Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results

It is well known that the \textit{Serrin condition} is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of $\mathbb{R}^n$ with certain regularity. In this paper we investigate this fact for the vertical mean curvature equation in the product $ M^n \times \mathbb{R}$. Precisely, given a $\mathscr{C}^2$ bounded domain $\Omega$ in $M$ and a function $H = H (x, z) $ continuous in $\overline{\Omega}\times\mathbb{R}$ and non-decreasing in the variable $z$, we prove that the \textit{strong Serrin condition} $(n-1)\mathcal{H}_{\partial\Omega}(y) \geq n\sup\limits_{z\in\mathbb{R}} |H(y,z)| \ \forall \ y\in\partial\Omega$, is a necessary condition for the solvability of the Dirichlet problem in a large class of Riemannian manifolds within which are the Hadamard manifolds and manifolds whose sectional curvatures are bounded above by a positive constant. As a consequence of our results we deduce Jenkins-Serrin and Serrin type sharp solvability criteria.


Introduction
We denote by M a complete Riemannian manifold of dimension n ≥ 2 and let Ω be a domain in M . The focus of our work is the prescribed mean curvature equation for vertical graphs in M × R, this is, where H is a continuous function over Ω × R and non-decreasing in the variable z, W = 1 + ∇u(x) 2 and the quantities involved are calculated with respect to the metric of M . In a coordinates system (x 1 , . . . , x n ) in M , it follows that where (σ ij ) is the inverse of the metric (σ ij ) of M , u i = n j=1 σ ij ∂ j u are the coordinates of ∇u and ∇ 2 ij u(x) = ∇ 2 u(x) ∂ ∂xi , ∂ ∂xj . We will denote by Q the operator defined by Qu = Mu − nH(x, u).
We notice that the matrix of the operator M is given by A = 1 W g, where g is the induce metric on the graph of u. This implies that the eigenvalues of A are positive and depends on x and on ∇u. Hence, M is locally uniformly elliptic. Furthermore, if Ω is bounded and u ∈ C 1 (Ω), then M is uniformly elliptic in Ω (see [19] for more details).
It has been proved in chronological order by Finn [9], Jenkins-Serrin [14] and Serrin [18], that the very well known Serrin condition is a necessary condition for the solvability of the Dirichlet problem for equation (1) in bounded domains of R n .
Dirichlet problems for equations whose solutions describe hypersurfaces of prescribed mean curvature has been also studied outside of the Euclidean space. Several works have considered a Serrin type condition that provides some existence theorem in their respective context (see [1], [2], [7], [8], [13], [16], [17] and [19] as examples). However, non-existence theorem has been only investigated in a few cases that we summarize below.
For instance, P.-A Nitsche [17] was concerned with graph-like prescribed mean curvature hypersurfaces in hyperbolic space H n+1 . In the half-space setting, he studied radial graphs over the totally geodesic hypersurface S = {x ∈ R n+1 ; (x 0 ) 2 + · · · + (x n ) 2 = 1}. He established an existence result if Ω is a bounded domain of S of class C 2,α and H ∈ C 1 (Ω) is a function satisfying sup Ω |H| ≤ 1 and |H(y)| < H C (y) everywhere on ∂Ω, where H C denotes the hyperbolic mean curvature of the cylinder C over ∂Ω. Furthermore he showed the existence of smooth boundary data such that no solution exists in case of |H(y)| > H C (y) for some y ∈ ∂Ω under the assumption that H has a sign. We observe that his results does not provide Serrin type solvability criterion.
Also, E. M. Guio-R. Sa Earp [12,13] considered a bounded domain Ω contained in a vertical totally geodesic hyperplane P of H n+1 and studied the Dirichlet problem for the mean curvature equation for horizontal graphs over Ω, that is, hypersurfaces which intersect at most only once the horizontal horocycles orthogonal to Ω. They considered the hyperbolic cylinder C generated by horocycles cutting ortogonally P along the boundary of Ω and the Serrin condition, H C (y) ≥ |H(y)| ∀ y ∈ ∂Ω. They obtained a Serrin type solvability criterion for prescribed mean curvature H = H(x) and also proved a sharp solvability criterion for constant H. To the best of our knowledge, no other Serrin-type solvability criterion has been proved outside of the Euclidean setting.
In this paper we generalize the aforementioned non-existence result in the M × R context. More precisely, we prove the following 1 : Theorem 1 (main theorem). Let Ω ⊂ M be a bounded domain whose boundary is of class C 2 . Let H ∈ C 0 (Ω × R) be a function either non-positive or non-negative and non-decreasing in the variable z. Let us assume that there exists y 0 ∈ ∂Ω such that Suppose also that cut(y 0 )∩Ω = ∅. Furthermore, assume that the radial curvature over the radial geodesics issuing from y 0 and intersecting Ω is bounded above by for all x ∈ Ω.
The statement ensures that the strong Serrin condition is a necessary condition for the solvability of the Dirichlet problem for equation (1). Some direct consequences inferred from our main non-existence theorem are stated as follows.
Corollary 2. Let M be a Cartan-Hadamard manifold and Ω ⊂ M a bounded domain whose boundary is of class C 2 . Let H ∈ C 0 (Ω × R) be a function either non-negative or non-positive and non-decreasing in the variable z. Suppose there exists y 0 ∈ ∂Ω such that Corollary 3. Let M be a simply connected and compact manifold whose sectional curvature satisfies 1 K0 and whose boundary is of class C 2 . Let H ∈ C 0 (Ω × R) be a function either non-negative or non-positive and nondecreasing in the variable z. Suppose there exists y 0 ∈ ∂Ω such that We remark that the assumption in the above statement guarantees that the injectivity radius of M is greater than π 2 √ K0 .

Sharp solvability criteria
We now want to highlight Serrin type solvability criteria derived from the combination of our non-existence results with existence results obtained by others [19,1] and by the authors [3,4].
Firstly, we observe that the combination of corollary 2 with the existence theorem from Aiolfi-Ripoll-Soret [1, Th. 1 p. 72] for the minimal case shows that the sharp solvability criterion of Jenkins-Serrin [14, Th. 1 p. 171] also holds in Cartan-Hadamard manifolds: Theorem 4 (Sharp Jenkins-Serrin-type solvability criterion). Let M be a Cartan-Hadamard manifold and Ω ⊂ M a bounded domain whose boundary is of class C 2,α for some α ∈ (0, 1). Then the Dirichlet problem for equation Theorem 5 (Sharp Serrin-type solvability criterion). Let M be a simply connected and compact manifold whose sectional curvature satisfies 1 4 and whose boundary is of class C 2,α for some α ∈ (0, 1). Then the Dirichlet problem for equation (1) in Ω has a unique solution for every constant H and arbitrary continuous boundary data if, and only if, (n − 1)H ∂Ω ≥ n |H|. Theorem 6 (Serrin type solvability criterion 1). Let Ω ⊂ H n be a bounded domain with ∂Ω of class C 2,α for some α ∈ (0, 1). Let H ∈ C 1,α (Ω × R) be a function satisfying ∂ z H ≥ 0 e 0 ≤ H ≤ n−1 n em Ω × R. Then the Dirichlet problem for equation (1) has a unique solution u ∈ C 2,α (Ω) for every ϕ ∈ C 2,α (Ω) if, and only if, the strong Serrin condition (3) holds.
in addition to the strong Serrin condition (3), then the Dirichlet problem for equation (1) is solvable for arbitrary boundary data sufficient smooth. This result in combination with corollary 2 yields the following generalization in the C 2,α class of a theorem of Serrin [18,Th. p. 484] in the Euclidean space: Theorem 8 (Serrin type solvability criterion 2). Let M be a Cartan-Hadamard manifold and Ω ⊂ M a bounded domain whose boundary is of class C 2,α for some α ∈ (0, 1). Suppose that H ∈ C 1,α (Ω × R) is either non-negative or non-positive in Ω × R, ∂ z H ≥ 0 and Then the Dirichlet problem for equation (1) has a unique solution u ∈ C 2,α (Ω) for every ϕ ∈ C 2,α (Ω) if, and only if, the strong Serrin condition (3) holds.

Proof of the main non-existence theorem
The proof of theorem 1 is based in two results that will be proved in the sequel. The following fundamental proposition can trace its roots back to the work of Finn Proof. By way of contradiction, suppose that m = max In view of the function H is non-decreasing in z and m > 0, we have As a consequence of the maximum principle (see [10,Th. 10 Dividing the expression by t and passing to the limit as t goes to zero it follows that ∂u ∂N ≤ −∞. This is a contradiction since The next lemma plays a fundamental role in this paper. In this lemma we establish a height a priori estimate for solutions of equation Mu = nH(x, u) in Ω in those points of ∂Ω on which the strong Serrin condition (3) fails.

Lemma 11.
Let Ω ⊂ M be a bounded domain whose boundary is of class C 2 .
Then for each ε > 0 there exists a > 0 depending only on ε, H ∂Ω (y 0 ), the geometry of Ω and the modulus of continuity of H(x, k) in y 0 , such that Proof. We proceed the proof in two steps. Firstly, we will find an estimate for u(y 0 ) depending on k and sup ∂Ba(y0)∩Ω u for some a that does not depend on u.
Secondly, we will get an upper bound for sup ∂Ba(y0)∩Ω u in terms of sup ∂Ω\Ba(y0) u.
First of all note that from (5) there exists ν > 0 such that Let R 1 > 0 be such that ∂B R1 (y 0 ) ∩ Ω is connected and Note also that we can construct an embedded and oriented hypersurface S, tangent to ∂Ω at y 0 and whose mean curvature with respect to the normal pointing inwards Ω at y 0 satisfies .
We know that for some τ > 0 the map is a diffeomorphism for each 0 ≤ t < τ , and so S t := Φ t (S) is parallel to S. Let us consider the distance function d(x) = dist(x, S). Let 0 < R 2 < min{τ, R 1 } be such that We now fix a < R 2 . For 0 < ǫ < a we set We choose φ ∈ C 2 (ǫ, a) satisfying We also require that φ ′3 ν + φ ′′ =0 in (ǫ, a). Let v = max k, sup Let us fix x ∈ Ω ǫ . A straightforward computation yields Since v ≥ k and H is non-decreasing in z it follows that H(x, v) ≥ H(x, k). Hence, By means of the properties of φ we have and by the assumption on the sign of H we obtain Furthermore, where (a) follows directly from (10), (b) from (9), (c) from (7) and (d) from (8).
Using this estimate on (11) we have Let us now define φ explicitly by 2 We observe that φ satisfies P1-P4 and that φ ′3 ν + φ ′′ = 0 for each ǫ < t < a. Then, Qv < 0 in Ω ǫ . From proposition 10 we deduce that In particular, where γ y0 (ǫ) = exp y0 (ǫN y0 ). Since this estimate holds for each 0 < ǫ < a, we can pass to the limit as ǫ goes to zero to obtain Step 2.
We observe that, in fact, this estimate holds for each a such that ∂B a (y 0 ) ∩ Ω is connected. We use (15) in (13)  It is easy to see that lim a→0 ψ(a) = 0. Hence, for each ε > 0, a can be choose small enough to satisfy ψ(a) + 2a ν < ε.
Remark 12. In the case where H = H(x), u(y 0 ) < sup where a is chosen as before.
At last we are able to prove theorem 1.
Proof of the main non-existence theorem. Obviously we can suppose that H ≥ 0. Then, (n − 1)H ∂Ω (y 0 ) < nH(y 0 , k) for some k ∈ R since H is non-decreasing in z. Let ε > 0 and ϕ ∈ C ∞ (Ω) such that ϕ = k in ∂Ω \ B a (y 0 ) and ϕ(y 0 ) = k + ε. Hence, no solution of equation (1) in Ω could have ϕ as boundary values because such a function does not satisfy the estimate (6).