ and an open set A ⊂ Ω ℓ , the perimeter P er g (E, A) of E in A with respect to the metric g is defined as the total variation of the distributional gradient ∇ g I E of the characteristic function I E of E in A, i.e.
When it is clear from the context, we simply write P er(E, A) instead of P er g (E, A). If P er(E, Ω ℓ ) < ∞ and ∂E is C 1 away from a set of zero (n − 1)-dimensional Hausdorff measure (defined using the metric g), then We also have the general inequality P er g (E 1 ∪ E 2 , A) + P er g (E 1 ∩ E 2 , A) ≤ P er g (E 1 , A) + P er g (E 2 , A). (3.4) We refer readers to e.g. [2,11,29] for more details.
As in the Euclidean case, it is an easy consequence of Lebesgue's dominated convergence theorem that the perimeter function is lower semi-continuous, namely if I E j → I E a.e. in Ω ℓ then P er(E, Ω ℓ ) ≤ lim inf j→∞ P er(E j , Ω ℓ ).
The following lemma gives a relation between the perimeter functions defined using two conformal metrics.  n−2g with ln u ∈ L ∞ (B r (x 0 )) ∩ W 1,1 (B r (x 0 )). For any measurable set E, we have ). Proof. We will only prove the first inequality. The proof of the second inequality is the same. For any w ∈ X (B r (x 0 )) with |w|g ≤ 1 in B r (x 0 ), we have where we have used |u − 2n n−2 w| g = u − 2(n−1) n−2 |w|g ≤ u −1 2(n−1) n−2 L ∞ (Br(x 0 )) . Taking the supremum on the left hand side, we conclude the proof.
Proof. The proof is standard and is included for completeness. Let E j be a minimizing sequence so that P er(E j , Ω ℓ ) → J. Then {I E j } is a bounded sequence in BV (Ω ℓ ). By the compactness theorem of BV functions, we may assume, after extracting a subsequence, that I E j converges in L 1 (Ω ℓ ) and almost everywhere to some BV function, which takes value in {0, 1} and so is the characteristic function of some measurable set E with Ω 1 ⊂ E ⊂ Ω ∪ Ω 1 . By the lower semi-continuity of the total variation of BV functions with respect to L 1 convergence, we have that J = P er(E, Ω ℓ ).
It is convenient to denote the symmetric difference of two sets A and B as A∆B = (A \ B) ∪ (B \ A). Lemma 3.3. Under the notations of Lemma 3.2, the set E is a local almost minimizer of P erg, namely there exist r 1 = r 1 (Ω) > 0 and C = C(n, Ω, ln u L ∞ (Ω ℓ ) ) > 0 such that P erg(E, B r (x 0 )) ≤ P erg(F, B r (x 0 )) + C(r + ess osc for all x 0 ∈ ∂E, r ∈ (0, r 1 ] and F such that Proof. Fix some x 0 ∈ ∂E and r 1 < 1 2 min i =j dist(∂Ω i , ∂Ω j ) so that the ball B r 1 (x 0 ) intersects at most one component of ∂Ω. We will only consider the case either B r 1 (x 0 ) ⊂ Ω or B r 1 (x 0 ) ∩ ∂Ω 1 = ∅. The other cases can be dealt with in the same way. Note that the above implies B r 1 (x 0 ) ⊂ Ω ∪ Ω 1 .
In view of known results for almost minimizing sets, we have the following two consequences of Lemma 3.3 when the u is suitably Dini continuous. In the sequel, we suppose that ω : (0, Then there exists a set S of Hausdorff dimension at most n − 8 such that ∂E \ S is a C 1 -hypersurface (containing the reduced boundary ∂ * E of E).
Proof. This is a direct consequence of Lemma 3.3 and known regularity for almost minimizing sets (see e.g. [ Pick a ∈ ∂E ∩ ∂Ω i . By Corollary 3.5, ∂E is a C 1 -hypersurface near x 0 . We then work in a local coordinate system x = (x ′ , x n ) = (x 1 , . . . , x n ) near x 0 ∼ = 0 such that the hypersurface ∂Ω i and ∂E are represented as 0 }, f (0) = f (0) = 0 and ∂f (0) = ∂f (0) = 0. The fact that ∂Ω 1 has non-negative mean curvature and that E solves the minimization problem J in (3.5) imply in the weak sense, where e(x ′ , ψ(x ′ ), ∂ψ(x ′ )) denotes the volume density of the metric induced by g on the graph of a function ψ, and s and p are dummy variables for ψ and ∂ψ.
∂Ω i coincides with ∂E and has zero mean curvature near x 0 . A standard argument then shows that the mean curvature of ∂Ω i is zero on all of ∂Ω i .
We conclude this section with the following proposition.
Note that in the above statement, the assumption ln u ∈ C 1 (Ω ℓ ) is used to defined the mean curvatures of ∂Ω and of ∂E \ S.
Proof. The result follows from Corollary 3.4, Lemma 3.6 and standard regularity theories for quasilinear elliptic equations.

An obstruction result and proof of Theorem 1.2
The next ingredient of the proof of Theorem 1.2 is the following obstruction result, which first appeared in [25] under stronger regularity assumptions.
Proof of the claim: Let w = 2 n−2 ln u so that Let U 2 be the set of symmetric n × n matrices whose eigenvalues belong to Γ 2 . Then U 2 is a convex cone with vertex at the zero matrix (see e.g. [24, Lemma B.1]). Let U * 2 be the cone dual to U 2 , i.e. U * 2 = symmetric n × n matrices a = (a ij ) : 1≤i,j≤n Note that U * 2 is a subcone of the cone of positive definite symmetric n × n matrices. Arguing as in the proof of [24, Proposition 2.5] and using that −λ(A u ) ∈Γ 2 in the viscosity sense, we have for any a ∈ C ∞ c (Ω; U * 2 ) that (This can be viewed as the weak sense of the inclusion −S(w) ∈Ū 2 .) Having (3.13) at hand, we can apply the proof of [24, Proposition 2.6] to obtain the claim. Since the situation in [24] was more sophisticated due to the generality considered there, we provide the details here for the readers' convenience.
Pulling this back to B, we get Sending ℓ → ∞ and using the relation between H and H u , we arrive at (3.11).
Proof of Theorem 1.2. Suppose by contradiction that such a function u exists. By Proposition 3.7, either a component ∂Ω i 0 of ∂Ω has zero mean curvature with respect to g, or there is a set E such that ∂E ⊂ Ω and, for some set S Hausdorff dimension at most n−8, ∂E \ S is a W 2,p -hypersurface with zero mean curvature with respect to g for every 1 ≤ p < ∞.
(In particular, if n ≤ 7, then S is empty.) We consider these cases in turn.
Case 1: ∂Ω i 0 has zero mean curvature with respect to g. Let Σ = ∂Ω i 0 andg be the metric on Σ induced byg. Since λ(A u ) ∈Γ 2 , by Corollary 3.9 we have ∆gu − n − 2 4(n − 1) |H Σ | 2 u − 1 (n − 2)u |∇gu| 2 ≥ 0 on Σ (3.17) in the weak sense whereH Σ is the mean curvature of Σ with respect to the flat ambient metric. Testing (3.17) against a constant function, we obtain that u is constant on Σ and H Σ = 0 a.e. on Σ. This gives a contradiction since there is no smooth closed minimal hypersurfaces in Euclidean space.
Case 2: There is a set E such that ∂E ⊂ Ω and, for some set S Hausdorff dimension at most n − 8, ∂E \ S is a W 2,p -hypersurface with zero mean curvature with respect to g for every 1 ≤ p < ∞ Let Σ = ∂E andg be the metric on Σ \ S induced byg. As before, we have ∆gu − n − 2 4(n − 1) |H Σ | 2 u − 1 (n − 2)u |∇gu| 2 ≥ 0 on Σ \ S (3.18) in the weak sense. If S is empty, we have as in the previous case thatH Σ = 0 a.e. on Σ. Since Σ is a W 2,p -hypersurface for all α ∈ (0, 1) and p ∈ (1, ∞), the regularity theory for minimal surfaces implies that Σ is smooth, which again gives a contradiction.
Suppose in the rest of the proof that S is non-empty, which implies n ≥ 8. Since S has Hausdorff dimension at most n − 8, for any ǫ > 0, one can find finite many (Euclidean) balls B r 1 (x 1 ), . . . , B rm (x m ) with m ≥ 1, x i ∈ S and 0 < r i < r 0 /4 such that S ⊂ , there is some j = j(x) such that x ∈ B 2r j (x j ) \ B r j (x j ) and |dζ j |g < 2 r j . Thus, by (3.8), Now multiplying (3.18) by 1 − ζ and integrating over Σ \ S and sending ǫ → 0, we get u is constant on Σ \ S andH Σ = 0 a.e. on Σ \ S. As in the case when S is empty, this implies that Σ \ S is smooth. Take a Euclidean ball B containing Σ such that ∂B touch Σ at some point, say x 0 . As in the proof of Corollary 3.5, this implies that x 0 ∈ Σ \ S. Since, near x 0 , Σ has zero mean curvature and ∂B has positive mean curvature with respect tog, we obtain a contradiction to the strong maximum principle. The proof is complete.