[Paper Review] Homogenization of the Oscillating Dirichlet Boundary Condition in General Domains

William M. Feldman|arXiv (Cornell University)|Mar 7, 2013

Advanced Mathematical Modeling in Engineering4 citations

TL;DR

This paper establishes the homogenization of fully nonlinear elliptic equations with periodic oscillations in both the operator and the Dirichlet boundary condition on general smooth bounded domains. Using a novel comparison principle with partial boundary data, it proves local uniform convergence of solutions despite potentially discontinuous homogenized boundary data, extending prior results from half-spaces to general domains under minimal assumptions on the operators.

ABSTRACT

We prove the homogenization of the Dirichlet problem for fully nonlinear elliptic operators with periodic oscillation in the operator and of the boundary condition for a general class of smooth bounded domains. This extends the previous results of Barles and Mironescu in half spaces. We show that homogenization holds despite a possible lack of continuity in the homogenized boundary data. The proof is based on a comparison principle with partial Dirichlet boundary data which is of independent interest.

Motivation & Objective

• To extend the homogenization of oscillating Dirichlet boundary conditions from half-spaces to general smooth bounded domains.
• To establish convergence of solutions to a homogenized limit problem even when the homogenized boundary data is discontinuous.
• To prove the existence and uniqueness of the homogenized solution using a comparison principle with partial boundary data.
• To identify conditions under which the homogenized boundary data is continuous, particularly excluding a set of small Hausdorff dimension.

Proposed method

• Blow-up analysis near boundary points to derive a cell problem in the half-space, modeling the local behavior of the solution near the boundary.
• Define the homogenized boundary data as the asymptotic limit of solutions to the cell problem, depending on the normal direction and the operator's periodic structure.
• Use a comparison principle with partial boundary data to control the upper and lower relaxed limits of the approximating solutions.
• Establish continuity of the homogenized boundary data on a residual set of full measure by leveraging the regularity of the domain and convergence of operators and normals.
• Apply barrier arguments and relaxed limits to show that the upper and lower limits of the solutions coincide in the interior, implying local uniform convergence.
• Leverage the interior homogenization result for fully nonlinear operators to link the behavior away from the boundary to the homogenized equation.

Experimental results

Research questions

• RQ1 Does homogenization of the Dirichlet problem hold for fully nonlinear elliptic equations with oscillating operators and boundary conditions in general domains, even when the homogenized boundary data is discontinuous?
• RQ2 Can a comparison principle with partial boundary data be used to establish convergence when the boundary data oscillates at small scales?
• RQ3 What conditions on the domain and operators ensure that the homogenized boundary data is continuous at most boundary points?
• RQ4 How does the geometry of the domain, particularly the normal vector and tangential derivatives, influence the continuity of the homogenized boundary condition?
• RQ5 To what extent can the results be extended to oscillating Neumann boundary conditions using similar techniques?

Key findings

• The solutions $ u^511 $ converge locally uniformly in $ \Omega $ to a limit $ \overline{u} $, even when the homogenized boundary data $ \overline{g} $ is discontinuous.
• The homogenized boundary data $ \overline{g}(x) = \overline{\mu}(g(x,\cdot), F^x, \nu_x) $ is continuous at all points $ x \in \partial\Omega \setminus \Gamma $, where $ \Gamma $ is a set of Hausdorff dimension less than $ \beta_0 $.
• The convergence is established via a comparison principle with partial boundary data, which is a key technical contribution independent of the main result.
• The upper and lower relaxed limits of $ u^\varepsilon $ coincide in $ \Omega $, implying local uniform convergence, due to the control of boundary layer effects through the choice of $ R $ and $ \eta $.
• The proof relies on barrier constructions and the stability of viscosity solutions under convergence of operators and boundary data, even in the absence of equicontinuity.
• The result generalizes prior work by Barles and Mironescu from half-spaces to general domains, removing the need for rotational invariance or continuity assumptions on the homogenized boundary data.