تقرير
Geometric Variations of an Allen-Cahn Energy on Hypersurfaces
| العنوان: | Geometric Variations of an Allen-Cahn Energy on Hypersurfaces |
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| المؤلفون: | Marx-Kuo, Jared, Silva, Érico Melo |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Differential Geometry, Mathematics - Analysis of PDEs |
| الوصف: | We introduce an Allen-Cahn type functional, $\text{BE}_{\epsilon}$, that defines an energy on separating hypersurfaces, $Y$, of closed Riemannian Manifolds. We establish $\Gamma$-convergence of $\text{BE}_{\epsilon}$ to the area functional, and compute first and second variations of this functional under hypersurface pertrubations. We then compute an explicit expansion for the variational formula as $\epsilon \to 0$. A key component of this proof is the invertibility of the linearized Allen-Cahn equation about a solution, on the space of functions vanishing on $Y$. We also relate the index and nullity of $\text{BE}_{\epsilon}$ to the Allen-Cahn index and nullity of a corresponding solution vanishing on $Y$. We apply the second variation formula and index theorems to show that the family of $2p$-dihedrally symmetric solutions to Allen-Cahn on $S^1$ have index $2p - 1$ and nullity $1$. Comment: Updated 7-16-23. Modified statement of corollary 2.2 Minor notation clarifications |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2304.11775 |
| رقم الأكسشن: | edsarx.2304.11775 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |