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Transactions of the American Mathematical Society

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


Author: Chung Ling Yu
Journal: Trans. Amer. Math. Soc. 244 (1978), 1-35
MSC: Primary 35J55; Secondary 30G20
DOI: https://doi.org/10.1090/S0002-9947-1978-0506608-X
MathSciNet review: 506608
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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:

$\displaystyle \frac{{\partial u}} {{\partial x}} - \frac{{\partial v}} {{\parti... ...\partial u}} {{\partial y}} + \frac{{\partial v}} {{\partial x}} = cu + dv + g.$ ( (M))

Theorem 1. Let the coefficients of (M) be Hölder continuous on $ \left\vert z \right\vert \leqslant 1$. Let $ (u, v)$ be a solution of (M) in $ \left\vert z \right\vert < 1$. If u is continuous on $ \left\vert z \right\vert \leqslant 1$ and Hölder continuous with index $ \alpha $ on $ \left\vert z \right\vert = 1$, then $ (u, v)$ is Höolder continuous with index $ \alpha $ on $ \left\vert z \right\vert \leqslant 1$.

Theorem 2. Let the coefficients of (M) be continuous on $ \left\vert z \right\vert \leqslant 1$ and satisfy the condition

$\displaystyle \int_0^y {b(x,t)dt + \int_0^x {d(t,y)dt = \int_0^y {b(0,t)dt + \int_0^x {d(t,0)dt}}}}$ ( (N))

for $ \left\vert z \right\vert \leqslant 1$. And let $ {\left\Vert f \right\Vert _p} = {\sup _{0 \leqslant r < 1}}\{ (1/2\pi )\int_{ ... ...i }^\pi {{{\left\vert {f(r{e^{i\theta }})} \right\vert}^p}d\theta {\} ^{1/p}}} $. Then to each p, $ 0 < p < \infty $, there correspond two constants $ {A_p}$ and $ {B_p}$ such that

\begin{displaymath}\begin{array}{*{20}{c}} {{{\left\Vert v \right\Vert}_p} \leqs... ...Vert u \right\Vert}_1} + {B_p},} & {0 < p < 1,} \\ \end{array} \end{displaymath}

hold for every solution $ (u, v)$ of (M) in $ \left\vert z \right\vert < 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\vert z \right\vert \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\vert z \right\vert < 1$. Let $ (u, v)$ be a solution of (M) in $ \{ \left\vert z \right\vert < 1\} \cap \{ y > 0\} $. If u is continuous in $ \{ \left\vert z \right\vert < 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\vert z \right\vert < 1\} $.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0506608-X
Keywords: First order elliptic equations, pseudoanalytic functions, Cauchy-Riemann equations, Privoloff's theorem, Schwarz reflection principle, Kolmogorov's theorem, the theorem of M. Riesz for conjugate functions
Article copyright: © Copyright 1978 American Mathematical Society

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