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Smoothness Properties of Bounded Solutions of Dirichlet's Problem for Elliptic Equations in Regions with Corners on the Boundary

Published online by Cambridge University Press:  20 November 2018

A. Azzam
A. Azzam*
Affiliation:
Department of Mathematics University of WindsorWindsorOnt. N9B 3P4
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We study here the smoothness of solutions of the Dirichlet problem for elliptic equations in a region G with a piece-wise smooth boundary. The smoothness of the solution given depends on the smoothness of the coefficients of the equation, the boundary, the boundary function and the values of the angles on the boundary and the values of the coefficients of the second derivatives at the corners.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 23 , Issue 2 , 01 June 1980 , pp. 213 - 226
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Agmon, S., Douglis, A., Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations, satisfying general boundary conditions. Comm. Pure Appl. Math N 12 (1959), 623-727.Google Scholar
2. Browder, F., Estimates and existence theorems for elliptic boundary value problems. Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 365-372.Google Scholar
3. Fufaev, V. V., On the Dirichlet problem for regions with corners, Dolk, Akad. Nauk SSSR 131 (1960), 37-39.Google Scholar
4. Kellog, O. D., On the derivatives of harmonic functions on the boundary. Trans. Amer. Math. Soc. 33 (1931), 486-510.Google Scholar
5. Kondratev, V. A., Boundary value problems for elliptic equations in domains with conical or angular points. Tran. of the Moscow Mat. Soc. 16 (1967), 227-313.Google Scholar
6. Kondratev, V. A., The smoothness of a solution of Dirichlefs problem for second order elliptic equations in a region with a piecewise smooth boundary. Diff?rents!alnye Uravneneya 10 (1970), 1831-1843.Google Scholar
7. Ladyzhenskaya, O. A., Ural'tseva, N. N., Linear and quasilinear elliptic equations. Academic Press, New York and London 1968.Google Scholar
8. Lehman, R.Sherman, Developments at an analytic corner of the solutions of elliptic partial differential equations. J. Math. Mech. 8 (1959), 727-760.Google Scholar
9. Lewy, H., Developments at the confluence of analytic boundary conditions. Univ. California Publ. Math. (N.S.) 1, (1950) 247-280.Google Scholar
10. Miranda, C., Partial differential equations of elliptic type. Springer-Verlag New York 1970.Google Scholar
11. Nikolskii, S. M., Harmonic functions on rectangular regions. Mat. Sbornik (N.S.), 43 (85) (1957), 127-144.Google Scholar
12. Sobolev, S. L., Applications of functional analysis in mathematical physics. Providence, A.M.S., 1963.Google Scholar
13. Volkov, E. A., On the differentiability properties of solutions of boundary value problems for the Laplace and Poisson equations on a rectangle. Trudy Mat. Inst. Steklov, 77 (1965), 89-112.Google Scholar
14. Volkov, E. A., Differential properties of solutions of boundary value problems for the Laplace equation on polygons. Trudy Mat. Inst. Steklov, 77 (1965), 113-142.Google Scholar
15. Wasow, W., Asymptotic development of the solution of Dirichlefs problem at analytic corners. Duke Math. J. 24 (1957), 47-56.Google Scholar
16. Wigley, Neil M., Asymptotic expansion at a corner of solutions of mixed boundary value problems. J. Math. Mech. 13 (1964), 549-576.Google Scholar
17. Wigley, Neil M., Mixed boundary value problems in plane domains with corners. Math. Z. 115 (1970) 33-52.Google Scholar