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

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* .

**[1]**S. Agmon, A. Douglis, and L. Nirenberg,*Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I*, Comm. Pure Appl. Math.**12**(1959), 623–727. MR**0125307****[2]**S. Agmon, A. Douglis, and L. Nirenberg,*Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II*, Comm. Pure Appl. Math.**17**(1964), 35–92. MR**0162050****[3]**Lipman Bers,*Theory of pseudo-analytic functions*, Institute for Mathematics and Mechanics, New York University, New York, 1953. MR**0057347****[4]**Lipman Bers, Fritz John, and Martin Schechter,*Partial differential equations*, Lectures in Applied Mathematics, Vol. III, Interscience Publishers John Wiley & Sons, Inc. New York-London-Sydney, 1964. MR**0163043****[5]**F. D. Gakhov,*Boundary value problems*, Translation edited by I. N. Sneddon, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. MR**0198152****[6]**P. R. Garabedian,*Analyticity and reflection for plane elliptic systems*, Comm. Pure Appl. Math.**14**(1961), 315–322. MR**0136848****[7]**Hans Lewy,*On the reflection laws of second order differential equations in two independent variables*, Bull. Amer. Math. Soc.**65**(1959), 37–58. MR**0104048**, 10.1090/S0002-9904-1959-10270-6**[8]**N. I. Muskhelishvili,*Singular integral equations*, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR**0355494****[9]**Umberto Neri,*Singular integrals*, Lecture Notes in Mathematics, Vol. 200, Springer-Verlag, Berlin-New York, 1971. Notes for a course given at the University of Maryland, College Park, Md., 1967. MR**0463818****[10]**Walter Rudin,*Real and complex analysis*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210528****[11]**I. N. Vekua,*Novye metody rešeniya èlliptičeskih uravneniĭ*, OGIZ, Moscow-Leningrad,], 1948 (Russian). MR**0034503****[12]**I. N. Vekua,*Generalized analytic functions*, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962. MR**0150320****[13]**Chung Ling Yu,*Reflection principle for systems of first order elliptic equations with analytic coefficients*, Trans. Amer. Math. Soc.**164**(1972), 489–501. MR**0293110**, 10.1090/S0002-9947-1972-0293110-0**[14]**-,*Integral representations, Cauchy problem and reflection principles under nonlinear boundary conditions for systems of first order elliptic equations with analytic coefficients*(to appear).

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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