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Boundary value problems for two types of degenerate elliptic systems in R 4

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Abstract

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic system of the first order equations in R 4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system’s solution, the boundary value problem as stated above is transformed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R 4 are derived.

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References

  1. S R Athreya, M T Barlow, R F Bass, E A Perkins. Degenerate stochastic differential equations and super-Markov chains, Probab Theory Related Fields, 2002, 123: 484–520.

    Article  MathSciNet  MATH  Google Scholar 

  2. S Čanić, E H Kim. A class of quasilinear degenerate elliptic problems, J Differential Equations, 2003, 189: 71–98.

    Article  MathSciNet  MATH  Google Scholar 

  3. C L Epstein, R Mazzeo. Analysis of degenerate diffusion operators arising in population biology, In: From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev Math, 2013, 28: 203–216.

    Article  MathSciNet  MATH  Google Scholar 

  4. R P Gilbert, Z Y Hou, X W Meng. Vekua theory in higher dimensional complex space: the Π−operator in Cn, Complex Var, 1993, 21: 99–103.

    Article  MathSciNet  Google Scholar 

  5. K Gürlebeck, W Sprössig. Quaternionic Analysis and Elliptic Boundary Value Problems, Birkhäuser, Basel, 1990.

    Book  MATH  Google Scholar 

  6. R P Gilbert, J L Buchanan. First Order Elliptic Systems: a Function Theoretic Approach, Academic Press, New York, 1983.

    MATH  Google Scholar 

  7. G N Hile. Elliptic systems in the plane with first order terms and constant coefficients, Comm Partial Differential Equations, 1978, 3: 949–977.

    Article  MathSciNet  MATH  Google Scholar 

  8. S Huang, Y Y Qiao, G C Wen. Real and Complex Clifford Analysis, Springer, 2005.

    MATH  Google Scholar 

  9. M V Keldysh. On some cases of degenerate elliptic equations on the boundary of a domain, Doklady Acad Nauk USSR, 1951, 77: 181–183.

    Google Scholar 

  10. X W Meng. T-operator in Cn and its applications, In: Integral Equations and Boundary Value Problems, World Scientific Publishing, Singapore, 1991.

    Google Scholar 

  11. E Obolashvili. Partial Differential Equations in Clifford Analysis, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol 96, 1998.

    Google Scholar 

  12. K Taira. Degenerate elliptic boundary value problems with asymptotically linear nonlinearity, Rend Circ Mat Palermo, 2011, 60: 283–308.

    Article  MathSciNet  MATH  Google Scholar 

  13. I N Vekua. Generalized Analytic Functions, Pergamon Press, Oxford, 1962.

    MATH  Google Scholar 

  14. G C Wen. Linear and Nonlinear Elliptic Complex Equations, Shanghai Scientific and Technical Publishers, 1986. (in Chinese)

    MATH  Google Scholar 

  15. G C Wen. Recent Progress in Theory and Applications of Modern Complex Analysis, Science Press, Beijing, 2010.

    MATH  Google Scholar 

  16. P W Yang. Hölder continuity of TG and Riemann-Hilbert boundary value problem in quaternionic analysis, Acta Math Sinica, 2003, 46: 993–998.

    MATH  Google Scholar 

  17. Z Zhao. Singular Integral Equation, Beijing Normal University Press, 1984.

    Google Scholar 

Download references

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Authors and Affiliations

  1. School of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, China

    Li-ping Wang

  2. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Guo-chun Wen

Authors
  1. Li-ping Wang
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  2. Guo-chun Wen
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Correspondence to Li-ping Wang.

Additional information

Supported by the National Science Foundation of China (11401162, 11571089, 11401159, 11301136) and the Natural Science Foundation of Hebei Province (A2015205012, A2016205218, A2014205069, A2014208158) and Hebei Normal University Dr. Fund (L2015B03).

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Wang, Lp., Wen, Gc. Boundary value problems for two types of degenerate elliptic systems in R 4 . Appl. Math. J. Chin. Univ. 31, 469–480 (2016). https://doi.org/10.1007/s11766-016-3285-3

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  • DOI: https://doi.org/10.1007/s11766-016-3285-3

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