EXISTENCE AND REGULARITY OF WEAK SOLUTIONS TO THE PRESCRIBED MEAN CURVATURE EQUATION FOR A NONPARAMETRIC SURFACE

It is known that for the parametric Plateau’s problem, weak solutions can be obtained as critical points of a functional (see [2, 6, 7, 8, 10, 11]). The nonparametric case has been studied for H = H(x,y) (and generally H = H(x1, . . . ,xn) for hypersurfaces in Rn+1) by Gilbarg, Trudinger, Simon, and Serrin, among other authors. It has been proved [5] that there exists a solution for any smooth boundary data if the mean curvature H ′ of ∂ satisfies H ′ ( x1, . . . ,xn )≥ n n−1 ∣∣H (x1, . . . ,xn)∣∣ (1.3)


Introduction
The prescribed mean curvature equation with Dirichlet condition for a nonparametric surface X : → R 3 , X(u, v) = (u, v, f (u, v)) is the quasilinear partial differential equation where is a bounded domain in R 2 , h : × R → R is continuous and g ∈ H 1 ( ).
The nonparametric case has been studied for H = H (x, y) (and generally H = H (x 1 , . . ., x n ) for hypersurfaces in R n+1 ) by Gilbarg, Trudinger, Simon, and Serrin, among other authors.It has been proved [5] that there exists a solution for any smooth boundary data if the mean curvature H of ∂ satisfies for any (x 1 , . . ., x n ) ∈ ∂ , and H ∈ C 1 ( , R) satisfying the inequality for any ϕ ∈ C 1 0 ( , R) and some > 0. They also proved a non-existence result (see [5,Corollary 14.13]): if H (x 1 , . . ., x n ) < (n/(n − 1))|H (x 1 , . . ., x n )| for some (x 1 , . . ., x n ) and the sign of H is constant, then for any > 0 there exists g ∈ C ∞ ( ) such that g ∞ ≤ and that Dirichlet's problem is not solvable.
We remark that the solutions obtained in [5] are classical.In this paper, we find weak solutions of the problem by variational methods.
We prove that for prescribed h there exists an associated functional to h, and under some conditions on h and g we find that this functional has a global minimum in a convex subset of H 1 ( ), which provides a weak solution of (1.1).We denote by H 1 ( ) the usual Sobolev space, [1].

The associated variational problem
Given a function f ∈ C 2 ( ), the generated nonparametric surface associated to this function is the graph of The mean curvature of this surface is where E, F , and G are the coefficients of the first fundamental form [4,9].For prescribed h, weak solutions of (1.1) can be obtained as critical points of a functional.
Proposition 2.1.Let J h : H 1 ( ) → R be the functional defined by

Behavior of the functional J h
In this section, we study the behavior of the functional J h restricted to T .For simplicity we write We will assume that h is bounded.

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Lemma 3.1.The functional A : T → R is continuous and convex.
Proof.Continuity can be proved by a simple computation.Let a, b ≥ 0 such that a + b = 1.By Cauchy inequality, it follows that and convexity holds.

Remark 3.2.
As A is continuous and convex, then it is weakly lower semicontinuous in T .
Lemma 3.3.The functional B is weakly lower semicontinuous in T .
Proof.Since h is bounded, we have From the compact immersion H 1 0 ( ) → L 1 ( ) and the continuity of Nemytskii operator associated to H in L 1 ( ), we conclude that B is weakly lower semicontinuous in T (see [3,12]).

Weak solutions as critical points of J h
Let us assume that g ∈ W 1,∞ , and consider for each k > 0, the following subset of T : M k is nonempty, closed, convex, bounded, then it is weakly compact.
Remark 4.1.As g ∈ W 1,∞ , taking p > 2 we obtain, for any f ∈ M k : Let ρ be the slope of J h in M k defined by (see [7,11]), then the following result holds. Proof.
Remark 4.3.Let J h be weakly semicontinuous and let M k be a weakly compact subset of T , then J h achieves a minimum f 0 in M k .By Lemma 4.2, ρ(f 0 , M k ) = 0.
As in [7], if f 0 has zero slope, we call it a ρ-critical point.The following result gives sufficient conditions to assure that if f 0 is a ρ-critical point, then it is a critical point of J h .Theorem 4.4.Let f 0 ∈ M k such that ρ(f 0 , M k ) = 0, and assume that one of the following conditions holds: We will prove that dJ h (f 0 Suppose that dJ h (f 0 )( ϕ) = 0, then dJ h (f 0 )(f 0 − g) < 0. If (i) holds, we immediately get a contradiction.On the other hand, if (ii) holds, there exists r > 1 such that g

Let us assume that ((∇(f
As a particular case, we may take h(u, v, z) = c(z − g(u, v)) for any c ≥ 0.

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Indeed, in all the examples the inequality dJ h (f )(f −g) ≥ 0 holds for any f ∈ M k , since Then the result follows immediately in example (b).In examples (a) and (c), being where c 1 is the Poincaré's constant associated to .Thus, the result holds for c ≤ 1/2 kc 2 1 .
Remark 4.5.As in the preceding examples, it can be proved that if dJ h (f )(f − g) ≥ 0 for any f ∈ M k , then g is a weak solution of (1.1).Indeed, if dJ h (g) = 0, from Theorem 4.4 it follows that ρ(g, M k ) > 0. As J h achieves a minimum in every M k , we may take k ≥ k n → 0, and

holds, and then dJ
and then it follows easily that dJ h (g) = 0.
Furthermore, for constant g we can see that if

Multiple solutions
In this section, we study the multiplicity of weak solutions of (1.1).Consider N k is a nonempty, closed, bounded, and convex subset of T , therefore N k is weakly compact.
Then we obtain the following theorem, which is a variant of the mountain pass lemma.
Theorem 5.1.Let f 0 ∈ N k be a local minimum of J h and assume that J h (f 1 ) < J h (f 0 ) for some ) We remark that f is not a local minimum of J h .This kind of f is called an unstable critical point.
The proof of Theorem 5.1 follows from Theorem 3 in [7] and Lemmas 5.2, 5.3, and 5.4 below.

Lemma 5.2. The functional
where N h is the Nemytskii operator associated to h.
and N h : L 2 → L 2 continuous, the result holds.
for n ≥ n 0 .Operating in the same way with ρ Proof.As f n ∈ N k , we may suppose that f n → f weakly.Let n = f n − f .We will see that n → 0. Indeed, (5.6) (5.9) Example 5.5.Now we will show with an example that problem (1.1) may have at least three ρ-critical points in N k .Let g = g 0 be a constant, and h(u, v, z) = −c(z−g 0 ) for some constant c > 0.Then, g 0 is a minimum of J h in M k 1 for k 1 small enough, and a local minimum in M k for any k ≥ k 1 .
Moreover, taking = B R , f (u, v) = g 0 + R 2 − (u 2 + v 2 ), it follows that and taking k = 2 √ πR it holds that f ∈ N k .Hence, if R is big enough, it follows that g 0 is not a global minimum in N k .Furthermore, we see that the proof of Lemma 4.2 may be repeated in N k , and then the minimum of J h in N k is a ρ-critical point.From Theorem 5.1 there is a third ρ-critical point which is not a local minimum of J h .