RADIALLY SYMMETRIC SOLUTIONS OF AN ANISOTROPIC MEAN CURVATURE EQUATION MODELING THE CORNEAL SHAPE

. We prove existence and uniqueness of classical solutions of the anisotropic prescribed mean curvature problem where a,b > 0 are given parameters and B is a ball in R N . The solution we ﬁnd is positive, radially symmetric, radially decreasing and concave. This equation has been proposed as a model of the corneal shape in the recent papers [13, 14, 15, 18, 17], where however a linearized version of the equation has been investigated.


1.
Introduction. This note is devoted to the study of the existence, the uniqueness and the qualitative properties of classical solutions of the anisotropic prescribed mean curvature problem      −div ∇u where a > 0 and b > 0 are given constants and B = B(x 0 , R) is the open ball in R N of center x 0 and radius R. This problem has been recently proposed in [13,14,15,18,17] as a mathematical model for the geometry of the human cornea: we refer to these articles for further references on the subject. However, in all these papers a simplified version of (1) has been investigated, where the curvature operator div ∇u/ 1 + |∇u| 2 298 CHIARA CORSATO, COLETTE DE COSTER AND PIERPAOLO OMARI has been replaced by its linearization ∆u around 0. In particular, it has been proved in [14] that, if with I n (n = 0, 1) the n-order modified Bessel functions of the first kind, and B is a unit disk in R 2 , then the (physically relevant) problem has a unique radially symmetric solution which is the uniform limit of a sequence of successive approximations. We stress that, in the one-dimensional case, these limitations on the parameters have later been removed in [18], where it has also been pointed out the interest of studying the complete model (1). Like in [6], dealing with the one-dimensional case, we tackle here the fully nonlinear problem (1) and we prove the existence of a unique solution for the whole range of positive parameters a, b and for any radius R. Precisely, we first prove a uniqueness result for a more general problem in Ω, where the ball B is replaced by any bounded domain Ω in R N with a Lipschitz boundary ∂Ω. Then we establish the existence of a classical radially symmetric solution of (1), by solving the problem This radial solution is therefore the unique solution of (1). We also prove that it is positive, radially decreasing and concave. Our result is stated in the following theorem. Then there exists a unique solution u ∈ C 2 (B) of (1), which in addition satisfies: It is well-known that in general the study of mean curvature problems requires much care because of the possible occurrence of derivative blow-up phenomena, even in the onedimensional case (see, e.g., [5,12] and the references therein). However, in this case, we can show that an a priori bound in C 1 for a class of solutions of can be obtained by an elementary argument which exploits the structure of the equation and the qualitative properties -positivity, monotonicity and concavity -of the solutions themselves. These estimates enable us to use a shooting method on a modification of equation (4) in order to prove the existence of a solution of (3) and hence of a radially symmetric solution of (1).
The proof of the uniqueness of solutions of (2) is instead based on converting, by a suitable change of variable, the original problem into a variational inequality, for which the uniqueness of solutions can be easily established by using a monotonicity argument.
The numerical solution of the quasilinear problem (3), but limited to the dimension N = 1, has been considered in [6] and [16], by using different approximation techniques.
We wish to mention that part of our results extends to the N -dimensional problem in a general domain: achieving this conclusion however requires a quite different approach, apparently even in the "simple" case of an annulus. This topic is discussed in [7].
We finally recall that anisotropic prescribed mean curvature equations have been recently considered, driven by different motivations, in [8, 9,  Proof. The proof consists of two steps. Step for all w ∈ C 1 (Ω) with min Ω w > 0 and w = 1 on ∂Ω. Indeed, it is easy to verify that, if u ∈ C 2 (Ω) is a solution of (2), then v = exp(−bu) satisfies Pick any w ∈ C 1 (Ω), with min Ω w > 0 and w = 1 on ∂Ω, multiply the equation in (6) by w − v and integrate by parts. The convexity and the differentiability in Step 2. Uniqueness. Let us show that problem (2) has at most one solution u ∈ C 2 (Ω). Suppose that u 1 , u 2 ∈ C 2 (Ω) are solutions of (2). Then, as v 1 = exp(−bu 1 ), v 2 = exp(−bu 2 ) satisfy (5), we have in particular Summing up and rearranging we get The strict monotonicity of the logarithm function yields v 1 = v 2 and hence u 1 = u 2 .
Step 1. A modified problem. Set c = exp(2b 2 /a) − 1 and define a function ϕ : R → R by Note that if |s| > c is bounded, bounded away from 0 and satisfies, for all s ∈ R, Let us introduce the initial value problem

is a solution of (8) if and only if it is a solution of
In addition, as Step 2. Global existence, uniqueness and continuous dependence. We are going to show that, for any given d ∈ R, the initial value problem (8), or equivalently (9), has a unique solution v ∈ C 2 ([0, R]), which, moreover, depends continuously on the initial datum d and satisfies (10).
The local existence and uniqueness of solutions of (8) can be proved as follows. One first observes that, for any δ > 0 small enough, the operator S, defined by is a contraction in the space C 1 ([0, δ]), endowed with the usual norm; here the global Lipschitz continuity of ϕ −1 is in particular exploited. The C 2 -regularity up to 0 of the fixed point is verified directly; thus it gives rise to a local solution of (8). The continuous dependence of local solutions on the initial datum d is a consequence of the continuous dependence of the fixed points of S on the parameter d.
Let us now denote by g : the function which appears at the right-hand side of the equation in (9). Since g is locally Lipschitz continuous in [δ, R] × R × R and grows linearly in (s, ξ) ∈ R × R uniformly in r ∈ [δ, R], any local solution of (9) can be uniquely continued to [0, R]. Finally, the continuous dependence of these global solutions on the initial datum d is a standard consequence of the uniqueness.
Step 3. Qualitative properties. We shall show that, for any given d < b/a, the solution v of (9), defined on [0, R], satisfies conditions (ii) and (iii).
Let us prove (iii). Assume by contradiction that there exists r ∈ ]0, R] such that v (r) ≥ 0.
Step 4. Solvability. The map T : where v is the solution of (9), is continuous and satisfies, according to condition (ii), In order to show that v is the desired solution of (3), we prove that v also satisfies v ∞ ≤ exp(2b 2 /a) − 1 = c, or equivalently, by (iii), v (R) ≥ −c. From the equation in (9) we easily get and hence v (r) v (r) 1 + v (r) 2 ≤ −b v (r). Integrating this inequality over [0, R], we obtain that is |v (R)| < exp(2b 2 /a) − 1 = c. This concludes the proof.
Proof of Theorem 1.1. Proposition 2.1 guarantees that problem (1) has at most one solution u ∈ C 2 (B). Proposition 2.2 ensures that problem (3) has a solution v ∈ C 2 ([0, R]). Setting u(x) = v(|x − x 0 |) for all x ∈ B, a simple calculation shows that u ∈ C 2 (B) is the unique solution of (1) which satisfies the cited conditions. Thus Theorem 1.1 follows.