A nonlinear problem for the Laplace equation with a degenerating Robin condition

作者：Musolino Paolo^*; Mishuris Gennady

来源：Mathematical Methods in the Applied Sciences, 2018, 41(13): 5211-5229.

DOI：10.1002/mma.5072

• boundary value problems for second-order elliptic equations
• integral equations methods
• Laplace operator
• Neumann problem
• Robin problem
• singularly perturbed problem

We investigate the behavior of the solutions of a mixed problem for the Laplace equation in a domain . On a part of the boundary , we consider a Neumann condition, whereas in another part, we consider a nonlinear Robin condition, which depends on a positive parameter in such a way that for =0 it degenerates into a Neumann condition. For small and positive, we prove that the boundary value problem has a solution u(,). We describe what happens to u(,) as 0 by means of representation formulas in terms of real analytic maps. Then, we confine ourselves to the linear case, and we compute explicitly the power series expansion of the solution.

• 出版日期2018-9-15