Mixed Riemann-Hilbert boundary value problem
with simply connected fibers

Let ${\rm\Delta}=\{z\in{\mathbb{C}};|z|<1\}$ be the open unit disc in the complex plane ${\mathbb{C}}$ and let $\partial{\rm\Delta}=\{\xi\in{\mathbb{C}};|\xi|=1\}$ be the unit circle. Let $L$ be a closed arc on $\partial{\rm\Delta}$, let $\mathring{L}$ denote its interior with respect to $\partial{\rm\Delta}$, and let $a:L\rightarrow\partial{\rm\Delta}$ be a smooth function.

Recall that the interior ${\rm Int}(\gamma)$ of a Jordan curve $\gamma\subseteq{\mathbb{C}}$ is the bounded component of ${\mathbb{C}}\setminus\gamma$. We orient $\gamma$ positively with respect to ${\rm Int}(\gamma)$. Jordan curve $\gamma\subset{\mathbb{C}}$ is starshaped with respect to $0$, if for any point $w$ in the interior of $\gamma$ the line segment which connects points $0$ and $w$ lies in the interior of $\gamma$, and it is strongly starshaped with respect to $0$, [9], if there exists a positive continuous function $R$ on the unit circle such that

and

Let $\{\gamma_{\xi}\}_{\xi\in\partial{\rm\Delta}\setminus\mathring{L}}$ be a smooth family of smooth Jordan curves in ${\mathbb{C}}$ which all contain point $0$ in their interiors. In this paper we study the existence and properties of holomorphic solutions of the nonlinear mixed Riemann-Hilbert problem, that is, the Cherepanov boundary value problem with simply connected fibers. The problem asks for a continuous function $f$ on $\overline{{\rm\Delta}}$, holomorphic on ${\rm\Delta}$, such that

and

That is, $f$ solves a linear Riemann-Hilbert problem on $L$ and a nonlinear Riemann-Hilbert problem with simply connected fibers on $\partial{\rm\Delta}\setminus\mathring{L}$. See also [1, 2, 13, 14].

The problem with circular fibers $\gamma_{\xi}$ and $L$ a finite union of disjoint arcs was considered by Obnosov and Zulkarnyaev in [14], and by the author in [5]. The structure of the family of solutions of problem (3-4) is well known in the cases where either $L=\partial{\rm\Delta}$ or $L=\emptyset$. If $L=\partial{\rm\Delta}$, we consider a homogeneous linear Riemann-Hilbert problem. In this case the essential information on the problem is given by the winding number $W(a)$ of function $a$. It is well known [11, 17, 18] that if the winding number $W(a)$ is nonnegative, the space of solutions of (3) is a vector subspace of $A^{\alpha}({\rm\Delta})$, $0<\alpha<1$, of real dimension $2W(a)+1$.

If $L$ is empty, we have a nonlinear Riemann-Hilbert problem with smooth simply connected fibers which all contain $0$ in their interiors. This problem was considered and solved in [8, 16, 17, 18]. In particular, it was proved that the family of solutions with exactly $m$ zeros on ${\rm\Delta}$, $m\in{\mathbb{N}}\cup\{0\}$, forms a manifold in space $A^{\alpha}({\rm\Delta})$ of dimension $2m+1$, and this manifold is compact if and only if $m=0$. We assume from now on that neither $L=\emptyset$ nor $L=\partial{\rm\Delta}$.

Let $0<\alpha<1$ and let $G\subset{\mathbb{C}}$ be a compact subset. We denote by $C^{\alpha}(G)$ the algebra over ${\mathbb{C}}$ of Hölder continuous complex functions on $G$ and by $C_{{\mathbb{R}}}^{\alpha}(G)$ the algebra over ${\mathbb{R}}$ of real Hölder continuous functions on $G$. Using the norm

the algebras $C^{\alpha}(G)$ and $C_{{\mathbb{R}}}^{\alpha}(G)$ become Banach algebras. For $G=\overline{{\rm\Delta}}$ or $G=\partial{\rm\Delta}$ and $k\in{\mathbb{N}}\cup\{0\}$ we also define spaces $C^{k,\alpha}(G)$ and $C_{{\mathbb{R}}}^{k,\alpha}(G)$ of $k$ times continuously differentiable functions on $G$, whose all $k$-th derivatives belong to space $C^{\alpha}(G)$ or space $C_{{\mathbb{R}}}^{\alpha}(G)$.

We also need some algebras of holomorphic functions on ${\rm\Delta}$. By $A({\rm\Delta})$ we denote the disc algebra, that is, the algebra of continuous functions on $\overline{{\rm\Delta}}$ which are holomorphic on ${\rm\Delta}$, and by $A^{\alpha}({\rm\Delta})=A({\rm\Delta})\cap C^{\alpha}(\overline{{\rm\Delta}})$ the algebra of Hölder continuous functions on the closed disc which are holomorphic on ${\rm\Delta}$. Using appropriate norms, that is, the maximum norm $\|\cdot\|_{\infty}$ for $A({\rm\Delta})$ and the Hölder norm $\|\cdot\|_{\alpha}$ for $A^{\alpha}({\rm\Delta})$, these algebras become Banach algebras. Similarly we define $A^{k,\alpha}({\rm\Delta})=A({\rm\Delta})\cap C^{k,\alpha}(\overline{{\rm\Delta}})$ $(k\in{\mathbb{N}}\cup\{0\},0<\alpha<1)$.

Recall that Hilbert transform $H$ assigns to a real function $u$ on $\partial{\rm\Delta}$ a real function $Hu$ on $\partial{\rm\Delta}$ such that the harmonic extension of $f=u+i\,Hu$ to ${\rm\Delta}$ is holomorphic on ${\rm\Delta}$ and real at $0$. It is known that $H$ is a bounded linear operator on $C_{{\mathbb{R}}}^{k,\alpha}(\partial{\rm\Delta})$ $(k\in{\mathbb{N}}\cup\{0\},0<\alpha<1)$, [18, §1.6.11], and hence the harmonic extension of $f=u+i\,Hu$ to ${\rm\Delta}$ belongs to $A^{k,\alpha}({\rm\Delta})$. Also, [18, §1.6.11], the Hilbert transform is a bounded linear operator on the Sobolev space $W^{k}_{p}(\partial{\rm\Delta})$ of $k$ times generalized differentiable functions with derivatives in $L^{p}(\partial{\rm\Delta})$ $(k\in{\mathbb{N}}\cup\{0\},1<p<\infty)$ equipped with the norm

Recall, [18, §1.6.14], that if $\partial{\rm\Delta}=T_{1}\cup T_{2}$ is a partition of $\partial{\rm\Delta}$ in two subarcs $T_{1}$ and $T_{2}$ and if $T_{0}\subseteq T_{1}$ is a compactly contained subarc of $T_{1}$, then for $k\in{\mathbb{N}}\cup\{0\}$, $1<p<\infty$, $0<\alpha<1$ there exists a constant $C=C(k,p,\alpha)$ such that

and

We will also need compact embedding result, [18, §1.1.8],

for $0<\alpha<\beta<1-\frac{1}{p}$, $1<p<\infty$, which holds on arcs in $\partial{\rm\Delta}$ as well.

Since $L\neq\partial{\rm\Delta}$ we can extend $a$ to $\partial{\rm\Delta}$ as a nowhere zero function of class $C^{k+1}$ so that the winding number $W(a)=0$. Therefore, [18, p. 25], we can write $\overline{a}$ in the form

where $r>0$ is a positive $C^{k,\alpha}$ function on $\partial{\rm\Delta}$ and $h\in A^{k,\alpha}({\rm\Delta})$. Thus the original problem (3-4) is equivalent to the problem

and

where $f_{\ast}=ie^{h}f$ and $\gamma^{\ast}_{\xi}=ie^{h(\xi)}\gamma_{\xi}$. Observe that the number of zeros of $f_{\ast}$ and $f$ are the same and that $0$ belongs to the interiors of all curves $\gamma^{\ast}_{\xi}$, $\xi\in\partial{\rm\Delta}\setminus\mathring{L}$. Also, since for each $\xi\in\partial{\rm\Delta}$ the transfomation

is a composition of a dilation and a rotation, the angle conditions from Theorem 1.2 stay the same.

Using a holomorphic automorphism of the unit disc we may even assume that $L=\{\xi\in\partial{\rm\Delta};{\rm Im}(\xi)\leq 0\}$ is the lower semicircle. From now on we will consider problem (12-13) with the addition that $L$ is the lower semicircle and instead of $f_{\ast}$ and $\{\gamma^{\ast}_{\xi}\}_{\xi\in\partial{\rm\Delta}\setminus\mathring{L}}$ we will still write $f$ and $\{\gamma_{\xi}\}_{\xi\in\partial{\rm\Delta}\setminus\mathring{L}}$.

We will consider smooth families of smooth Jordan curves $\{\gamma_{\xi}\}_{\xi\in\partial{\rm\Delta}\setminus\mathring{L}}$ in ${\mathbb{C}}$. Let $k\in{\mathbb{N}}$. The family of Jordan curves $\{\gamma_{\xi}\}_{\xi\in\partial{\rm\Delta}\setminus\mathring{L}}$ is a $C^{k}$ family parametrized by $\xi\in\partial{\rm\Delta}\setminus\mathring{L}$ if there exists a function $\rho\in C^{k}((\partial{\rm\Delta}\setminus\mathring{L})\times{\mathbb{C}})$ such that

and the gradient $\frac{\partial\rho}{\partial\overline{w}}(\xi,w)=\rho_{\overline{w}}(\xi,w)\neq 0$ for every $\xi\in\partial{\rm\Delta}\setminus\mathring{L}$ and $w\in\gamma_{\xi}$. We call $\rho$ a defining function for $C^{k}$ family of Jordan curves $\{\gamma_{\xi}\}_{\xi\in\partial{\rm\Delta}\setminus\mathring{L}}$. We will consider only bounded families of Jordan curves which all lie in some fixed disc $\overline{{\rm\Delta}(0,R)}$, $R>0$, and the space $C^{k}((\partial{\rm\Delta}\setminus\mathring{L})\times\overline{{\rm\Delta}(0,R)})$ is equipped with the standard $C^{k}$ norm.

Since we assume that $\gamma_{\pm 1}$ are strongly starshaped Jordan curves, we also assume that for $\rho$, the defining function for Jordan curves $\{\gamma_{\xi}\}_{\xi\in\partial{\rm\Delta}\setminus\mathring{L}}$, and $j=\pm 1$ we have

for some positive $C^{k}$ functions $R_{j}(z)$ on ${\mathbb{C}}$.

Using parametrization $\theta\mapsto e^{i\theta}$ of the unit circle we will also use the notation $\gamma_{\theta}$, $\rho(\theta,w)$ and $\rho_{\theta}(\theta,w)$ instead of $\gamma_{\xi}$, $\rho(\xi,w)$ and $\rho_{\xi}(\xi,w)$. Also, for a function $h$ on $\partial{\rm\Delta}$, we will write either $h(\xi)$ or $h(\theta)$, where $\xi=e^{i\theta}$. Observe that if $h$ is holomorphic on ${\rm\Delta}$ with well defined derivative on $\partial{\rm\Delta}$, then $\frac{\partial h}{\partial\theta}(\theta)=i\xi h^{\prime}(\xi)$ for $\xi=e^{i\theta}$.

In this section we prove regularity of continuous solutions of a specific form of problem (12-13), where the defining function $\rho\in C^{k}((\partial{\rm\Delta}\setminus\mathring{L})\times{\mathbb{C}})$ $(k\geq 3)$.

Let $f\in A({\rm\Delta})$ be a solution of (12-13). It is well known [6, 7, 8, 18] that $f$ restricted to $\partial{\rm\Delta}\setminus\{-1,1\}$ is in $C^{k-1,\alpha}$ for any $0<\alpha<1$. Hence we need information on the regularity of $f$ near points $\xi=\pm 1$. For $j=\pm 1$ we denote $f(j)=w_{j}\in{\mathbb{R}}\cap\gamma_{j}$.

Using Möbius tranformation from the unit disc ${\rm\Delta}$ to the upper half-plane ${\cal H}=\{z\in{\mathbb{C}};{\rm Im}(z)>0\}$ we consider the case where $f$ is bounded and continuous on $\overline{\cal H}$ and holomorphic on ${\cal H}$. Also, point $\xi=1$ is mapped into $t=0$ and point $\xi=-1$ into $\infty$. Now $f$ solves the problem

and

Also, using translation, we will assume that $f(0)=0\in{\mathbb{R}}\cap\gamma_{0}$.

Let $\pi\beta_{1}$ $(\beta_{1}\in(-1,1)\setminus\{0\})$ be the oriented angle of intersection of the real axis ${\rm Im}(w)=0$ and $\gamma_{0}$ at $w=f(0)=0$. The orientation of the real axis is positive with respect to the upper half-plane and the orientation of $\gamma_{0}$ is positive with respect to the interior of $\gamma_{0}$. Hence $\beta_{1}>0$, if the tangent vector to the real axis is rotated counterclockwise by angle $\pi\beta_{1}$ to get a tangent vector to $\gamma_{0}$ at point $0$, and $\beta_{1}<0$, if the tangent vector to the real axis is rotated clockwise by angle $\pi|\beta_{1}|$ to get a tangent vector to $\gamma_{0}$ at $0$.

The defining function $\rho$ can near $(0,0)$ for $t\geq 0$ be written as

where $g\in C^{1}({\mathbb{R}}\times{\mathbb{C}})$ such that $g(0,0)=g_{t}(0,0)=g_{w}(0,0)=g_{\overline{w}}(0,0)=0$.

Recall that $\rho(0,0)=0$ and that $\rho_{\overline{w}}(0,0)$ represents an outer normal to $\gamma_{0}$ at point $w=0$. So we have

for some real $\lambda>0$. We may assume $\lambda=\frac{1}{2}$.

Because

we have

for some $A,B,C\in{\mathbb{R}}$ and $D,E\in{\mathbb{C}}$.

Let us assume that we have a solution $f$ of the problem (19-20) of the form

where $\kappa$ is bounded and continuous on $\overline{\cal H}$, holomorphic on ${\cal H}$, and $0<s<1$ to be determined.

For $t\leq 0$ we have $t=(-1)|t|$ and from (19) we get

On the other hand for $t>0$ we have

We choose $0<s<1$ so that $\kappa$ solves boundary value problem with continuous boundary data. That is, we choose $s=1-\beta_{1}$, if $\beta_{1}>0$, and $s=-\beta_{1}=|\beta_{1}|$, if $\beta_{1}<0$.

Thus $\kappa$ solves the following Riemann-Hilbert problem

and

where, if $\beta_{1}>0$,

and, if $\beta_{1}<0$,

For such choice of $s$ are the defining function for problem (31-32)

and its partial $w$-derivative

continuous on ${\mathbb{R}}\times{\mathbb{C}}$.

On the other hand, the partial derivative of defining function (37) with respect to the $t$ variable is not continuous at $t=0$, but, as we will see, it still has certain $L^{p}$ regularity properties, which will imply regularity conditions on $\kappa$ and $f$.

We know that $\kappa$ is $C^{k-1,\alpha}$ on ${\mathbb{R}}\setminus\{0\}$ and we can differentiate (31-32) on ${\mathbb{R}}\setminus\{0\}$ to get

and

For $t>0$ and $\beta_{1}>0$ we have

and for $t>0$ and $\beta_{1}<0$ we have

The $t$-derivative of defining function (37) is $0$ for $t<0$.

Since $\beta_{1}\in(-1,1)\setminus\{0\}$ and $\kappa$ is bounded, we have that $\widetilde{\rho}_{t}(t,\kappa(t))$ is in $L^{p}_{\rm loc}({\mathbb{R}})$ for

A similar argument can be used for point $\xi=-1\in\partial{\rm\Delta}$. Let $\pi\beta_{-1}$ $(\beta_{-1}\in(-1,1)\setminus\{0\})$ be the orientied angle of intersection of $\gamma_{-1}$ and the real axis ${\rm Im}(w)=0$ at point $f(-1)$. Now $\beta_{-1}$ is positive, if a positive tangent vector to $\gamma_{-1}$ at $f(-1)$ is rotated counterclockwise to get a positive tangent vector to the real axis and negative otherwise. For $j=\pm 1$ we define $\delta_{j}=1-\beta_{j}$, if $\beta_{j}\in(0,1)$, and $\delta_{j}=|\beta_{j}|$, if $\beta_{j}\in(-1,0)$.

To transfer our observations to the boundary value problem (12-13) on the unit disc, let $\Psi\in A^{\frac{1}{2}}({\rm\Delta})$ be a biholomorphic map from ${\rm\Delta}$ to the upper half-disc ${\rm\Delta}_{+}$, which maps the lower semicircle $L$ on $[-1,1]$ so that $\Psi(\pm 1)=\pm 1$. Let $F(x)=\frac{1}{2}x(3-x^{2})$. Then $F(x)-1=-\frac{1}{2}(x-1)^{2}(x+2)$ and $F(x)+1=-\frac{1}{2}(x+1)^{2}(x-2)$. Hence function $\psi(\xi)=F(\Psi(\xi))$ is real on $L$, $\psi(\pm 1)=\pm 1$, and $C^{1}$ on $\partial{\rm\Delta}$.

Recall that $w_{j}$ is the positive intersection of $\gamma_{j}$ and the real axis, $j=\pm 1$. Now we consider only those solutions $f$ of the Cherepanov problem (12-13), which are of the form

where $\kappa$ is in $A({\rm\Delta})$.

We will define two (local) defining functions $\widetilde{\rho}_{1}(\xi,w)$ for $\xi\neq-1$ and $\widetilde{\rho}_{-1}(\xi,w)$ for $\xi\neq 1$. Let