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A variational approach to a nonlinear Steklov problem

Published online by Cambridge University Press:  14 November 2011

B. Ermens  and
J. Mawhin
B. Ermens
Affiliation:
Université de Louvain, Institut Mathématique, B-1348 Louvain-la Neuve, Belgium
J. Mawhin
Affiliation:
Université de Louvain, Institut Mathématique, B-1348 Louvain-la Neuve, Belgium
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Synopsis

We study the Steklov problem which consists in finding a complex function w(z) = U(z) + iV(z) holomorphic in the open unit disc G of the complex plane, continuous on its closure, such that V(0) = 0 and verifying on its boundary the condition

(d/ds)V(eis)+g(s, U(eis))=h(s),

where g and h are given functions. Using the Hilbert transform, the problem is reduced to the search for periodic solutions of an equivalent singular integro-differential equation which is treated by the direct method of the calculus of variations. When g(s,.) is non-decreasing, we obtain a necessary and sufficient condition for the solvability. The case of a non-monotone nonlinearity is also considered.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 1-2 , 1990 , pp. 139 - 149
Copyright
Copyright © Royal Society of Edinburgh 1990

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