Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-05T12:16:29.048Z Has data issue: false hasContentIssue false
  • English
  • Français

Corner Behavior of Solutions of Semilinear Dirichlet Problems

Published online by Cambridge University Press:  20 November 2018

Neil M. Wigley
Neil M. Wigley*
Affiliation:
University of Windsor, Windsor, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In recent years there has been considerable attention paid to the behavior of solutions of elliptic boundary value problems in domains with piecewise smooth boundary. In two dimensions the study concerns the behavior of a solution near a corner, and in three (or more) dimensions two cases have been given considerable attention: a conical vertex on the boundary, or an edge.

The solution of such a problem may be singular at the nonsmooth boundary points. The standard example in two dimensions is a solution in polar coordinates of the Dirichlet problem near a corner of interior angle πα;u = r1/α sin θ/α is a function which is harmonic in the sector 0 < θ < πα, has zero boundary values near the corner, and yet at the origin has unbounded derivatives of order > 1/α unless 1/α is an integer.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 6 , 01 December 1985 , pp. 1025 - 1046
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Agmon, S., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Commun. Pure Appl. Math 17 (1959), 623–727.Google Scholar
2. Carter, D. S., Local behavior of plane gravity flows at the confluence of free boundaries and analytic fixed boundaries, J. Math. Mech. 10 (1961), 441–450.Google Scholar
3. Carter, D. S., Local behavior of plane gravity flows II. Infinite jets on fixed boundaries, J. Math. Mech. 73 (1964), 329–352.Google Scholar
4. Dziuk, G., Das Verhalten von Lösungen semilinearer elliptischer System an Ecken eines Gebietes, Math. Z. 159 (1978), 89–100.Google Scholar
5. Fox, L. and Sankar, R., Boundary singularities in linear elliptic differential equations, J. Inst. Math. Appl. 5 (1969), 340–350.Google Scholar
6. Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order (Springer-Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
7. Kawohl, B., On non-linear mixed boundary value problems for second order elliptic differential equations on domains with corners, Proc. Roy. Soc. Edinburgh Sect. A 87 (1980/81), 35–51.Google Scholar
8. Kondrat'ev, V. A., Boundary-value problems for elliptic equations in domains with conic or corner points, Trudy Moskov. Mat. Obshch. 16 (1967), 209–292.Google Scholar
9. Kondrat'ev, V. A. and Oleinik, O., Boundary-value problems for partial differential equations in non-smooth domains, Uspekhi Mat. Nauk 38:2 (1983), 3–76.Google Scholar
10. Lehman, R. S., Developments in the neighborhood of the beach of surface waves over an inclined bottom, Comm. Pure Appl. Math. 7 (1954), 393–439.Google Scholar
11. Lehman, R. S., Development of the mapping function at an analytic corner, Pac. J. Math. 7 (1957), 1437–1449.Google Scholar
12. Lehman, R. S., Developments at an analytic corner of solutions of elliptic partial differential equations, J. Math. Mech. 8 (1959), 727–760.Google Scholar
13. Lehman, R. S. and Lewy, H., Uniqueness of water waves on a sloping beach, Comm. Pure Appl. Math. 14 (1961), 521–546.Google Scholar
14. Lewy, H., Developments at the confluence of analytic boundary conditions, Univ. Calif. Publ. Math. 1 (1950), 247–280.Google Scholar
15. Tolksdorf, P., On a quasilinear boundary value problem in a domain with corners or conical points, Z. Angew. Math. Mech. 62 (1982), 308–309.Google Scholar
16. Veidinger, L., On the order of convergence of finite element methods for the Neumann problem, Stud. Sci. Math. Hung. 13 (1978), 411–422.Google Scholar
17. Wasow, W., Asymptotic development of the solution of Dirichlet's problem at an analyticcorner, Duke Math. J. 24 (1957), 47–56.Google Scholar
18. Wigley, N. M., Asymptotic expansions at a corner of solutions of mixed boundary value problems, J. Math. Mech. 13 (1964) 549–576.Google Scholar
19. Wigley, N. M., Development of the mapping function at a corner, Pac. J. Math. 75 (1965), 1435–1461.Google Scholar
20. Wigley, N. M., Mixed boundary value problems in plane domains with corners, Math. Z. 115 (1970), 33–52.Google Scholar