Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem

Area of Science:

• Mathematical analysis
• Partial differential equations
• Asymptotic analysis

Background:

• Boundary value problems for the Laplace equation are fundamental in physics and engineering.
• Perforated domains introduce complexities in analyzing solution behavior.
• Robin boundary conditions model various physical phenomena, including heat transfer and electromagnetism.

Purpose of the Study:

• To investigate the asymptotic behavior of solutions to a boundary value problem for the Laplace equation in a perforated domain.
• To analyze the impact of three interacting degeneracies: Robin to Neumann condition transition, infinite Robin datum, and collapsing hole size (epsilon -> 0).
• To represent the solution and its energy integral using real analytic maps and functions of singular perturbation parameters.

Main Methods:

• Asymptotic analysis of partial differential equations.
• Study of boundary value problems in domains with small holes.
• Analysis of singular perturbations and their impact on solutions.

Main Results:

• The study characterizes the asymptotic behavior of solutions as the hole size (epsilon) approaches zero.
• It quantifies the influence of the degenerating Robin boundary condition and infinite Robin datum on the solution.
• The solution and its energy integral are expressed in terms of real analytic maps and functions of the perturbation parameters.

Conclusions:

• The interaction of multiple singularities in boundary value problems can be systematically analyzed.
• Understanding these asymptotic behaviors is crucial for applications involving perforated domains with complex boundary conditions.
• The findings contribute to the field of non-smooth variational problems and their applications.