1 Introduction
Recently, numerous nonlinear counterparts of the classical Beltrami equation are successfully studied by many mathematicians. The directional dilatation coefficients provide fruitful and flexible tools for investigating the main features of regular solutions to these equations (see [Reference Andreian Cazacu2, Reference Golberg9, Reference Ryazanov, Srebro and Yakubov26]).
In this paper, we estimate the ratio
$|f(z)|/|z|$
under
$f(0)=0,$
which can be treated as asymptotic dilation or real differentiability condition (cf. [Reference Bojarski, Gutlyanskiĭ, Martio and Ryazanov6, Reference Golberg10]) at the origin and goes back to the classical Schwarz lemma bounds.
Let
$\mathbb {C}$
be the complex plane. In the complex notation
$f=u+iv$
and
$z=x+iy$
, the Beltrami equation in a domain
$G\subset \mathbb {C}$
has the form
$$ \begin{align} f_{\overline{z}}\,=\,\mu(z)f_{z}, \end{align} $$
where
$\mu \colon G\to \mathbb {C}$
is a measurable function and
$$ \begin{align*}f_{\overline{z}}=\frac{1}{2}(f_{x}+if_{y})\qquad \mathrm{and} \qquad f_{z}=\frac{1}{2}(f_{x}-if_{y})\end{align*} $$
are formal derivatives of f in
$\overline {z}$
and z, while
$f_{x}$
and
$f_{y}$
are partial derivatives of f in the variables x and y, respectively.
Various existence theorems for solutions of the Sobolev class have been recently established applying the modulus approach for a quite wide class of linear and quasilinear degenerate Beltrami equations (see, e.g., [Reference Gutlyanskiĭ, Ryazanov, Srebro and Yakubov14, Reference Gutlyanskiĭ, Ryazanov, Srebro and Yakubov15, Reference Martio, Ryazanov, Srebro and Yakubov23, Reference Ryazanov, Srebro and Yakubov26, Reference Sevost’yanov34, Reference Srebro and Yakubov35]).
Let
$\sigma \colon G \to \mathbb {C}$
be a measurable function and
$m\geqslant 0.$
We consider the following equation written in the polar coordinates
$(r,\theta )$
:
$$ \begin{align} f_{r}=\sigma(re^{i\theta})\,|f_{\theta}|^{m}\, f_{\theta}, \end{align} $$
where
$f_{\theta }$
and
$f_{r}$
are the partial derivatives of f by
$\theta $
and
$r,$
respectively. The equations of this type were studied in the works [Reference Golberg and Salimov11, Reference Golberg, Salimov and Stefanchuk12, Reference Salimov and Stefanchuk30–Reference Salimov and Stefanchuk33].
Applying the relations between these derivatives and the formal derivatives
$$ \begin{align} rf_{r}\,=\,zf_{z}+\overline{z}f_{\overline{z}}\,, \qquad f_{\theta}\,=\,i(zf_{z}-\overline{z}f_{\overline{z}}) \end{align} $$
(see, e.g., [Reference Astala, Iwaniec and Martin5, (21.51)]), one can rewrite equation (1.2) in the Cartesian form:
$$ \begin{align} f_{\overline{z}}\,=\,\frac{z}{\overline{z}}\,\frac{\widetilde{\sigma}(z)\,|zf_{z}-\overline{z}f_{\overline{z}}|^{m}-1}{\widetilde{\sigma}(z)\,|zf_{z}-\overline{z}f_{\overline{z}}|^{m}+1}\, f_{z}\,, \end{align} $$
where
$\widetilde {\sigma }(z)=i\sigma (z)|z|$
.
Under
$m=0$
, equation (1.4) reduces to the standard linear Beltrami equation (1.1) with the complex coefficient
$$ \begin{align*} \mu(z)\,=\,\frac{z}{\overline{z}}\,\frac{i\sigma(z)\,|z|-1}{i\sigma(z)\,|z|+1}\,. \end{align*} $$
Picking
$m=0$
and
$\sigma =-i/|z|$
in (1.4), we arrive at the classical Cauchy–Riemann system. For
$m>0$
, equation (1.4) provides a partial case of the general nonlinear system of equations (7.33) given in [Reference Astala, Iwaniec and Martin5, Section 7.7].
Next, we consider an equation of another type, namely
$$ \begin{align} f_{\theta}=\sigma(re^{i\theta})\,|f_{r}|^{m}\, f_{r}. \end{align} $$
Applying the relations (1.3), one can rewrite equation (1.5) by
$$ \begin{align} f_{\overline{z}}\,=\,\frac{z}{\overline{z}}\,\frac{1+i\sigma(z)\,|z|^{-m-1}|zf_{z}+\overline{z}f_{\overline{z}}|^{m}}{1-i\sigma(z)\,|z|^{-m-1}|zf_{z}+\overline{z}f_{\overline{z}}|^{m}}\, f_{z}. \end{align} $$
Assuming
$m=0$
, equation (1.6) also becomes the standard linear Beltrami equation (1.1) with
$$ \begin{align*} \mu(z)\,= \,\frac{z}{\overline{z}}\,\frac{1+i\sigma(z)/|z|}{1-i\sigma(z)/|z|}\, \,. \end{align*} $$
Choosing
$m=0$
and
$\sigma =i|z|$
in (1.6), we arrive again at the classical Cauchy–Riemann system. Later on, we assume that
$m>0.$
A mapping
$f \colon G \to \mathbb C$
is called regular at a point
$z_0\in G,$
if f has the total differential at this point and its Jacobian
$J_f=|f_z|^{2}-|f_{\bar {z}}|^{2}$
does not vanish (cf. [Reference Lehto and Virtanen22, I.1.6]). A homeomorphism f of Sobolev class
$W_{\mathrm {{loc}}}^{1, 1}$
is called regular, if
$J_{f}>0$
a.e. By a regular solution of equation (1.6), we call a regular homeomorphism
$f \colon G \to \mathbb C,$
which satisfies (1.6) a.e. in
$G.$
The nonlinear equations (1.4) and (1.6) provide partial cases of the nonlinear system of two real partial differential equations (see [Reference Lavrent’ev and Šabat21, (1)] and [Reference Lavrent’ev19, Reference Lavrent’ev20]). Note that various nonlinear systems of similar partial differential equations studied from different aspects and features can be found in [Reference Adamowicz1, Reference Aronsson3–Reference Astala, Iwaniec and Martin5, Reference Bojarski and Iwaniec7, Reference Carozza, Giannetti, di Napoli, Sbordone and Schiattarella8, Reference Guo and Kar13, Reference Kruschkal and Kühnau17–Reference Lavrent’ev and Šabat21, Reference Romanov25, Reference Šabat27, Reference Šabat28].
2 Auxiliary results
Later on, we use the following notations:
$$ \begin{align*} B_r=\{z\in\mathbb C: |z|< r\}\,, \quad \mathbb B=\{z\in\mathbb C: |z| < 1\} \end{align*} $$
and
$$ \begin{align*} \gamma_r=\{z\in\mathbb C: |z|=r\}\,,\, \ \ \mathbb{A}(0,r_1,r_2)=\{z\in \mathbb{C}: \, r_1<|z|<r_2 \}. \end{align*} $$
The area of set
$f(B_r)$
we denote by
$S_f(r)=|f(B_r)|.$
Let
$f:\mathbb {B}\to \mathbb {C}$
be a regular homeomorphism of the Sobolev class
$W_{\mathrm {loc }}^{1,1}$
, and let
$p>1$
. By the p-angular dilatation of the mapping f with respect to the point
$z_0=0$
, we call a quantity
$$ \begin{align*}D_{p,f}(z) = D_{p,f}(re^{i\theta}) = \frac{|f_{\theta}(re^{i\theta})|^p}{r^p J_f(re^{i\theta})}\,,\end{align*} $$
where
$z = re^{i\theta }$
and
$J_f$
is the Jacobian of f.
For
$D_{p,f}(z)$
and
$p>1$
, denote
$$ \begin{align} d_{p,f}(r)=\left( \frac{1}{2\pi r} \int\limits_{\gamma_r}\, D_{p,f}^{\frac{1}{p-1}}(z) \, |dz| \right)^{p-1}\,. \end{align} $$
The following statement provides a differential inequality for the area functional
$S_f(r)=|f(B_r)|$
(see Lemma 2.1 in [Reference Golberg, Salimov and Stefanchuk12]).
Proposition 2.1 Let
$f:\mathbb {B}\to \mathbb {C}$
be a regular homeomorphism of the Sobolev class
$W_{\mathrm {loc }}^{1,1}$
that possesses the N-property, and let
$p> 1$
. Then
$$ \begin{align*}S_f'(r) \geqslant 2\pi^{\frac{2-p}{2}} r^{1-p} d_{p,f}^{-1}(r) S_f^{\frac{p}{2}}(r)\end{align*} $$
for almost all (a.a.)
$r \in [0, 1)$
.
Proposition 2.2 Let
$f:\mathbb {B}\to \mathbb {C}$
be a regular homeomorphism of the Sobolev class
$W_{\mathrm {loc }}^{1,1}$
that possesses the N-property, and let
$p> 1$
and
$K>0$
. If
$$ \begin{align} d_{p,f}(r)\leqslant K \quad \text{for a.a.} \quad r\in (0,1)\,, \end{align} $$
then
$$ \begin{align} S'_f(r)\geqslant 2\pi^{\frac{2-p}{2}}\, K^{-1}\, r^{1-p}\, S_f^{\frac{p}{2}}(r) \end{align} $$
for a.a.
$r\in [0,1).$
Lemma 2.3 Let
$f:\mathbb {B}\to \mathbb {C}$
be a regular homeomorphism of the Sobolev class
$W_{\mathrm {loc }}^{1,1}$
that possesses the N-property,
$1<p <2$
, and
$K>0$
. If
$d_{p,f}(r)\leqslant K$
for a.a.
$r\in (0,1)$
, then for
$r\in [0,1)$
,
$$ \begin{align} |f(B_r)|\geqslant\, C(p,K)\, r^2\,, \end{align} $$
where
$ C(p,K)=\pi K^{\frac {2}{p-2}} $
.
Proof Let
$0<\varepsilon <r<1$
. By (2.3), for a.a.
$t\in [0,1)$
,
$$ \begin{align*}\frac{S'_f(t)\, dt }{S^{\frac{p}{2}}_f(t)}\geqslant 2\pi^{\frac{2-p}{2}}\, K^{-1}\, t^{1-p}\,dt, \end{align*} $$
and integrating over the segment
$[\varepsilon , r]$
, we obtain
$$ \begin{align*}\int\limits_\varepsilon^r \frac{S'_f(t) }{S^{\frac{p}{2}}_f(t)} \, dt \geqslant 2\pi^{\frac{2-p}{2}}\, K^{-1}\, \int\limits_\varepsilon^r t^{1-p}\,dt\,. \end{align*} $$
Hence,
$$ \begin{align*}\int\limits_\varepsilon^r \phi'_p(t)\, dt \geqslant \frac{2\pi^{\frac{2-p}{2}}\,}{(2-p)K} \left(r^{2-p}-\varepsilon^{2-p}\right), \end{align*} $$
where
$\phi _p(t)=\frac {2}{2-p}\, S^{\frac {2-p}{2}}_f(t)\,.$
This function is nondecreasing on
$[0, 1)$
, and
$$ \begin{align*}\int\limits_\varepsilon^r \phi'_p(t)\, dt \leqslant \phi_p(r)- \phi_p(\varepsilon)\,=\frac{2}{2-p}\, \left(S^{\frac{2-p}{2}}_f(r)-S^{\frac{2-p}{2}}_f(\varepsilon)\right) \end{align*} $$
(see Theorem IV.7.4 in [Reference Saks29]). Combining the last two inequalities, we have
$$ \begin{align*}S^{\frac{2-p}{2}}_f(r)\geqslant S^{\frac{2-p}{2}}_f(r)-S^{\frac{2-p}{2}}_f(\varepsilon) \geqslant \frac{\pi^{\frac{2-p}{2}}\,}{K} \left(r^{2-p}-\varepsilon^{2-p}\right)\,. \end{align*} $$
Finally, letting
$\varepsilon \to 0$
, we get the estimate (2.4).
The following result is an analog of the well-known Ikoma–Schwartz lemma on estimating the limsup (see Corollary 3 in [Reference Ikoma16]).
Lemma 2.4 Let
$f: \mathbb {B}\to \mathbb {C}$
be a regular homeomorphism of the Sobolev class
$W_{\mathrm {loc }}^{1,1}$
that possesses the N-property and normalized by
$f(0)= 0$
, and
$1<p <2$
and
$K>0$
. If
$d_{p,f}(r)\leqslant K$
for a.a.
$r\in (0,1)$
, then
$$ \begin{align*}\limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}\geqslant K^{-\frac{1}{2-p}}\,.\end{align*} $$
Proof Denote
$\mathcal {M}_f(r)=\max \limits _{|z|=r}|f(z)|$
. Since
$f(0)=0$
, we have
$$ \begin{align} S_f(r)\leqslant \pi \, \mathcal{M}^2_f(r)\,. \end{align} $$
Thus, applying Lemma 2.3, we obtain
$$ \begin{align*}\limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}=\limsup\limits_{r\to 0} \frac{\mathcal{M}_f(r)}{r}\geqslant \limsup\limits_{r\to 0} \sqrt{\frac{S_f(r)}{\pi r^2}}\geqslant K^{-\frac{1}{2-p}}\,.\\[-43pt] \end{align*} $$
In particular, by Lemma 2.4, we come to the following statement.
Corollary 2.5 If
$K>0$
and
$D_{p,f}(z)\leqslant K$
for a.a.
$z\in \mathbb {B}$
, then
$ \limsup \limits _{z\to 0} \, \frac {|f(z)|}{|z|}\geqslant K^{-\frac {1}{2-p}}\,.$
Next, we prove the following statement.
Theorem 2.6 Let
$f: \mathbb {B}\to \mathbb {C}$
be a regular homeomorphism of the Sobolev class
$W_{\mathrm {loc }}^{1,1}$
that possesses the N-property and normalized by
$f(0)= 0$
, and let
$1<p <2$
. Suppose that
$$ \begin{align*}\kappa_0=\liminf\limits_{\varepsilon\to 0} \left( \frac{1}{\pi\varepsilon^2}\int\limits_{B_{\varepsilon}}D_{p,f}^ {\frac{1}{p-1}}(z)\,dx\,dy\right)^{p-1}\,. \end{align*} $$
1) If
$\kappa _0\in (0, \infty )$
, then
$$ \begin{align*}\limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}\geqslant c_p \, \kappa_0^{-\frac{1}{2-p}}\,,\end{align*} $$
where
$c_p$
is a positive constant depending on the parameter p.
2) If
$\kappa _0=0$
, then
$$ \begin{align*}\limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}=\infty\,.\end{align*} $$
Proof Let
$\varepsilon \in \left (0, 1 \right )$
. By Proposition 2.1, for a.a.
$r\in [0,1)$
,
$$ \begin{align*}\frac{S'_f(r)\, dr }{S^{\frac{p}{2}}_f(r)}\geqslant \frac{2\pi^{\frac{2-p}{2}}}{r^{p-1}d_{p,f}(r) }\,dr; \end{align*} $$
therefore, integrating over the segment
$[\frac {\varepsilon }{2}, \varepsilon ]$
, and arguing similarly to the proof of Lemma 2.3, we reach
$$ \begin{align*}S^{\frac{2-p}{2}}_f(\varepsilon)\geqslant S^{\frac{2-p}{2}}_f(\varepsilon)-S^{\frac{2-p}{2}}_f\left(\frac{\varepsilon}{2}\right) \geqslant \pi^{\frac{2-p}{2}}(2-p) \int\limits_{\frac{\varepsilon}{2}}^{\varepsilon} \frac{dr}{r^{p-1}d_{p,f}(r) }. \end{align*} $$
Hence,
$$ \begin{align} S_f(\varepsilon)\geqslant \pi(2-p)^{\frac{2}{2-p}} \left(\int\limits_{\frac{\varepsilon}{2}}^{\varepsilon} \frac{dr}{r^{p-1}d_{p,f}(r) } \right)^{\frac{2}{2-p}} \,. \end{align} $$
Noting
$$ \begin{align*}\frac{\varepsilon}{2}=\int\limits_{\frac{\varepsilon}{2}}^{\varepsilon} \frac{1}{\left(\int\limits_{\gamma_{r}}D_{p,f}^{\frac{1}{p-1}}(z)\,|dz|\right)^{\frac{p-1}{p}}} \left(\int\limits_{\gamma_{r}}D_{p,f}^{\frac{1}{p-1}}(z)\,|dz|\right)^{\frac{p-1}{p}}dr, \end{align*} $$
one obtains by Holder’s inequality with exponents p and
$p^{'}=p/(p-1),$
$$ \begin{align*}\left(\frac{\varepsilon}{2}\right)^{p}\leqslant\int\limits_{\frac{\varepsilon}{2}}^{\varepsilon} \frac{dr}{\left(\int\limits_{\gamma_{r}}D_{p,f}^{\frac{1}{p-1}}(z)\,|dz|\right)^{p-1}} \left(\int\limits_{\frac{\varepsilon}{2}}^{\varepsilon} \int\limits_{\gamma_{r}}D_{p,f}^{\frac{1}{p-1}}(z)\,|dz|\,dr\right)^{p-1}. \end{align*} $$
Due to (2.1), we rewrite the last inequality in the following form:
$$ \begin{align*}\left(\frac{\varepsilon}{2}\right)^{p}\leqslant \frac{1}{(2\pi)^{p-1}} \int\limits_{\frac{\varepsilon}{2}}^{\varepsilon}\frac{dr}{r^{p-1}d_{p,f}(r)} \left(\int\limits_{\frac{\varepsilon}{2}}^{\varepsilon}\int\limits_{\gamma_{r}}D_{p,f}^{\frac{1}{p-1}}(z)\,|dz|\,dr\right)^{p-1}. \end{align*} $$
Now, by the Fubini theorem, we have
$$ \begin{align*}\int\limits_{\frac{\varepsilon}{2}}^{\varepsilon}\frac{dr}{r^{p-1}d_{p,f}(r)}\geqslant (2\pi)^{p-1}\left(\frac{\varepsilon}{2}\right)^p\left(\, \int\limits_{A(0, \frac{\varepsilon}{2}, \varepsilon)}D_{p,f}^ {\frac{1}{p-1}}(z)\,dx\,dy\right)^{1-p}\,. \end{align*} $$
Since
$ A(0, \frac {\varepsilon }{2}, \varepsilon )\subset B_{\varepsilon }$
,
$$ \begin{align} \int\limits_{\frac{\varepsilon}{2}}^{\varepsilon}\frac{dr}{r^{p-1}d_{p,f}(r)}\geqslant (2\pi)^{p-1}\left(\frac{\varepsilon}{2}\right)^p\left( \int\limits_{B_{\varepsilon}}D_{p,f}^ {\frac{1}{p-1}}(z)\,dx\,dy\right)^{1-p}. \end{align} $$
Combining (2.6) and (2.7), we obtain
$$ \begin{align*}S_f(\varepsilon)\geqslant \pi\left( \frac{2-p}{2} \right)^{\frac{2}{2-p}} \varepsilon^2 \left(\frac{1}{\pi\varepsilon^2}\int\limits_{B_{\varepsilon}}D_{p,f}^ {\frac{1}{p-1}}(z)\,dx\,dy\right)^{-\frac{2(p-1)}{2-p}} \,, \end{align*} $$
and by (2.5),
$$ \begin{align*}\frac{\mathcal{M}_f(\varepsilon)}{\varepsilon}\geqslant\sqrt{\frac{S_f(\varepsilon)}{\pi\varepsilon^2}}\geqslant c_p \left( \frac{1}{\pi\varepsilon^2}\int\limits_{B_{\varepsilon}}D_{p,f}^ {\frac{1}{p-1}}(z)\,dxdy\right)^{-\frac{p-1}{2-p}} \,, \end{align*} $$
where
$c_p=\left (\frac {2-p}{2}\right )^{\frac {1}{2-p}}$
.
Thus,
$$ \begin{align*}\limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}=\limsup\limits_{\varepsilon\to 0} \frac{\mathcal{M}_f(\varepsilon)}{\varepsilon}\geqslant c_p \, \kappa_0^{-\frac{1}{2-p}}. \end{align*} $$
This completes the proof of Theorem 2.6.
3 Application to nonlinear Beltrami equations
In this section, we present theorems on the asymptotic behavior of regular homeomorphic solutions to a nonlinear Beltrami equation of the form (1.6).
Theorem 3.1 Let
$f: \mathbb {B}\to \mathbb {C}$
be a regular homeomorphic solution of equation (1.6) which belongs to Sobolev class
$W_{\mathrm {loc }}^{1,2}$
, and normalized by
$f(0) = 0$
. Assume that
$C>0$
and the coefficient
$\sigma : \mathbb {B}\to \mathbb {C}$
satisfies the following condition:
$$ \begin{align} \int\limits_{\gamma_r} \frac{|\sigma(z) |^{m+2} }{\, \left(\mathrm{Im}\,\sigma(z)\right)^{m+1} }\, |dz| \leqslant C\, r^{2} \end{align} $$
for a.a.
$r\in (0,1)$
. Then
$$ \begin{align} \limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}\geqslant \left(\frac{2\pi}{C}\right)^{\frac{1}{m}}\,. \end{align} $$
Proof Let
$p=\frac {m+2}{m+1}$
,
$1<p<2$
. It is well known that any homeomorphism
$W_{\mathrm {loc }}^{1,2}$
possesses the N-property (see [Reference Martio and Ziemer24, Theorem 4.4]). Since f is a regular homeomorphic solution of equation (1.6), we get
$$ \begin{align*}J_f(re^{i\theta})= \frac{1}{r} \, \mathrm{Im}\, \left(\overline{f_r}\, f_\theta \right)=\frac{1}{r} \, |f_r|^{m+2} \, \mathrm{Im}\, \sigma(re^{i\theta})> 0 \ \text {a.e.}\end{align*} $$
and
$$ \begin{align*}D_{p,f}(re^{i\theta}) = \frac{|f_{\theta}(re^{i\theta})|^p}{r^p J_f(re^{i\theta})}= \frac{|\sigma (re^{i\theta}) |^{\frac{m+2}{m+1}} }{r^{\frac{1}{m+1}} \, \mathrm{Im}\,\sigma (re^{i\theta})}\,. \end{align*} $$
Therefore, in view of the condition (3.1), we have
$$ \begin{align*}d_{p,f}(r)= \left( \frac{1}{2\pi r^2} \int\limits_{\gamma_r}\, \frac{|\sigma(z) |^{m+2} }{\, \left(\mathrm{Im}\,\sigma(z)\right)^{m+1} } \, |dz| \right)^{\frac{1}{m+1}} \leqslant \left(\frac{C}{2\pi}\right)^{\frac{1}{m+1}} \end{align*} $$
for a.a.
$r\in (0,1)$
.
Corollary 3.2 Let
$f: \mathbb {B}\to \mathbb {C}$
be a regular homeomorphic solution of equation (1.6) which belongs to Sobolev class
$W_{\mathrm {loc }}^{1,2}$
, and normalized by
$f(0) = 0$
and
$K>0$
. Assume that the coefficient
$\sigma : \mathbb {B}\to \mathbb {C}$
satisfies the following condition:
$$ \begin{align} \frac{|\sigma (z) |^{m+2} }{\, \left(\mathrm{Im}\,\sigma(z)\right)^{m+1} }\, \leqslant K\, |z| \end{align} $$
for a.a.
$z\in \mathbb {B}$
. Then
$$ \begin{align} \limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}\geqslant K^{-\frac{1}{m}}\,.\end{align} $$
Proof Indeed, in view of the condition (3.3), we have
$$ \begin{align*}\int\limits_{\gamma_r} \frac{|\sigma(z) |^{m+2} }{\, \left(\mathrm{Im}\,\sigma(z)\right)^{m+1} }\, |dz| \leqslant \int\limits_{\gamma_r} \, K|z|\, |dz|=2\pi K\, r^{2}\,.\end{align*} $$
Next, applying Theorem 3.1 with the parameter
$C=2\pi K$
, we get the estimate (3.4).
We illustrate the last bound by the following example.
Example 3.3 Fix
$k>0$
and consider the equation
$$ \begin{align} f_\theta=\frac{i}{k^m}\,r |f_r|^m \, f_r \end{align} $$
in the unit disk
$\mathbb {B}$
.
Let
$f=kre^{i\theta }$
. Obviously, the mapping f belongs to the Sobolev class
$W^{1,2}(\mathbb {B})$
. The partial derivatives of f with respect to
$\theta $
and r are
$$ \begin{align*}f_\theta=ki\,re^{i\theta}, f_r=k\, e^{i\theta},\end{align*} $$
and
$J_f(re^{i\theta })= \frac {1}{r} \, \mathrm {Im}\, \left (\overline {f_r}\, f_\theta \right )=k^2>0\,. $
Now, we show that the mapping
$f=k\, re^{i\theta }$
is a solution of equation (3.5). Clearly,
$$ \begin{align*}\sigma=\frac{f_\theta}{|f_r|^m f_r}=\frac{i}{k^m}\,r \,.\end{align*} $$
Thus, (3.1) holds, since
$$ \begin{align*}\int\limits_{\gamma_r} \frac{|\sigma(z) |^{m+2} }{\, \left(\mathrm{Im}\,\sigma(z)\right)^{m+1} }\, |dz| =C \, r^2\,, \end{align*} $$
where
$C=\frac {2\pi }{k^m}$
.
On the other hand,
$ \lim \limits _{z\to 0} \, \frac {|f(z)|}{|z|}=k\,.$
Remark 3.4 The estimate (3.2) is sharp. It is easy to check that it is attained for a mapping
$f=\left (\frac {2\pi }{C}\right )^{\frac {1}{m}}z$
.
By Theorem 2.6, similarly to the proof of Theorem 3.1, we obtain the following statement.
Theorem 3.5 Let
$f\colon \mathbb {B}\to \mathbb {C}$
be a regular homeomorphic solution of equation (1.6) which belongs to Sobolev class
$W^{1,2}_{\mathrm {loc}},$
and normalized by
$f(0)=0.$
Suppose that
$$ \begin{align*}\sigma_{0}=\liminf\limits_{\varepsilon\rightarrow 0}\frac{1}{\pi \varepsilon^2}\int\limits_{B_{\varepsilon}} \frac{|\sigma(z)|^{m+2}}{|z| \left( \mathrm{Im}\, \sigma(z)\right)^{m+1}}\,dxdy. \end{align*} $$
1) If
$\sigma _{0}\in (0,\infty )$
, then
$$ \begin{align*}\limsup\limits_{z\rightarrow 0}\frac{|f(z)|}{|z|}\geqslant c_{m}\,\sigma_{0}^{-\frac{1}{m}}, \end{align*} $$
where
$c_{m}$
is a positive constant depending on the parameter
$m.$
2) If
$\sigma _{0}=0$
, then
$$ \begin{align*}\limsup\limits_{z\rightarrow 0}\frac{|f(z)|}{|z|}=\infty. \end{align*} $$
Example 3.6 Let
$k>0$
and
$\alpha \in (1,m+1)$
. Consider the equation
$$ \begin{align} f_{\theta}=i k r^{\alpha}|f_{r}|^{m}f_{r} \end{align} $$
in the unit disk
$\mathbb {B}.$
The mapping
$$ \begin{align*}f=k^{-\frac{1}{m}}\beta^{\frac{m+1}{m}}r^{\frac{m+1-\alpha}{m}} e^{i\theta}\,, \quad \beta=\frac{m}{m+1-\alpha}\,,\end{align*} $$
belongs to the Sobolev class
$W^{1,2}_{\mathrm {loc}}(\mathbb {B})$
. Its partial derivatives with respect to r and
$\theta $
are
$$ \begin{align*}f_{\theta}=ik^{-\frac{1}{m}}\beta^{\frac{m+1}{m}}r^{\frac{m+1-\alpha}{m}} e^{i\theta},\quad f_{r}=k^{-\frac{1}{m}}\beta^{\frac{1}{m}}r^{\frac{1-\alpha}{m}} e^{i\theta}\,.\end{align*} $$
It is easy to see that the mapping
$f=k^{-\frac {1}{m}}\beta ^{\frac {m+1}{m}}r^{\frac {m+1-\alpha }{m}} e^{i\theta }$
is a regular homeomorphic solution of equation (3.6). Clearly,
$$ \begin{align*}\sigma=\frac{f_{\theta}}{|f_{r}|^{m}f_{r}}=i k r^{\alpha}.\end{align*} $$
The condition
$\sigma _0=0$
in Theorem 3.5 is fulfilled, since
$$ \begin{align*}\lim\limits_{\varepsilon\rightarrow 0}\frac{1}{\pi \varepsilon^2}\int\limits_{B_{\varepsilon}} \frac{|\sigma(z)|^{m+2}}{|z| \left( \mathrm{Im}\, \sigma(z)\right)^{m+1}}\,dxdy =0.\end{align*} $$
By a direct calculation,
$|f(z)|/|z|\to \infty $
as
$z\to 0$
.
