Generalized conjugate function theorems for solutions of firstorder elliptic systems on the plane
Author:
Chung Ling Yu
Journal:
Trans. Amer. Math. Soc. 244 (1978), 135
MSC:
Primary 35J55; Secondary 30G20
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 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 . And let . Then to each p, , there correspond two constants and such that hold for every solution of (M) in with . If , the theorem holds for . Furthermore, if b and d do not satisfy the condition (N) in , then we can relax the condition , and still have the above inequalities. 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:
http://dx.doi.org/10.1090/S0002994719780506608X
PII:
S 00029947(1978)0506608X
Keywords:
First order elliptic equations,
pseudoanalytic functions,
CauchyRiemann 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
