Solutions to the prescribed mean curvature equation

with Dirichlet conditions by variational methods

P. Amster, P. De Nápoli and M. C. Mariani Universidad de Buenos Aires

Abstract:

We apply variational methods in order to prove that the nonlinear system (1) admits at least one regular solution.

Introduction

We consider the problem of finding u ∈ H^{1}(Ω, IR^{3}) such that

(1) \{ \begin{matrix}△u = 2H(u)u_{x} ∧ u_{y} in Ω \\ u = γ in ∂Ω \end{matrix}

where H is a given continuously differentiable function and Ω is an open subset of IR^{2} with C^{1} boundary.

The system of differential equations above is the Dirichlet problem for the prescribed mean curvature equation in isothermal coordinates, also called H-system, which arises in the generalized Plateau’s problem (see [O]).

Hildebrant, Wente (see [W1,W2]) and others have studied the problem for constant H . For this case, Brezis-Coron and Struwe have shown by variational methods (see [BC], [S]) that if the boundary data is small and non constant, then there are at least two weak solutions. The problem has been also studied by fixed point methods (see [AMR]).

The goal of this work is to show that also for non constant H , the problem may be solved by variational methods, under rather general conditions on H and the boundary data.

Our main result is the following:

Theorem 1 Let γ ∈ H^{1/2}(∂Ω, IR^{3}) ∩ L^{∞}(∂Ω, IR^{3}) , R = ∥γ − c∥_{L^{∞}(∂Ω,IR^{3})} for some constant c, and B ⊂ IR^{3} the closed ball of radius R centered at c. Then, if H ∈ C^{1}(B) satisfies

∥H∥_{L^{∞}(B)} < \frac{1}{R}

then problem (1) admits a weak solution u ∈ H^{1}(Ω, IR^{3}) ∩ L^{∞}(Ω, IR^{3}) , with ∥u − c∥_{L^{∞}(B)} ≤ R.

Remark: By a result of Bethuel ([B]) weak solutions of the mean curvature equation are C^{2,α}(Ω) for H bounded and Lipschitz continuous. Moreover, by a result of Chanillo and Li ([CL]) the solutions are continuous up to the boundary.

Variational formulation of the problem

For simplicity, we may assume that c = 0 and extend H to a function of class C^{1}(IR^{3}) such that ∥H∥_{L^{∞}(IR^{3})} < \frac{1}{R} .

We define the vector field q : IR^{3} → IR^{3} :

q(x, y, z) = \frac{4}{3}(∫_{0}^{x} H(t, y, z)dt, ∫_{0}^{y} H(x, t, z)dt, ∫_{0}^{z} H(x, y, t)dt)

It is immediate that div q = 4H and |q(x, y, z)| ≤ \frac{4}{3}∥H∥_{L^{∞}}|(x, y, z)|.
We’ll work in the Banach space H^{1}(Ω, IR^{3}) ∩ L^{∞}(Ω, IR^{3}) , with the norm:
∥u∥ = ∥u∥_{H^{1}} + ∥u∥_{L^{∞}}

and consider energy functional

E(u) = D(u) + Q(u)

where D is the Dirichlet’s integral and Q is the functional

Q(u) = ∫_{Ω} q(u) · (u_{x} ∧ u_{y})

We’ll use the estimatives

(2)  |Q(u)| ≤ ∫_{Ω} |q(u)| · |u_{x} ∧ u_{y})| ≤ \frac{2}{3}∥H∥_{L^{∞}}∥u∥_{L^{∞}} ∫_{Ω} |∇u|^{2}

(3)  E(u) ≥ (1 − \frac{2}{3}∥H∥_{L^{∞}}∥u∥_{L^{∞}}) ∫_{Ω} |∇u|^{2}

Technical lemmas

In this section we present some technical facts. Lemma 2 Q is continuous in H^{1} ∩ L^{∞}. Lemma 3 We consider the functional

I(u, h) = ∫_{Ω} H(u) · (u_{x} ∧ u_{y}) · h

where u ∈ H^{1} ∩ L^{∞} and h ∈ L^{∞}.

Then I is continuous. \underline{Proof}

Lemma 2 and Lemma 3 follow from simple computations. Lemma 4

The functional E is Gateaux differentiable in H^{1}∩L^{∞} for variations in H_{0}^{1}∩L^{∞}, and

dE(u)(h) = 2 ∫_{Ω} ∇u · ∇h + ∫_{Ω} 4H(u) · (u_{x} ∧ u_{y}) · h

for every u ∈ H^{1} ∩ L^{∞}, h ∈ H_{0}^{1} ∩ L^{∞}. \underline{Proof}

Let us first take u, h ∈ C^{1} and define a function U : Ω × IR → IR^{3} by:

U(x, y, t) = u(x, y) + t · h(x, y)

We’ll write u^{t}(x, y) = U(x, y, t) .

We recall that the field q in IR^{3} is associated to the 2-form ω given by:

ω(X, Y) = (X ∧ Y) · q(u)

with

dω = div(q)du^{1} ∧ du^{2} ∧ du^{3}

Applying Stokes theorem to the cylinder C = Ω × [0, ε] , we obtain:

∫_{C} d(U^{∗}ω) = ∫_{∂C} U^{∗}ω

Moreover,

d(U^{∗}ω) = U^{∗}(dω) = U^{∗}(divq(u)du^{1} ∧ du^{2} ∧ du^{3})

= 4H(U) · \frac{∂(U^{1}, U^{2}, U^{3})}{∂(x, y, t)}dx ∧ dy ∧ dt = 4H(u^{t}) · (u_{x}^{t} ∧ u_{y}^{t}) · h

We have:

U^{∗}ω = ω(\frac{∂U}{∂x} , \frac{∂U}{∂y})dx ∧ dy = q(u^{t}) · (u_{x}^{t} ∧ u_{y}^{t})

on Ω × {ε} and Ω × {0} and

U^{∗}ω = ω(\frac{∂U}{∂σ} , \frac{∂U}{∂t})dσ ∧ dt = q(u^{t}) · (\frac{∂U}{∂σ} ∧ h)dσ ∧ dt

on the lateral surface ∂Ω × [0, ε] , where σ is the unit tangent vector on ∂σ. Thus,

∫_{0}^{ε} ∫_{Ω} 4H(u^{t}) · (u_{x}^{t} ∧ u_{y}^{t}) · hdx ∧ dy ∧ dt =

∫_{Ω×{ε}} q(u^{ε}) · (u_{x}^{ε} ∧ u_{y}^{ε})dx ∧ dy ∧ dt − ∫_{Ω×{0}} q(u) ∧ (u_{x} ∧ u_{y}) · h

+ ∫_{∂Ω×[0,ε]} q(u) · (\frac{∂U}{∂σ} ∧ h)dσ ∧ dt

Assuming that h = 0 on the boundary of Ω , we get:

∫_{0}^{ε} ∫_{Ω} 4H(u^{t}) · (u_{x}^{t} ∧ u_{y}^{t}) · h = Q(u^{ε}) − Q(u).

By density, this formula holds for u ∈ H^{1} ∩ L^{∞} and h ∈ H_{0}^{1} ∩ L^{∞}. By dominated convergence we conclude that

dQ(u)(h) = ∫_{Ω} 4H(u) · (u_{x} ∧ u_{y}) · h

which completes the proof.

Remark: By lemmas 2 and 3 dE is continuous, i.e. E ∈ C^{1}(H_{0}^{1} ∩ L^{∞}) .

Corollary 5

If u is a critical point of functional E for variations in H_{0}^{1} ∩ L^{∞} then u is a weak solution of (1).

Proof of Theorem 1

We fix R′ > R such that ∥H∥_{L^{∞}}R′ < 1 and consider the nonempty set:

K = {u ∈ H^{1}(Ω, IR^{3}) : u = γ in ∂Ω, ∥u∥ ≤ R′}

Then we obtain

Lemma 6

E achieves a minimum in K. \underline{Proof}

By (3), the functional E is bounded from below in K. Let E_{0} = inf_{v∈K} E(v) and consider a minimizing sequence (u^{n})

By (3) and Poincaré’s inequality u^{n} is bounded in H^{1} , and then we may suppose that

u^{n} → \underline{u} in H^{1}weakly

and

u^{n} → \underline{u} a.e.

By the continuity of q we conclude that:

q(u^{n}) → q(\underline{u}) a.e.

and since |q(u^{n})| ≤ \frac{4}{3}∥H∥_{L^{∞}} , we may also assume that:

q(u^{n}) → q(\underline{u}) in L^{∞} weak*
Set θ^{n} = u^{n} − u ∈ H_{0}^{1} . Then ∥θ^{n}∥_{L^{∞}} ≤ 2R′, and
E(u^{n}) = ∫_{Ω} |∇\underline{u}|^{2} + 2 ∫_{Ω} ∇u · ∇θ^{n}

+ ∫_{Ω} |∇θ^{n}|^{2} + ∫_{Ω} q(u^{n}) · (\underline{u}_{x} + θ_{x}^{n}) ∧ (\underline{u}_{y} + θ_{y}^{n})

Thus,

∫_{Ω} ∇u · ∇θ^{n} → 0

Moreover, being

∫_{Ω} q(u^{n}) · (\underline{u}_{x} ∧ θ_{y}^{n}) = − ∫_{Ω} θ_{y}^{n} · (q(u^{n}) ∧ \underline{u}_{x})

then

∫_{Ω} q(u^{n}) · (\underline{u}_{x} ∧ θ_{y}^{n}) → 0,

since θ_{y}^{n} → 0 weakly in L^{2} , and

|q(u^{n}) ∧ \underline{u}_{x}|^{2} ≤ (\frac{4}{3}|u^{n}|)^{2} · |\underline{u}_{x}|^{2} ≤ (\frac{4}{3}R′) |\underline{u}_{x}|^{2}.

In the same way, we conclude that

∫_{Ω} q(u^{n}) · (θ_{x}^{n} ∧ \underline{u}_{y}) → 0

We also have the estimative:

| ∫_{Ω} q(u^{n}) · θ_{x}^{n} ∧ θ_{y}^{n}| ≤ ∫_{Ω} 4∥H∥_{L^{∞}}∥u^{n})∥_{L^{∞}}|θ_{x}^{n} ∧ θ_{y}^{n}|

≤ \frac{2}{3}∥H∥_{L^{∞}}R′ ∫_{Ω} |∇θ^{n}|^{2} ≤ \frac{2}{3} ∫_{Ω} |∇θ^{n}|^{2}

Finally as q(u^{n}) → q(\underline{u}) weak* in L^{∞}:

∫_{Ω} q(u^{n}) · (\underline{u}_{x} ∧ \underline{u}_{y}) = ∫_{Ω} q(\underline{u}) · (\underline{u}_{x} ∧ \underline{u}_{y}) + o(1)

Then,

E(u^{n}) ≥ E(u) + o(1) + \frac{1}{3} ∫_{Ω} |∇θ^{n}|^{2}

and we can conclude that E(u) = E_{0} and ∫_{Ω} |∇θ^{n}|^{2} → 0 . Lemma 7

Let u ∈ K satisfy E(u) = inf_{v∈K} E(v) . Then u is a weak solution of (1).

\underline{Proof}

Let us fix a nonnegative test function η ∈ D(Ω) . Then u(1 − εη) ∈ K for ε ≥ 0 small enough.

As the function φ(ε) = E(u(1 − ε · η)) has a minimum in ε = 0 , then φ′(0) = dE(u)(−u · η) ≥ 0 , i.e.

2 ∫_{Ω} ∇u · ∇(−η · u) + ∫_{Ω} 4H(u)(u_{x} ∧ u_{y}) · (−ηu) ≥ 0 ∀ η ∈ D(Ω)

By Green’s formula,

∫_{Ω} △(|u|^{2}) · η = − ∫_{Ω} ∇(|u|^{2})∇(η) = − ∫_{Ω} 2(∇u · u) · ∇η

It follows that

−\frac{1}{2}△|u|^{2} + |∇u|^{2} + 2H(u)u · u_{x} ∧ u_{y} ≤ 0 in D′(Ω)

Then, as

|2H(u)u · (u_{x} ∧ u_{y})| ≤ ∥H∥_{L^{∞}}R′|∇u|^{2}

it follows that

−△|u|^{2} ≤ 0 in D′(Ω)

and then by Stampacchia’s maximum principle (see [GT]) we get:

sup_{Ω}|u| = sup_{∂Ω}|u| = R

Since R < R′, u is interior in K. It follows that dE(u)(v) = 0 for any v ∈ H_{0}^{1} ∩ L^{∞}, and by corollary 5 u is a weak solution of (1).

References

[AMR] Amster P. Mariani, M.C, Rial, D.F: Existence and uniqueness of H-System’s solutions with Dirichlet conditions. To appear in Nonlinear Analysis, Theory, Methods, and Applications.

[B] Bethuel, F: Un résultat de regularité pour les solutions de l’equation des surfaces à courboure moyenne prescrite. R.Acad. Sci. Paris Sér. I Math 314 (1992) n^{o} 13, 1003-100.

[B-C] Brezis, H., Coron, J., Multiple Solutions of H-Systems and Rellich’s conjecture. Comm.Pure Appl. Math 37, 1984, 149-187.

[CL] Chanillo S., Li Y.: Continuity of Solutions of Uniformly Elliptic Equations in IR^{2} . Manuscripta math. 77,415-433 (1992)

[GT] Gilbarg, D., Trudinger N.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag 1977

[O] Osserman, R: A Survey of Minimal Surfaces. Van Nostrand Reinhold Company, 1969

[S] Struwe, M.: Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, p. 180.

[W1] Wente, H.: An Existence Theorem for Surfaces of Constant Mean Curvature. Journal of Mathematical Analysis and Applications 26, p. 318-344 (1969)

[W2] Wente, H.: The differential equation ΔX = 2H(X_{u} ∧ X_{v}) with vanishing boundary values. Proceedings of the American Mathematical Society 50 (1975), 131-7.

P.Amster^{∗}, P. De Nápoli and M. C. Mariani^{∗} Dpto. de Matemática, Fac. de Cs. Exactas y Naturales, Universidad de Buenos Aires.

∗ Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)

Address for correspondence: Prof. P. Amster and M. C. Mariani, Dpto. de Matemática - Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad Universitaria (1428) Buenos Aires, Argentina

E-mail: pamster@dm.uba.ar - mcmarian@dm.uba.ar