On a Nonlinear Boundary Integral Equation
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The aim of this article is to study the Laplacian in a bounded set of R2 with smooth boundary with a nonlinear boundary condition. If the solution of this problem is represented by a double layer potential, a nonlinear boundary integral equation is obtained. The linear part of this equation is interpreted as a pseudo-differential operator; and via the symbolic computation of this latter, we can give the properties of this operator. In the end, the existence of the solution will be a consequence of applying Krasnosel’skii fixed point theorem.
Abstract
The aim of this article is to study the Laplacian in a bounded set of R2 with smooth boundary with a nonlinear boundary condition. If the solution of this problem is represented by a double layer potential, a nonlinear boundary integral equation is obtained. The linear part of this equation is interpreted as a pseudo-differential operator; and via the symbolic computation of this latter, we can give the properties of this operator. In the end, the existence of the solution will be a consequence of applying Krasnosel’skii fixed point theorem.
참고문헌 (9)
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- M. Krasnosel’skii, "On the continuity of the operator Fu(x) = f(x, u(x))", Dokl. Acad.Nauk.SSSR, 77 (1951)
- M. Krasnosel’skii, "Topological methods in the theory of nonlinear integral equations" (1964), Mac Millan
- K.Ruotsalainen, "On the boundary element method for some nonlinear boundary value problems", Numer. Math., 53, pp.299~314 (1988)
- D. R. Smart, "Fixed point theorems" (1980), Cambridge University Press
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