Existence and uniqueness of H-systems’s solutions with Dirichlet conditions

P. Amster, M. C. Mariani and D. F. Rial

Abstract: We give conditions on the boundary data in order to obtain a solution for the problem (1) below and we prove that in certain cases the solutions are isolated. The condition ∥H∥_{∞}∥g∥_{∞} ≤ 1 is not necessary.

Introduction

We consider the Dirichlet problem in the unit disc B = {(u, v) ∈ R^{2}; u^{2}+v^{2} < 1} for a vector function X : \overline{B} ⟶ R^{3} which satisfies the equation of prescribed mean curvature

(1) \{ \begin{matrix}ΔX = 2H(u, v)X_{u} ∧ X_{v}  in B \\ X = g  in ∂B \end{matrix}

where X_{u} = \frac{∂X}{∂u} , X_{v} = \frac{∂X}{∂v} , ∧ denotes the exterior product in R^{3} and H : \overline{B} ⟶ R is a given continuous function.

The problem above arises in the Plateau and Dirichlet problems for the prescribed mean curvature equation that has been studied in [1-3-4-5-6-7-8].

The main results are the following theorems:

Theorem 1 Let be 2 < p < ∞, and g ∈ W^{2,p}(B, R^{3}) such that g − a is small enough for a certain a ∈ R, then there exists a solution X ∈ W^{2,p} of (1).

Theorem 2 Let X be a solution of (1) in W^{2,p}, 1 < p < ∞. Then: i) If p ≤ 2 and X ∈ W^{1,∞} then X is isolated in (W^{1,∞} ∩ W^{2,p}, ∥ · ∥_{1,∞}) . ii) If p > 2 then X is isolated in W^{1,p}. Moreover, if \overline{Ω} is a bounded subset of

W^{1,p} the number of solutions of (1) in \overline{Ω} is finite.

Remark

For H constant Hildebrandt found a solution of (1) in W^{1,2} , for the case |H|∥g∥_{∞} ≤ 1 [3]. Theorem 1 gives a solution for p > 2 when g is close to a constant a. Since the equation in B depends only on the derivatives of X, we may suppose a = 0

1. Solution by fixed point methods

The systems (2) and (3) are equivalent to (1) with X = X_{0} + Y

(2) \{ \begin{matrix}ΔX_{0} = 0 in B \\ X_{0} = g  in ∂B \end{matrix}

(3) \{ \begin{matrix}ΔY = F (X_{0}, Y)  in B \\ Y = 0  in ∂B \end{matrix}

where F is given by

F (X_{0}, Y) = 2H(u, v)[(X_{0u} ∧ Y_{v} + Y_{u} ∧ X_{0v}) + (Y_{u} ∧ Y_{v} + X_{0u} ∧ X_{0v})]

For g ∈ W^{2,p} (B, R^{3}), (2) admits a unique solution in W^{2,p} (B, R^{3}). We can rewrite (3) as:

\{ \begin{matrix}L(X_{0})Y = F_{1}(X_{0}, Y)  in B \\ Y = 0  in ∂B \end{matrix}

where L(X_{0}) is the linear operator

L(X_{0})Y = ΔY − 2H(u, v)(X_{0u} ∧ Y_{v} + Y_{u} ∧ X_{0v})

and F_{1} is defined by

F_{1}(X_{0}, Y) = 2H(u, v)(Y_{u} ∧ Y_{v} + X_{0u} ∧ X_{0v}).

Remark L(X_{0}) is strictly elliptic and for p > 2 its coefficients are bounded since X_{0} ∈ W^{1,∞} (B, R^{3}) and H ∈ C(\overline{B}) .

We will use the following technical lemmas:

Lemma 4 Let be X_{0} ∈ W^{2,p}(B, R^{3}) , then

i) If 2 < p < ∞,

(4)  ∥F_{1}(X_{0}, Y_{1})∥_{p/2} ≤ 2∥H∥_{∞}(∥X_{0}∥_{1,p}^{2} + ∥Y_{1}∥_{1,p}^{2})

(5)  ∥F_{1} (X_{0}, Y_{1}) − F_{1} (X_{0}, Y_{2})∥_{p/2} ≤ 4∥H∥_{∞}R (∥Y_{1} − Y_{2}∥_{1,p})

for Y_{1}, Y_{2} ∈ B_{R}(0) ⊂ W^{1,p}.

ii) If p > 1 , X_{0} , Y_{1} , Y_{2} ∈ W^{1,p} ∩ W^{1,∞}

(6)  ∥F_{1}(X_{0}, Y_{1})∥_{p} ≤ 2∥H∥_{∞}(∥X_{0}∥_{1,∞}.∥X_{0}∥_{1,p} + ∥Y_{1}∥_{1,∞} .∥Y_{1}∥_{1,p})

(7)  ∥F_{1} (X_{0}, Y_{1}) − F_{1} (X_{0}, Y_{2})∥_{p} ≤ 4∥H∥_{∞}R (∥Y_{1} − Y_{2}∥_{1,p})

for Y_{1}, Y_{2} ∈ B_{R}(0) ⊂ W^{1,∞}. \underline{Proof}

∥F_{1}(X_{0}, Y_{1})∥_{p/2} ≤ 2∥H∥_{∞}(∥Y_{1u} ∧ Y_{1v}∥_{p/2} + ∥X_{0u} ∧ X_{0v}∥_{p/2})

and

∥Y_{1u} ∧ Y_{1v}∥_{p/2}^{p/2} ≤ ∫ |Y_{1u}|^{p/2}|Y_{1v}|^{p/2} ≤ ∥Y_{1u}∥_{p}^{p/2}∥Y_{1v}∥_{p}^{p/2}

The same inequality holds for X_{0} , and in order to prove (5), we also have that

∥Y_{1u} ∧ Y_{1v} − Y_{2u} ∧ Y_{2v}∥_{p/2} ≤ ∥Y_{1u} ∧ (Y_{1v} − Y_{2v})∥_{p/2} + ∥Y_{2v} ∧ (Y_{1u} − Y_{2u})∥_{p/2} and the proof follows.

In the same way we obtain (6) and (7), using that ∥a ∧ b∥_{p} ≤ ∥a∥_{∞}.∥b∥_{p} ♣

Lemma 5 If ∥X_{0}∥_{1,p} ≤ δ, 2 < p < ∞ with δ small enough, then there exists a constant C independent of X_{0} verifying

∥Y ∥_{2,p/2} ≤ C∥L(X_{0})Y ∥_{p/2} for all Y ∈ W^{2,p/2} ∩ W_{0}^{1,p/2} \underline{Proof}

Writing L(X_{0})Y = ΔY − R(X_{0})Y it follows that

∥L(X_{0})Y ∥_{p/2} ≥ ∥ΔY ∥_{p/2} − ∥R(X_{0})Y ∥_{p/2} ≥ \frac{1}{C_{1}} ∥Y ∥_{2,p/2} − ∥R(X_{0})Y ∥_{p/2}

where C_{1} is the constant provided by Lemma 9.17 in [2].

As ∥R(X_{0})Y ∥_{p/2} ≤ 4∥H∥_{∞}∥X_{0}∥_{1,p}∥Y ∥_{1,p} ≤ 4\overline{c}∥H∥_{∞}∥X_{0}∥_{1,p}∥Y ∥_{2,p/2} (where \overline{c} is the constant of Sobolev inmersion W^{2,p/2} ↪ W^{1,p}), the proof follows. ♣ Remark

The same Lemma 9.17 in [2] shows that if ∥g∥_{2,p} is small, then ∥X_{0} − g∥_{2,p} is small, and then also ∥X_{0}∥_{1,p}.

Proposition 6 Let be X_{0} ∈ W^{2,p} (B, R^{3}) with p > 2 , then the following problem

(8) \{ \begin{matrix}L(X_{0})Y = F_{1}(X_{0}, \overline{Y})  in B \\ Y = 0  in ∂B \end{matrix}

define a continuous map T : \overline{Y} ⟶ Y in W_{0}^{1,p}. Furthermore, if ∥X_{0}∥_{1,p} is small enough, there exists a number R such that T is a contraction in B_{R}(0) ⊂ W^{1,p}. \underline{Proof}

From Theorem 9.15 in [2], problem (8) admits a unique solution Y ∈ W^{2,p/2} , so the map T is well defined. Also we have from (5) that if \overline{Y} , \overline{Z} ∈ B_{R} ⊂ W^{1,p} then

∥Y − Z∥_{2,p/2} ≤ C(∥L(X_{0})(Y −Z)∥_{p/2} = C∥F_{1}(X_{0}, \overline{Y})−F_{1}(X_{0}, \overline{Z})∥_{p/2} ≤ 4C∥H∥_{∞}R∥\overline{Y} −\overline{Z}∥_{1,p}

and using the Sobolev inmersion of W^{2,p/2} in W^{1,p} the continuity of T follows. Also we have from (4)

∥Y ∥_{1,p} ≤ \overline{c} ∥Y ∥_{2,p/2} ≤ C\overline{c} ∥L(X_{0})Y ∥_{p/2} ≤ 2C\overline{c}∥H∥_{∞}(∥X_{0}∥_{1,p}^{2} + ∥\overline{Y} ∥_{1,p}^{2})

Choosing ∥X_{0}∥_{1,p} and R small enough, we obtain

4C∥H∥_{∞}R < 1

∥Y ∥_{1,p} ≤ R

and T is a contraction in B_{R}. ♣ Remark

Note that if Y is a fixed point of T , then Y ∈ W^{2,p}, and we obtain Theorem 1 as an immediate consequence of Proposition 6.

2. Local unicity of the solutions Now we will prove theorem 2. Let X_{0} be a solution of (1). If Y is another solution of (1) then Z = Y − X_{0}

satisfies

\{ \begin{matrix}L(X_{0})Z = 2H(u, v)Z_{u} ∧ Z_{v} in B \\ Z = 0  in ∂B \end{matrix}

We consider the associated problem

\{ \begin{matrix}L(X_{0})Z = 2H(u, v)\overline{Z}_{u} ∧ \overline{Z}_{v} in B \\ Z = 0  in ∂B \end{matrix}

For ii), in the same way as before, we obtain

∥Z∥_{1,p} ≤ 2C(X_{0})\overline{c}∥H∥_{∞}∥\overline{Z}∥_{1,p}^{2}

and if \overline{Z}_{1}, \overline{Z}_{2} ∈ B_{R}(0) ⊂ W^{1,p} then

∥Z_{1} − Z_{2}∥_{1,p} ≤ 4C(X_{0})\overline{c}∥H∥_{∞}R∥\overline{Z}_{1} − \overline{Z}_{2}∥_{1,p}

If we choose R verifying

4C(X_{0})\overline{c}∥H∥_{∞}R < 1

the map \overline{T} : \overline{Z} ⟶ Z is a contraction in B_{R}, so X_{0} is isolated. Remark From Sobolev inmersions \overline{T} is compact in W^{1,p} and being any solution of (1) a fixed point of \overline{T} , we conclude that the number of solutions in B_{R} is finite for every R.

In the same way we prove i) using (6) and (7). ♣

Remark The number R in the theorems above can be estimated in terms of the Sobolev inmersions constants, ∥H∥_{∞} and ∥g∥_{2,p}.

References

[1] H. Brezis, J. Coron, Multiple solutions of H systems and Rellich’s conjecture, Comm. Pure Appl. Math. 37 (1984), 149-187.

[2] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer- Verlag (1983).

[3] S. Hildebrandt, On the Plateau problem for surfaces of constant mean curvature. Comm. Pure Appl. Math. 23 (1970) 97-114.

[4] E. Lami Dozo, M.C. Mariani, A Dirichlet problem for an H system with variable H . Manuscripta Math. 81 (1993), 1-14.

[5] M. C. Mariani, D. Rial, Solutions to the mean cuvature equation by fixed point methods. Bulletin of The Belgian Mathematical Society - Simon Stevin (1997).

[6] M. Struwe, Plateau ’s problem and the calculus of variations, Lecture Notes Princeton Univ. Press (1988).

[7] M. Struwe, Multiple solutions to the Dirichlet problem for the equation of prescribed mean curvature, Research papers published in honor of J. Moser’s 60 Birthday, Rabinowitz, P and Ehnder, E. (Eds.), Boston, Academic Press, 639-666 (1990).

[8] Wang Guofang: The Dirichlet problem for the equation of prescribed mean curvature, Analyse Nonlinéaire 9 (1992), 643-655.

P.Amster and D. F. Rial Dpto. de Matemática Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad

Universitaria (1428) Capital, Argentina M. C. Mariani Dpto. de Matemática Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad

Universitaria (1428) Capital, Argentina CONICET

Address for correspondence: Prof. M. C. Mariani, Dpto. de Matemática Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad Universitaria (1428) Capital, Argentina

E-mail: mcmarian@dm.uba.ar