Dirichlet problems in lens and lune. (English) Zbl 1388.30046
Summary: In this article, we obtain the explicit solvability of the Dirichlet problem for the Cauchy-Riemann operator, for a Beltrami operator with constant coefficient, and for the Bitsadze/Laplace operator in a lens and two complementary lunes. Using the parqueting-reflection technique, Cauchy-Pompeiu integral formulas are established. Then the expressions of both the solutions and the solvability conditions are explicitly obtained. The boundary behavior of resulting integral operators is discussed.
MSC:
| 30E25 | Boundary value problems in the complex plane |
| 30E20 | Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane |
| 35J56 | Boundary value problems for first-order elliptic systems |
| 35J57 | Boundary value problems for second-order elliptic systems |
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