Elliptic Superlevel Sets in the Class OO_C: Boundary-Layer Geometry and Stability of the Return Map

Abstract

We study elliptic superlevel geometry and return dynamics for the class 𝒪_C. Let C ⊂ ℝ^N be a compact strictly convex core, and Ω∈ 𝒪_C a bounded domain whose boundary is a global normal graph over ∂ C with inward normals returning to int C. For −Δu = f in Ω (u=0 on ∂Ω), with f ≥ 0, f not≡ 0, supp f Subset int C, we analyze the superlevel sets Ω_t = {u > t} for t ↓ 0. We prove that for small t > 0, ∂Ω_t is a global normal graph given by ∂Ω_t = {c + d_t(c)ν(c) : c ∈ ∂ C}, satisfying |d_t − d₀|_C¹ = O(t) and d_t = d₀ − t b₀ + o_C¹(t), where b₀ depends on the Hopf derivative and transversality. Using normal-graph formulas, we show Ω_t ∈ 𝒪_C. The return maps F_t : ∂ C → ∂ C are well-defined, satisfying |F_t − F₀|_L^∞ = O(t) and F_t = F₀ + t 𝒜₀ + o_L^∞(t). For any sequence t_n ↓ 0, the non-autonomous system x_n+1 = F_{t_n}(x_n) is asymptotically autonomous with background map F₀. These results bridge elliptic boundary-layer deformation and return-map dynamics relative to a convex core.

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BibTeX RIS

BibTeX

@misc{morsalani2026elliptic,
  title = {{Elliptic Superlevel Sets in the Class $\OO_C$: Boundary-Layer Geometry and Stability of the Return Map}},
  author = {Mohamed El Morsalani and Mohammd Barkatou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.16912},
  eprint = {2607.16912},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.16912},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Elliptic Superlevel Sets in the Class OO_C: Boundary-Layer Geometry and Stability of the Return Map
AU  - Mohamed El Morsalani
AU  - Mohammd Barkatou
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2607.16912
UR  - https://arxiv.org/abs/2607.16912
AB  - We study elliptic superlevel geometry and return dynamics for the class 𝒪_C. Let C ⊂ ℝ^N be a compact strictly convex core, and Ω∈ 𝒪_C a bounded domain whose boundary is a global normal graph over ∂ C with inward normals returning to int C. For −Δu = f in Ω (u=0 on ∂Ω), with f ≥ 0, f not≡ 0, supp f Subset int C, we analyze the superlevel sets Ω_t = {u > t} for t ↓ 0. We prove that for small t > 0, ∂Ω_t is a global normal graph given by ∂Ω_t = {c + d_t(c)ν(c) : c ∈ ∂ C}, satisfying |d_t − d_0|_C^1 = O(t) and d_t = d_0 − t b_0 + o_C^1(t), where b_0 depends on the Hopf derivative and transversality. Using normal-graph formulas, we show Ω_t ∈ 𝒪_C. The return maps F_t : ∂ C → ∂ C are well-defined, satisfying |F_t − F_0|_L^∞ = O(t) and F_t = F_0 + t 𝒜_0 + o_L^∞(t). For any sequence t_n ↓ 0, the non-autonomous system x_n+1 = F_t_n(x_n) is asymptotically autonomous with background map F_0. These results bridge elliptic boundary-layer deformation and return-map dynamics relative to a convex core.
ER  -

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