Uniqueness of approximate solutions of the Beltrami equations
We introduce a notion of an approximate solution to the Beltrami equations, obtain some properties of such solutions and show that the approximate solution is unique up to pre-composition with a conformal mapping.
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nasplib_isofts_kiev_ua-123456789-1239062025-02-09T21:58:03Z Uniqueness of approximate solutions of the Beltrami equations Kolomoitsev, Yu.S. Ryazanov, V.I. We introduce a notion of an approximate solution to the Beltrami equations, obtain some properties of such solutions and show that the approximate solution is unique up to pre-composition with a conformal mapping. 2009 Article Uniqueness of approximate solutions of the Beltrami equations / Yu.S. Kolomoitsev, V.I. Ryazanov // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2009. — Т. 19. — С. 116-124. — Бібліогр.: 49 назв. — англ. 1683-4720 https://nasplib.isofts.kiev.ua/handle/123456789/123906 517.5 en Труды Института прикладной математики и механики application/pdf Інститут прикладної математики і механіки НАН України |
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We introduce a notion of an approximate solution to the Beltrami equations, obtain some properties of such solutions and show that the approximate solution is unique up to pre-composition with a conformal mapping. |
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Kolomoitsev, Yu.S. Ryazanov, V.I. |
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Kolomoitsev, Yu.S. Ryazanov, V.I. Uniqueness of approximate solutions of the Beltrami equations Труды Института прикладной математики и механики |
| author_facet |
Kolomoitsev, Yu.S. Ryazanov, V.I. |
| author_sort |
Kolomoitsev, Yu.S. |
| title |
Uniqueness of approximate solutions of the Beltrami equations |
| title_short |
Uniqueness of approximate solutions of the Beltrami equations |
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Uniqueness of approximate solutions of the Beltrami equations |
| title_fullStr |
Uniqueness of approximate solutions of the Beltrami equations |
| title_full_unstemmed |
Uniqueness of approximate solutions of the Beltrami equations |
| title_sort |
uniqueness of approximate solutions of the beltrami equations |
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Інститут прикладної математики і механіки НАН України |
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2009 |
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https://nasplib.isofts.kiev.ua/handle/123456789/123906 |
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Uniqueness of approximate solutions of the Beltrami equations / Yu.S. Kolomoitsev, V.I. Ryazanov // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2009. — Т. 19. — С. 116-124. — Бібліогр.: 49 назв. — англ. |
| series |
Труды Института прикладной математики и механики |
| work_keys_str_mv |
AT kolomoitsevyus uniquenessofapproximatesolutionsofthebeltramiequations AT ryazanovvi uniquenessofapproximatesolutionsofthebeltramiequations |
| first_indexed |
2025-12-01T05:12:42Z |
| last_indexed |
2025-12-01T05:12:42Z |
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1850281532478980096 |
| fulltext |
ISSN 1683-4720 Труды ИПММ НАН Украины. 2009. Том 19
UDK 517.5
c©2009. Iu.S. Kolomoitsev, V.I. Ryazanov
UNIQUENESS OF APPROXIMATE SOLUTIONS
OF THE BELTRAMI EQUATIONS
We introduce a notion of an approximate solution to the Beltrami equations, obtain some properties of
such solutions and show that the approximate solution is unique up to pre-composition with a conformal
mapping.
1. Introduction. Let D be a domain in the complex plane C, i.e., a connected and
open subset of C, and let µ : D → C be a measurable function with |µ(z)| < 1 a.e. The
Beltrami equation is the equation of the form
fz = µ(z) · fz (1)
where fz = ∂f = (fx + ify)/2, fz = ∂f = (fx − ify)/2, z = x + iy, and fx and fy are
partial derivatives of f in x and y, correspondingly. The function µ is called the complex
coefficient and
Kµ(z) =
1 + |µ(z)|
1− |µ(z)| (2)
the maximal dilatation or in short the dilatation of the equation (1). The Beltrami
equation (1) is said to be degenerate if ess sup Kµ(z) = ∞.
There are numerous old and recent works devoted to the existence problem for degene-
rate Beltrami equations, see e.g. [2], [7]–[12], [18], [21]–[23], [25], [28]–[30], [32], [38]–
[42], [48]–[49]. In almost all these works one actually proves just the existence of the
approximate solution for (1). However, the problem of uniqueness of solutions for (1) is
insufficiently known explored. To the moment it is known the Stoilow factorization only
for narrow special cases of solutions and µ. In this paper we show that if Kµ ∈ L1
loc, then
the approximate solution of Beltrami equation (1) is unique up to pre-composition with
a conformal mapping.
Given z0 ∈ D, the tangential dilatation of (1) with respect to z0 is
KT
µ (z, z0) =
∣∣∣1− z−z0
z−z0
µ(z)
∣∣∣
2
1− |µ(z)|2 ,
see [40]–[41], cf. the corresponding terms and notations in [3]–[5], [18], [25] and [34].
Recall also that a function f : D → C is absolutely continuous on lines, abbr.
f ∈ACL, if, for every closed rectangle R in D whose sides are parallel to the coordinate
axes, f |R is absolutely continuous on almost all line segments in R which are parallel to
the sides of R. In particular, f is ACL (possibly modified on a set of Lebesgue measure
zero) if it belongs to the Sobolev class W 1,1
loc of locally integrable functions with locally
116
Uniqueness of approximate solutions of the Beltrami equations
integrable first generalized derivatives and, conversely, if f ∈ ACL has locally integrable
first partial derivatives, then f ∈ W 1,1
loc , see e.g. 1.2.4 in [31]. Note that, if f ∈ ACL, then
f has partial derivatives fx and fy a.e. and, for a sense-preserving ACL homeomorphism
f : D → C, the Jacobian Jf (z) = |fz|2 − |fz|2 is nonnegative a.e. In this case, the
complex dilatation µf of f is the ratio µ(z) = fz/fz, if fz 6= 0 and µ(z) = 0 otherwise,
and the dilatation Kf of f is Kµ(z), see (2). Note that |µ(z)| ≤ 1 a.e. and Kµ(z) ≥ 1
a.e.
Recall that, given a family of paths Γ in C, a Borel function ρ : C→ [0,∞] is called
admissible for Γ, abbr. ρ ∈ admΓ, if
∫
γ
ρ(z) |dz| ≥ 1 (3)
for each γ ∈ Γ. The modulus of Γ is defined by
M(Γ) = inf
ρ∈adm Γ
∫
C
ρ2(z) dxdy . (4)
Given a domain D and two sets E and F in C, ∆(E,F, D) denotes the family of all
paths γ : [a, b] → C which join E and F in D, i.e., γ(a) ∈ E, γ(b) ∈ F and γ(t) ∈ D for
a < t < b. Motivated by the ring definition of quasiconformality in [16], we introduced
the following notion in [40]. Let D be a domain in C, z0 ∈ D, and Q : D → [0,∞]
a measurable function. A homeomorphism f : D → C is called a ring Q−homeo-
morphism at the point z0 if
M(∆(fC1, fC2, fD)) ≤
∫
A
Q(z) · η2(|z − z0|) dxdy (5)
for every circular ring A ⊂ D centered at z0,
A = A(z0, r1, r2) = {z ∈ C : r1 < |z − z0| < r2}, 0 < r1 < r2 < ∞ ,
and every measurable function η : (r1, r2) → [0,∞] such that
r2∫
r1
η(r) dr = 1 (6)
and where C1 = {z ∈ C : |z − z0| = r1} and C2 = {z ∈ C : |z − z0| = r2}.
Now, given a domain D in C and a measurable function Q : D → [0,∞], we say that
a homeomorphism f : D → C is a ring Q−homeomorphism at a boundary point
z0 of the domain D if
M(∆(fC1, fC2, fD)) ≤
∫
A∩D
Q(z) · η2(|z − z0|) dxdy (7)
117
Iu.S. Kolomoitsev, V.I. Ryazanov
for every ring A = A(z0, r1, r2) and every continua C1 and C2 in D which belong to
the different components of the complement to the ring A in C containing z0 and ∞,
correspondingly, and for every measurable function η : (r1, r2) → [0,∞] satisfying the
condition (6).
An ACL homeomorphism f : D → C is called a strong ring solution of the Beltrami
equation (1) with a complex coefficient µ if f satisfies (1) a.e., f−1 ∈ W 1,2
loc (f(D)) and f is
a ring Q–homeomorphism at every point z0 ∈ D with Q(z) = Qz0(z) := KT
µ (z, z0) ≤ Kµ(z).
In fact, if Q ∈ L1
loc(D), then similarly to [44] one can prove that the single condition (5)
implies f ∈ ACL, furthermore, f ∈ W 1,1
loc (D), Jf (z) 6= 0 a.e., see e.g. [45].
Following to [8], we call a homeomorphism f ∈ W 1,1
loc (D) a regular solution of (1)
if f satisfies (1) a.e. and Jf (z) 6= 0 a.e.
Note that above the condition f−1 ∈ W 1,2
loc (f(D)) implies that f has (N−1)-property
and a.e. point z is a regular point for the mapping f, i.e., f is differentiable at z with
Jf (z) 6= 0, see e.g. [26], p.121, 128–130 and 150, and Theorem 1 in [33]. Conversely, if
f ∈ W 1,1
loc (D), Kf ∈ L1
loc(D) and Jf (z) 6= 0 a.e., then f−1 ∈ W 1,2
loc (f(D)), see e.g. [19].
Moreover, by [19] gw = 0 = gw for a.e. w where Jg(w) = 0, g = f−1. Note also that the
condition Kµ ∈ L1
loc(D) is necessary for a homeomorphic ACL solution f of (1) to have
the property g = f−1 ∈ W 1,2
loc (f(D)) because this property implies that
∫
C
Kµ(z) dxdy ≤ 4
∫
C
dxdy
1− |µ(z)|2 = 4
∫
f(C)
|∂g|2 dudv < ∞
for every compact set C ⊂ D. The change of variables is correct here, say by Lemmas
III.2.1 and III.3.2 and Theorems III.3.1 and III.6.1 in [26], cf. also I.C(3) in [1].
For n ∈ N, define µn : D → C by letting µn(z) = µ(z) if |µ(z)| ≤ 1 − 1/n and 0
otherwise. Let fn : D → C be a homeomorphic ACL solution of (1) with µn instead of
µ. We call a homeomorphism f an approximate solution of (1) if there exists such
a sequence {fn} converged to f uniformly on each compact set in D. We call such a
sequence {fn} an approximating sequence for f .
In the classical case when ‖µ‖∞ < 1, equivalently, when Kµ ∈ L∞(D), every ACL
homeomorphic solution f of the Beltrami equation (1) is in the class W 1,2
loc (D) together
with its inverse mapping f−1, and hence f is a strong ring solution of (1) by Theorem 1
below. In the case ‖µ‖∞ = 1 with Kµ ≤ Q ∈ BMO, again f−1 ∈ W 1,2
loc (f(D)) and
f belongs to W 1,s
loc (D) for all 1 ≤ s < 2 but not necessarily to W 1,2
loc (D), see e.g. [38].
However, there is a varity of degenerate Beltrami equations for which strong ring solutions
exist as shown in the paper [42]. The inequalities (5) and (7), which holds for the strong
ring solutions, is an important tool in deriving various local and boundary properties of
such solutions, see e.g. [27], [37] and [46], cf. also [36].
2. Preliminaries. We consider the extended complex plane C as a metric space with
the spherical (chordal) metric:
s(z, ζ) =
|z − ζ|√
1 + |z|2
√
1 + |ζ|2
, z 6= ∞ 6= ζ ; s(z,∞) =
1√
1 + |z|2
.
118
Uniqueness of approximate solutions of the Beltrami equations
The kernel of a sequence of open sets Ωn ⊆ C, n = 1, 2, . . . is the open set
Ω0 = Kern Ωn : =
∞⋃
m=1
Int
( ∞⋂
n=m
Ωn
)
where Int A denotes the set consisting of all inner points of A, in other words, Int A is
the union of all open disks in A with respect to the spherical distance.
Proposition 2.1. Let hn : D → D′
n, D′
n = hn(D), be a sequence of homeomorphisms
given in a domain D ⊆ C. If hn converge as n → ∞ locally uniformly with respect to
the spherical (chordal) metric to a homeomorphism h : D → D′ ⊆ C, then D′ = h(D) ⊆
KernD′
n.
This is Proposition 3.6 in [8]. Later on, we apply also the following useful.
Remark 2.1. It’s well known that every metric space is L∗-space, i.e. a space with a
convergence, see e.g. Theorem 2.1.1 in [24], and in the compact spaces the Uhryson axiom
says: xn → x0 as n →∞ if and only if, for every convergent subsequence xnk
→ x∗, the
equality x∗ = x0 holds, see the definition 20.1.3 in [24].
To prove that an approximate solution is a strong ring solution we need the following
two auxiliary statements. The next proposition can be found as Theorem 2.16 in [42], cf.
the corresponding result for inner points in [39].
Proposition 2.2. Let f : D → C be a sense-preserving homeomorphism of the class
W 1,2
loc (D) such that f−1 ∈ W 1,2
loc (f(D)). Then at every point z0 ∈ D the mapping f is a
ring Q-homeomorphism with Q(z) = KT
µ (z, z0) where µ(z) = µf (z).
The following proposition was proved in [43] as Theorem 4.1.
Proposition 2.3. Let fn : D → C, n = 1, 2, . . . be a sequence of ring Q-homeo-
morphisms at a point z0 ∈ D. If fn converges locally uniformly to a homeomorphism
f : D → C, then f is also a ring Q-homeomorphism at z0.
We also need the following convergence theorem for the Beltrami equations, see
Theorem 3.1 in [43].
Proposition 2.4. Let D be a domain in C and let fn : D → C be a sequence of
sense-preserving ACL homeomorphisms with complex dilatations µn such that
1 + |µn(z)|
1− |µn(z)| ≤ Q(z) ∈ L1
loc(D) ∀ n = 1, 2, . . . (8)
If fn → f uniformly on each compact set in D, where f : D → C is a homeomorphism,
then f ∈ ACL and ∂fn and ∂fn converge weakly in L1
loc(D) to ∂f and ∂f , respectively.
Moreover, if in addition µn → µ a.e., then ∂f = µ∂f a.e.
Remark 2.2. In fact, it is easy to show that under the condition (8) fn as well as
f belong to W 1,1
loc (D). Moreover, if in addition Q ∈ Lp
loc(D), then fn and f belong to
W 1,s
loc (D), ∂fn → ∂f and ∂fn → ∂f weakly in Ls
loc(D), where s = 2p/(1 + p), see e.g.
Lemma 2.2 in [7].
119
Iu.S. Kolomoitsev, V.I. Ryazanov
3. On convergence of inverse homeomorphisms.
Lemma 3.1. Let D be a domain in C and let fn : D → C be a sequence of
homeomorphisms from D into C such that fn → f as n → ∞ locally uniformly with
respect to the spherical metric to a homeomorphism f from D into C. Then f−1
n → f−1
locally uniformly in f(D), too.
Proof. Set gn = f−1
n and g = f−1. The locally uniform convergence gn → g is
equivalent to the so-called continuous convergence, meaning that gn(wn) → g(w0) for
every convergent sequence wn → w0 in f(D), see e.g. [13], p.268. So, let wn ∈ f(D),
n = 0, 1, 2, . . . and wn → w0 as n → ∞. Let us show that zn := g(wn) → z0 := g(w0)
as n → ∞. By Remark 2.1 it suffices to prove that for every convergent subsequence
znk
→ z∗ as k →∞, the equality z∗ = z0 holds. Let D0 be a subdomain of D such that
z0 ∈ D0 and D0 is a compact subset of D. Then by Proposition 2.1 f(D0) ⊆ Kern fnk
(D0)
and hence w0 together with its neighborhood belongs to fnk
(D0) for all k ≥ K. Thus,
with no loss of generality we may assume that wnk
∈ fnk
(D0), i.e. znk
∈ D0 for all
k = 1, 2, . . . and, consequently, z∗ ∈ D. Then, by the continuous convergence fn → f , we
have that fnk
(znk
) → f(z∗), i.e. fnk
(gnk
(wnk
)) = wnk
→ f(z∗). The latter implies that
w0 = f(z∗), i.e. z∗ = z0. The proof is complete. ¤
4. Properties of approximate solutions. In this section we show that an approximate
solution to (1) is its regular solution and also a strong ring solution for any complex
coefficient µ with Kµ ∈ L1
loc.
Theorem 4.1. Let µ : D → C be a measurable function with |µ(z)| < 1 a.e. and
Kµ ∈ L1
loc(D). Then any approximate solution to the Beltrami equation (1) is a regular
solution.
Proof. Let f be an approximate solution of the Beltrami equation (1) and let {fn}
be its approximating sequence. Then f ∈ W 1,1
loc by Proposition 2.4.
Now, set gn = f−1
n and g = f−1. By Lemma 3.1 we have that gn → g locally uniformly
in f(D). Moreover, by a change of variables which is permitted because fn and gn are
in W 1,2
loc , see e.g. Lemmas III.2.1 and III.3.2 and Theorems III.3.1 and III.6.1 in [26], cf.
also I.C(3) in [1], we obtain that for large n
∫
B
|∂gn|2 dudv =
∫
gn(B)
dxdy
1− |µn(z)|2 ≤
∫
B∗
Kµ dxdy < ∞ (9)
where B∗ and B are relatively compact domains in D and f(D), respectively, such that
g(B̄) ⊂ B∗. The relation (9) implies that the sequence gn is bounded in W1,2(B), and
hence f−1 ∈ W1,2
loc(f(D)), see e.g. Lemma III.3.5 in [35] or Theorem 4.6.1 in [15]. The
latter condition brings in turn that f has (N−1)−property, see e.g. Theorem III.6.1 in
[26], and hence Jf (z) 6= 0 a.e., see Theorem 1 in [33]. Thus, f is a regular solution of (1).
¤
Theorem 4.2. Let µ : D → C be a measurable function with |µ(z)| < 1 a.e. and
Kµ ∈ L1
loc(D). Then any approximate solution to the Beltrami equation (1) is a strong
ring solution.
120
Uniqueness of approximate solutions of the Beltrami equations
Proof. Let {fn} be an approximating sequence for f . By Proposition 2.2 the mapping
fn is a ring Q-homeomorphism with Q(z) = KT
µ (z, z0) where µ(z) = µf (z). Then by
the Proposition 2.3 we obtain that f is a ring Q–homeomorphism with Q(z) = KT
µ (z, z0)
at every point z0 ∈ D. We have already shown under the proof of Theorem 4.1 that
f ∈ W 1,1
loc and f−1 ∈ W1,2
loc(f(D)). Thus, f is a strong ring solution of (1). ¤
5. Factorization theorem.
Theorem 5.1. Let f : D → C be an approximate solution to the Beltrami equation
(1) with measurable µ : D → C such that |µ(z)| < 1 a.e. and
1 + |µ(z)|
1− |µ(z)| ≤ Q(z) ∈ L1
loc ∀ n = 1, 2, . . .
Suppose g is another approximate solution to (1) in D. Then there is a conformal mapping
h : f(D) → C such that
g = h ◦ f.
Proof. Let {fn} and {gn} be approximating sequences for f and g, correspondingly.
Set hn = gn ◦ f−1
n . By uniqueness theorem for the uniformly elliptic Beltrami equations,
see [26, p.183], hn is a conformal mapping for any n ∈ N. Next, by Lemma 3.1 we have
that hn = gn ◦ f−1
n → g ◦ f−1 = h as n →∞ locally uniformly in f(D). Thus, it remains
to apply the Weierstrass theorem on the uniform convergence of sequences of analytic
functions, see e.g., [17, p.17], from which we conclude that h is the conformal mapping.
¤
6. The main corollaries and conjectures. Following to [20], we say that a function
ϕ : D → R of the class L1
loc has finite mean oscillation at a point z0 ∈ D, write
ϕ ∈ FMO(z0), if
lim
ε→0
−
∫
B(z0,ε)
|ϕ(z)− ϕε(z0)| dxdy < ∞
where
ϕε(z0) = −
∫
B(z0,ε)
ϕ(z) dxdy
is the average of ϕ over the disk B(z0, ε) = {z ∈ C : |z − z0| < ε} with small ε > 0. We
also write ϕ ∈ FMO(D), or simply ϕ ∈ FMO, if ϕ ∈ FMO(z0) for all z0 ∈ D.
Applying Theorem 11.6 from [29], we obtain the following corollary of Theorem 5.1.
Corollary 6.1. Let µ : D → C be a measurable function with |µ(z)| < 1 a.e. and
Kµ ∈ L1
loc. Suppose that every point z0 ∈ D has neighborhood Uz0 such that
KT
µ (z, z0) ≤ Qz0(z) a.e.
for some function Qz0(z) of finite mean oscillation at the point z0 in the variable z. Then
the Beltrami equation (1) has unique approximate solution up to pre-composition with a
conformal mapping.
121
Iu.S. Kolomoitsev, V.I. Ryazanov
Remark 6.1. In particular, we obtain that the conclusion of Corollary 6.1 holds if
either
lim
ε→0
−
∫
B(z0,ε)
∣∣∣1− z−z0
z−z0
µ(z)
∣∣∣
2
1− |µ(z)|2 dxdy < ∞ ∀ z0 ∈ D (10)
or
Kµ(z) =
1 + |µ(z)|
1− |µ(z)| ≤ Q(z) ∈ FMO (D) . (11)
Similarly, by Theorem 11.10 in [29] this is valid if Kµ ∈ L1
loc(D) and
δ(z0)∫
0
dr
rkT
z0
(r)
= ∞ ∀ z0 ∈ D (12)
where kT
z0
(r) is the average of the tangential dilatation KT
µ (z, z0) over the circle
C(z0, r) = {z ∈ C : |z − z0| = r}, δ(z0) < dist (z0, ∂D), and, in particular, if
kT
z0
(r) = O
(
log
1
r
)
as r →∞ ∀ z0 ∈ D . (13)
We complete our paper with the following two equivalent conjectures.
Conjecture 1. Let µ : D → C be a measurable function with |µ(z)| < 1 a.e. and
Kµ ∈ L1
loc(D) (for which the Beltrami equation (1) has at least one approximate solution).
Then any regular solution to (1) is an approximate solution to (1).
Conjecture 2. Let µ : D → C be a measurable function with |µ(z)| < 1 a.e. and
Kµ ∈ L1
loc(D) (for which the Beltrami equation (1) has at least one approximate solution).
Then a regular solution to (1) is unique up to pre-composition with a conformal mapping.
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Institute of Applied Mathematics and Mechanics, Donetsk
kolomus1@mail.ru, vlryazanov1@rambler.ru
Received 15.11.09
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титул
Научное издание
Том 19
содержание
Том 19
Донецк, 2009
Основан в 1997г.
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