Effective Hamiltonians in Perforated Domains: Inf-Max formula and Stability

We study the dependence of the effective Hamiltonian $\overline H_Ω(p)$ on the geometry of periodic perforated domains for convex Hamilton--Jacobi equations with state-constraint boundary conditions. We establish an inf--max representation over smooth test functions for domains with $C^1$ boundary. Using this representation, we derive a quantitative stability estimate in terms of the Hausdorff distance between domains and the reach of one boundary. This estimate yields convergence for non-nested domain approximations whose Hausdorff error is negligible relative to their boundary reach. Examples demonstrate the sharpness of the continuity modulus and show that Hausdorff convergence alone can fail to preserve the effective Hamiltonian, even for smooth domains and the quadratic Hamiltonian.