Non-local boundary energy forms for quasidiscs: Codimension gap and approximation

TL;DR

This work develops a rigorous framework for approximating non-local boundary energy forms of fractional Laplace type defined on fractal-like quasicircle boundaries by analogous forms on polygonal prefractal curves. Using the Kuwae–Shioya KS-generalized Mosco convergence across varying Hilbert spaces, the authors establish convergence of boundary energies, Dirichlet energies on approximating domains, and their natural, Lipschitz-trace couplings, via explicit averaging techniques to address the codimension gap. They construct polygonal approximations that yield quasidiscs with uniform geometric control and prove Mosco convergence for Dirichlet forms and for the sum of domain and boundary energies, enabling stability results for elliptic and parabolic problems with non-local Wentzell-type boundary conditions. The results provide explicit, provable prefractal-to-fractal convergence and have potential numerical implications for simulating boundary dynamics on fractal interfaces. Overall, the paper integrates non-local boundary energy forms, trace spaces, and Mosco convergence to bridge fractal boundaries and their polygonal approximations in a quantitative, applicative setting.

Abstract

We consider non-local energy forms of fractional Laplace type on quasicircles and prove that they can be approximated by similar energy forms on polygonal curves. The approximation is in terms of generalized Mosco convergence along a sequence of varying Hilbert spaces. The domains of the energy forms are the natural trace spaces, and we focus on the case of quasicircles of Hausdorff dimension greater than one. The jump in Hausdorff dimension results in a mismatch of fractional orders, which we compensate by a suitable choice of kernels. We provide approximations of quasidiscs by polygonal $(\varepsilon,\infty)$-domains with common parameter $\varepsilon>0$ and show convergence results for superpositions of Dirichlet integrals and non-local boundary energy forms.

Figures (2)

• Figure 3: Koch snowflake $\Gamma$ and polygon $\Gamma_0$.
• Figure 4: Koch snowflake $\Gamma$ and polygon $\Gamma_1$.

Theorems & Definitions (49)

• Remark 2.1
• Remark 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Remark 3.2
• Remark 3.3
• Lemma 3.3
• Lemma 3.4