Dimension reduction of fractional Sobolev seminorms on thin domains
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912966191874048 |
|---|---|
| author | Braides, Andrea Pinamonti, Andrea Solci, Margherita |
| author_facet | Braides, Andrea Pinamonti, Andrea Solci, Margherita |
| contents | We study the asymptotic behaviour of Gagliardo seminorms in $H^s$ defined on thin films $Ω_\e=ω\times(0,\e)$. The first relevant order is $\e^{1-2s}$, at which the corresponding limit captures the vertical fractional oscillations through one-dimensional sections. The second relevant order produces dimension-reduction regimes that undergo a qualitative transition at the critical exponent $s=\tfrac12$. For $s<\tfrac12$, the dominant contribution is driven by interactions at finite planar distance, and the dimension-reduction scale is $\e^2$. In this regime, the limit is a lower-dimensional \emph{fractional} energy with an effective gain of $\tfrac12$ in the differentiability index. At the critical exponent $s=1/2$, the dimension-reduction scale is $\e^{2}|\log\e|$, and the limit is {\em local}, with dominant interactions at scales between $\e$ and $1$, producing a Dirichlet-type limit on $ω$. For $s>\tfrac12$, the dominant contribution is instead driven by interactions at distances of order $\varepsilon$, the dimension-reduction scale is $\e^{3-2s}$, and the second-order $Γ$-limit is still local. We also study the case $s=s_\e\to 1^-$, showing a Bourgain--Brezis--Mironescu-type result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_13968 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dimension reduction of fractional Sobolev seminorms on thin domains Braides, Andrea Pinamonti, Andrea Solci, Margherita Analysis of PDEs Functional Analysis 49J45, 46E35, 35R11, 49J53, 74K35 We study the asymptotic behaviour of Gagliardo seminorms in $H^s$ defined on thin films $Ω_\e=ω\times(0,\e)$. The first relevant order is $\e^{1-2s}$, at which the corresponding limit captures the vertical fractional oscillations through one-dimensional sections. The second relevant order produces dimension-reduction regimes that undergo a qualitative transition at the critical exponent $s=\tfrac12$. For $s<\tfrac12$, the dominant contribution is driven by interactions at finite planar distance, and the dimension-reduction scale is $\e^2$. In this regime, the limit is a lower-dimensional \emph{fractional} energy with an effective gain of $\tfrac12$ in the differentiability index. At the critical exponent $s=1/2$, the dimension-reduction scale is $\e^{2}|\log\e|$, and the limit is {\em local}, with dominant interactions at scales between $\e$ and $1$, producing a Dirichlet-type limit on $ω$. For $s>\tfrac12$, the dominant contribution is instead driven by interactions at distances of order $\varepsilon$, the dimension-reduction scale is $\e^{3-2s}$, and the second-order $Γ$-limit is still local. We also study the case $s=s_\e\to 1^-$, showing a Bourgain--Brezis--Mironescu-type result. |
| title | Dimension reduction of fractional Sobolev seminorms on thin domains |
| topic | Analysis of PDEs Functional Analysis 49J45, 46E35, 35R11, 49J53, 74K35 |
| url | https://arxiv.org/abs/2603.13968 |