Generalized conjugate function theorems for solutions of first-order elliptic systems on the plane

Abstract:

Our essential aim is to generalize Privoloff’s theorem, Schwarz reflection principle, Kolmogorov’s theorem and the theorem of M. Riesz for conjugate functions to the solutions of differential equations in the $z = x + iy$ plane of the following elliptic type: \begin{equation} \tag {$\text {(M)}$} \frac {{\partial u}} {{\partial x}} - \frac {{\partial v}} {{\partial y}} = au + bv + f,\frac {{\partial u}} {{\partial y}} + \frac {{\partial v}} {{\partial x}} = cu + dv + g. \end{equation} Theorem 1. Let the coefficients of (M) be Hölder continuous on $\left | z \right | \leqslant 1$. Let $(u, v)$ be a solution of (M) in $\left | z \right | < 1$. If u is continuous on $\left | z \right | \leqslant 1$ and Hölder continuous with index $\alpha$ on $\left | z \right | = 1$, then $(u, v)$ is Höolder continuous with index $\alpha$ on $\left | z \right | \leqslant 1$. Theorem 2. Let the coefficients of (M) be continuous on $\left | z \right | \leqslant 1$ and satisfy the condition \begin{equation} \tag {$\text {(N)}$} \int _0^y {b(x,t)dt + \int _0^x {d(t,y)dt = \int _0^y {b(0,t)dt + \int _0^x {d(t,0)dt}}}} \end{equation} for $\left | z \right | \leqslant 1$. And let ${\left \| f \right \|_p} = {\sup _{0 \leqslant r < 1}}\{ (1/2\pi )\int _{ - \pi }^\pi {{{\left | {f(r{e^{i\theta }})} \right |}^p}d\theta {\} ^{1/p}}}$. Then to each p, $0 < p < \infty$, there correspond two constants ${A_p}$ and ${B_p}$ such that \[ \begin {array}{*{20}{c}} {{{\left \| v \right \|}_p} \leqslant {A_p}{{\left \| u \right \|}_p} + {B_p},} & {1 < p < \infty ,} \\ {{{\left \| v \right \|}_p} \leqslant {A_p}{{\left \| u \right \|}_1} + {B_p},} & {0 < p < 1,} \\ \end {array} \] hold for every solution $(u, v)$ of (M) in $\left | z \right | < 1$ with $v (0) = 0$. If $f \equiv g \equiv 0$, the theorem holds for ${B_p} = 0$. Furthermore, if b and d do not satisfy the condition (N) in $\left | z \right | \leqslant 1$, then we can relax the condition $v (0) = 0$, and still have the above inequalities. Theorem 3. Let the coefficients of (M) be analytic for x, y in $\left | z \right | < 1$. Let $(u, v)$ be a solution of (M) in $\{ \left | z \right | < 1\} \cap \{ y > 0\}$. If u is continuous in $\{ \left | z \right | < 1\} \cap \{ y \geqslant 0\}$ and analytic on $\{ - 1 < x < 1\}$, then $(u,v)$ can be continued analytically across the boundary $\{ - 1 < x < 1\}$. Furthermore, if the coefficients and u satisfy some further boundary conditions, then $(u, v)$ can be continued analytically into the whole of $\{ \left | z \right | < 1\}$.