Journal of Inequalities in Pure and
Applied Mathematics
TWO ITERATIVE SCHEMES FOR AN H-SYSTEM
PABLO AMSTER AND MARíA CRISTINA MARIANI
FCEyN - Departamento de Matemática
Universidad de Buenos Aires
Ciudad Universitaria, Pabellón I
(1428) Buenos Aires, Argentina
CONICET
EMail: pamster@dm.uba.ar
Department of Mathematical Sciences
New Mexico State University Las Cruces
NM 88003-0001, USA
EMail: mmariani@nmsu.edu
volume 6, issue 1, article 5,
2005.
Received 1 November, 2004;
accepted 15 December, 2004.
Communicated by: L. Debnath
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
243-04
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Abstract
Two iterative schemes for the solution of an H-system with Dirichlet boundary
data for a revolution surface are studied: a Newton imbedding type procedure,
which yields the local quadratic convergence of the iteration and a more simple
scheme based on the method of upper and lower solutions.
2000 Mathematics Subject Classification: 34B15, 35J25.
Key words: H-systems, Newton Imbedding, Upper and Lower solutions, Iterative
methods.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
A Newton Imbedding Type Procedure . . . . . . . . . . . . . . . . . . . 6
3
Upper and Lower Solutions for Problem (1.4) . . . . . . . . . . . . . 11
References
Two Iterative Schemes for an
H-system
Pablo Amster and
María Cristina Mariani
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1.
Introduction
The prescribed mean curvature equation with Dirichlet condition for a vector
function X : Ω −→ R3 is given by the following nonlinear system of partial
differential equations:
(
4X = 2H(X)Xu ∧ Xv in Ω
(1.1)
X = X0
on ∂Ω
where Ω ⊂ R2 is a bounded domain, ∧ denotes the exterior product in R3 ,
H : R3 −→ R is a given continuous function and X0 is the boundary data.
The parametric Plateau and Dirichlet problems have been studied by different authors (see [3, 4], [7] – [9]). Nonparametric and more general quasilinear
equations are considered in [1, 2, 6].
We shall consider the particular case of a revolution surface
X(u, v) = (f (u) cos v, f (u) sin v, g(u))
with f, g ∈ C 2 (I) such that f > 0 and g 0 > 0 over the interval I ⊂ R. Without
loss of generality we may assume that I = (0, L), and problem (1.1) becomes
 00
f − f = −2H(f, g)f g 0 in I





 g 00 = 2H(f, g)f f 0
in I
(1.2)

f (0) = α0
f (L) = αL





g(0) = β0
g(L) = βL
Two Iterative Schemes for an
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María Cristina Mariani
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where H : R2 −→ R is a given continuous function, and α0 , αL > 0, β0 < βL
are fixed real numbers.
It is easy to see that any solution of (1.2) verifies the equality
2
2
(f 0 ) + (g 0 ) = f 2 + c.
Hence, the isothermal condition
|Xu | − |Xv | = Xu Xv = 0
holds if and only if c = 0. In this case, H is the mean curvature of the surface
parameterized by X (see [8]).
We shall study problem (1.2) for a surface with connected boundary, namely
(1.3)
 00
f − f = −2H(f, g)f g 0





 g 00 = 2H(f, g)f f 0
in I

f (L) = αL ,





f (0) = g 0 (0) = 0.
g(L) = βL
in I
g 0 (0) = 0,
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g(L) = βL
e
problem (1.3) easily reduces to a single equation; indeed, if H(t)
=
the following integral formula holds for g:
Z L
e (s))ds.
g(t) = βL − 2
H(f
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María Cristina Mariani
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In particular, if H depends only on the radius f , from the equality
g 00 = 2H(f )f f 0 ,
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Thus, problem (1.3) is equivalent to the equation

0
 f 00 − f = −4H(f )f H(f
e 2 (f )
e ) = −2 H
(1.4)
 f =α
in I
on ∂I
e
with α(t) := αLL t. We remark that if αL = 0 then g 0 (L) = 2H(0)
= 0, which
corresponds to a surface without boundary.
The paper is organized as follows. In Section 2, a Newton imbedding type
procedure for problem (1.4) is considered, which yields the local quadratic convergence of the iteration. In Section 3 we construct a more simple convergent
scheme based on the existence of an ordered pair of a lower and an upper solution.
Two Iterative Schemes for an
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Pablo Amster and
María Cristina Mariani
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2.
A Newton Imbedding Type Procedure
Throughout this section we shall assume the following condition:
00
1 + Lπ Lπ − δ0
2
e
(2.1)
H
(x) ≤
for − k ≤ x ≤ k + αL ,
2
where δ0 <
π
L
and k satisfies:
kδ0 > L
1/2
0
2
e
2 H
(α) − α
.
L2
Remark 1. A straightforward application of Leray-Schauder degree theory
proves that if condition (2.1) holds then there exists at least one solution of
(1.4), which is unique in the set
Two Iterative Schemes for an
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Pablo Amster and
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K = {u ∈ H 2 (I) : −k ≤ u ≤ k + αL }.
00
0
2
2
e
e
Remark 2. As H
(0) = 0 and 2 H (α) − α 2 → 0 for αL → 0, we
L
deduce that for any H there exists a positive number α∗ such that (2.1) holds
when αL < α∗ .
In order to solve equation (1.4) in an iterative manner, we shall embed it in
a family of problems

0

2
 f 00 + λ 2 H
e
(f ) − f = 0
(1.4λ )

 f (0) = 0, f (L) = α .
L
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A simple computation shows that if Sλ is the semilinear operator given by
0
00
2
e
Sλ (f ) = f + λ 2 H (f ) − f ,
then the following estimate holds for any f, g ∈ K such that f = g on ∂I:
(2.2)
kf 0 − g 0 kL2 ≤
1
kSλ (f ) − Sλ (g)kL2 .
δ0
Hence, if f λ is the (unique) solution of (1.4λ ) in K, we have that
0
1
1
e 2 (α) − α .
k(f λ − α)0 kL2 ≤ kSλ (α)kL2 ≤ 2
H
2
δ0
δ0 L
0
1/2
e 2 (α) − αkL2 we obtain:
Thus, setting k0 = Lδ0 k2 H
−k0 ≤ f λ ≤ k0 + αL .
We first present a sketch of the method: given λ < 1, and assuming that
f λ ∈ K is known, we shall prove the existence of a positive ε and a recursive
sequence {fn } which converges quadratically to the unique solution of (1.4λ+ε )
in K. As ε can be chosen independently of f λ and λ, starting at f 0 = α, we
deduce the existence of a sequence
0 = λ0 < λ1 < · · · < λN = 1,
where f λk ∈ K is obtained iteratively from f λk−1 .
Two Iterative Schemes for an
H-system
Pablo Amster and
María Cristina Mariani
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Let λ < 1, and f λ ∈ K be a solution of (1.4λ ). Define the constants:
k − k0
,
L1/2
0
λ
2
λ
e
k1 = f − 2 H (f ) ,
2
000
2
e
k2 =
sup
H
(ξ)
−k≤ξ≤k+α
R=
L
and define a sequence {fn } in the following way:
• f0 := f λ .
Pablo Amster and
María Cristina Mariani
• fn+1 the unique element of H 2 (I) that solves the linear problem

0
00

2
2
 f 00 = (λ + ε) 1 − 2 H
e
e
(fn )
(fn ) (fn+1 − fn ) + fn − 2 H
n+1

 f (0) = 0,
n+1
Two Iterative Schemes for an
H-system
fn+1 (L) = αL .
We shall prove that if ε is small enough, then {fn } is well defined. More
precisely:
Theorem 2.1. Assume that (2.1) holds, and that H is twice continuously differentiable on [−k, k +αL ]. Then {fn } is well defined and converges quadratically
for the H 1 -norm to the unique solution of (1.4λ+ε ) in K for any ε ≤ 1 − λ satisfying:
RL3/2 k2
δ0 R
ε 1+
<
.
πδ0
k1
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Proof. As f0 ∈ K, f1 is well defined, and by an estimate analogous to (2.2) we
obtain:
00
1 0
0
00
2
e
kf1 − f0 kL2 ≤ (f1 − f0 ) − (λ + ε) 1 − 2 H
(f0 ) (f1 − f0 )
2
δ0
L
0
ε e 2 (f0 )
= f0 − 2 H
2
δ0 L
ε
= k1 ≤ R.
δ0
We shall assume as inductive hypothesis that fk is well defined for k = 1, . . . , n,
and that kfk0 − f00 kL2 ≤ R. Thus, fn ∈ K and fn+1 is well defined. Moreover,
for k = 1, . . . , n we have that
00
00
2
e
(fk+1 − fk ) − (λ + ε) 1 − 2 H
(fk ) (fk+1 − fk )
000
e 2 (ξk )(fk − fk−1 )2
= −(λ + ε) H
for some mean value ξk (x), and hence
k(fk+1 − fk )0 kL2 ≤
3/2
k2 (fk − fk−1 )2 2 ≤ k2 L k(fk − fk−1 )0 k2 2 .
L
L
δ0
δ0 π
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By induction,
k(fk+1 − fk )0 kL2 ≤
k2 L3/2
k(f1 − f0 )0 kL2
δ0 π
≤ A2
k −1
k(f1 − f0 )0 kL2 ,
2k −1
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k(f1 − f0 )0 kL2
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where
A=
εk1 k2 L3/2
< 1.
δ02 π
Hence,
0
k(fn+1 − f0 ) kL2 ≤
n
X
k(fk+1 − fk )0 kL2
k=0
1
k(f1 − f0 )0 kL2
1−A
εk1
≤
.
δ0 (1 − A)
≤
By hypothesis, we conclude that k(fn+1 −f0 )0 kL2 ≤ R. Thus, fn is well defined
for every n, and the inequality
k(fn+1 − fn )0 kL2 ≤ A2
n −1
k(f1 − f0 )0 kL2
holds, proving that {fn } is a Cauchy sequence in H 1 (I). Furthermore, if f =
limn→∞ fn , it is immediate that fn → f in H 2 (I), and f ∈ K solves (1.4λ+ε ).
Two Iterative Schemes for an
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María Cristina Mariani
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Remark 3. A uniform choice of ε can be obtained if we set
0
1/2
2
e
x − 2 H
k1 = L
sup
(x) .
−k ≤x≤k +α
0
0
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3.
Upper and Lower Solutions for Problem (1.4)
In this section we define a convergent sequence based on the existence of an
upper solution of the problem: namely, a nonnegative function β such that
0
e 2 (β),
(3.1)
β 00 − β ≤ −2 H
β(L) ≥ αL .
We remark that it suffices to consider this assumption, since 0 is a lower solution
of (1.4).
Theorem 3.1. Assume that β ≥ 0 satisfies (3.1) and that H is continuously
differentiable for 0 ≤ x ≤ kβk∞ . Set
00
e 2 (x)
H
C = 1 − 2 min
0≤x≤kβk∞
and define the sequences {fn± } given by:
• f0− ≡ 0
f0+ = β
±
• {fn+1
} the unique solution of the linear problem

0
 (f ± )00 − Cf ± = (1 − C)f ± − 2 H
e 2 (f ± )
n+1
n+1
n
n
 f ± = α on ∂I.
n+1
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Then {fn− } (respectively {fn+ }) is nondecreasing (nonincreasing), and converges pointwise to a solution of (1.2). Moreover, the respective limits f ± satisfy: 0 ≤ f − ≤ f + ≤ β.
Proof. Let us first note that C ≥ 0 (in fact, C ≥ 1), which implies that both
sequences are well defined. Furthermore,
from
0 the choice of C, it is immediate
e 2 (x) is nonincreasing for 0 ≤ x ≤
that the function ψ(x) = (1 − C)x − 2 H
kβk∞ . By definition,
(f1+ )00
−
Cf1+
e2
= (1 − C)β − 2 H
0
(β) ≥ β 00 − Cβ
and using the maximum principle it follows that f1+ ≤ β. On the other hand,
Two Iterative Schemes for an
H-system
Pablo Amster and
María Cristina Mariani
(f1+ )00 − Cf1+ = ψ(β) ≤ ψ(0) = 0
and as f1+ ≥ 0 on ∂I we deduce that f1+ ≥ 0 over I. Assume as inductive
hypothesis that
+
0 ≤ fn+ ≤ fn−1
.
Then
+
+
+
(fn+1
)00 − Cfn+1
= ψ(fn+ ) ≥ ψ(fn−1
) = (fn+ )00 − Cfn+
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and
+
+
)00 − Cfn+1
= ψ(fn+ ) ≤ ψ(0) = 0
(fn+1
+
0 ≤ fn+1
≤ fn+ . Thus, {fn+ } is nonincreasing
+
which implies that
and converges
pointwise to a function f ≥ 0. By standard apriori estimates, we have that
+
+
+
kfn+1
− αkH 2 ≤ ck(fn+1
− α)00 − C(fn+1
− α)kL2
+
= ckψ(fn ) − αkL2 ≤ M
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for some constant M . By compactness, {fn+ } admits a convergent subsequence
in C 1 (I), proving that f + ∈ C 1 (I). Furthermore,
Z x
+
0
+
0
+
+ ψ(fn+ ),
(fn+1 ) (x) − (fn+1 ) (0) =
Cfn+1
0
and by dominated convergence we conclude that
Z x
Z
+ 0
+ 0
+
+
(f ) (x) − (f ) (0) =
Cf + ψ(f ) =
0
x
+
e2
f −2 H
0
(f + ).
0
Thus, the result follows. The proof is analogous for {fn− }.
Two Iterative Schemes for an
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As a simple consequence we have:
Corollary 3.2. Assume there exists a number k ≥ αL such that
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1
e
H(k)H(k)
≤ .
4
Then β ≡ k is an upper solution, and the schemes defined in the previous
theorem converge.
n
e
Example 3.1. For H(x) = rx , we have that H(x)
=
tions of the previous corollary hold for
r
xn+2 ,
n+2
and the condi-
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αL ≤ k ≤
n+2
4r2
1
2n+2
.
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However, it is possible to find a sharper bound for αL , if we consider the
parabola
"
2 #
x−L
β(x) = αL 1 −
.
L
Indeed, in this case we have that β is an upper solution if and only if
−
4r2 2n+3
2αL
−
β
≤
−
β
L2
n+2
Two Iterative Schemes for an
H-system
or equivalently
φ(β) ≤
2αL
L2
4r 2 2n+2
for φ(x) = x n+2
x
− 1 . Note that for 0 ≤ x ≤ αL we have:
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φ(x) ≤ max{0, φ(αL )}.
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Thus, it suffices to assume that
0<
αL2n+2
n+2
≤
4r2
2
1+ 2
L
.
Remark 4. In the previous example, equation (1.4) is superlinear, namely:
4r2 2n+3
f
− f = 0,
f (0) = 0, f (L) = αL .
n+2
It can be proved (see e.g. [5]) that this problem admits infinitely many solutions.
More precisely, there exists k0 ∈ N such that for any j > k0 the problem has at
least two solutions crossing the line α(t) = αLL t exactly j times in (0, L).
f 00 +
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References
[1] P. AMSTER, M.M. CASSINELLI AND M.C. MARIANI, Solutions to general quasilinear elliptic second order problems, Nonlinear Studies, 7(2)
(2000), 283–289.
[2] P. AMSTER, M.M. CASSINELLI AND M.C. MARIANI, Solutions to
quasilinear equations by an iterative method, Bulletin of the Belgian Math.
Society, 7 (2000), 435–441.
[3] P. AMSTER, M.M. CASSINELLI AND D.F RIAL, Existence and uniqueness of H-System’s solutions with Dirichlet conditions, Nonlinear Analysis,
Theory, Methods, and Applications, 42(4) (2000), 673–677.
[4] H. BREZIS AND J.M. CORON, Multiple solutions of H systems and Rellich’s conjecture, Comm. Pure Appl. Math., 37 (1984), 149–187.
[5] A. CAPPIETO, J. MAWHIN AND F. ZANOLIN, Boundary value problems
for forced superlinear second order ordinary differential equations, Séminaire du Collége de France.
[6] D. GILBARG AND N.S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer-Verlag (1983).
Two Iterative Schemes for an
H-system
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[7] S. HILDEBRANDT, On the Plateau problem for surfaces of constant mean
curvature, Comm. Pure Appl. Math., 23 (1970), 97–114.
[8] M. STRUWE, Plateau’s Problem and the Calculus of Variations, Lecture
Notes, Princeton Univ. Press (1988).
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[9] GUOFANG WANG, The Dirichlet problem for the equation of prescribed
mean curvature, Analyse Nonlinéaire, 9 (1992), 643–655.
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