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 | References | Similar Articles | Additional Information
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 plane of the following elliptic type:
![]() | ( (M)) |
Theorem 1. Let the coefficients of (M) be Hölder continuous on . Let
be a solution of (M) in
. If u is continuous on
and Hölder continuous with index
on
, then
is Höolder continuous with index
on
.
Theorem 2. Let the coefficients of (M) be continuous on and satisfy the condition
![]() | ( (N)) |
for













Theorem 3. Let the coefficients of (M) be analytic for x, y in . Let
be a solution of (M) in
. If u is continuous in
and analytic on
, then
can be continued analytically across the boundary
. Furthermore, if the coefficients and u satisfy some further boundary conditions, then
can be continued analytically into the whole of
.
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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