Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiplicity results for mass constrained Allen-Cahn equations on Riemannian manifolds with boundary

Dario Corona, Stefano Nardulli, Ramon Oliver-Bonafoux, Giandomenico Orlandi, Paolo Piccione

Abstract

We present multiplicity results for mass constrained Allen-Cahn equations on a Riemannian manifold with boundary, considering both Neumann and Dirichlet conditions. These results hold under the assumptions of small mass constraint and small diffusion parameter. We obtain lower bounds on the number of solutions according to the Lusternik--Schnirelmann category of the manifold in case of Dirichlet boundary conditions and of its boundary in the case of Neumann boundary conditions. Under generic non-degeneracy assumptions on the solutions, we obtain stronger results based on Morse inequalities. Our approach combines topological and variational methods with tools from Geometric Measure Theory.

Multiplicity results for mass constrained Allen-Cahn equations on Riemannian manifolds with boundary

Abstract

We present multiplicity results for mass constrained Allen-Cahn equations on a Riemannian manifold with boundary, considering both Neumann and Dirichlet conditions. These results hold under the assumptions of small mass constraint and small diffusion parameter. We obtain lower bounds on the number of solutions according to the Lusternik--Schnirelmann category of the manifold in case of Dirichlet boundary conditions and of its boundary in the case of Neumann boundary conditions. Under generic non-degeneracy assumptions on the solutions, we obtain stronger results based on Morse inequalities. Our approach combines topological and variational methods with tools from Geometric Measure Theory.
Paper Structure (19 sections, 32 theorems, 147 equations, 3 figures)

This paper contains 19 sections, 32 theorems, 147 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Sketch of the proofs
  3. Proof of Theorem \ref{['theorem:ACH-Neumann']}
  4. The relative isoperimetric problem
  5. $\Gamma$-convergence
  6. Photography map
  7. Barycenter map
  8. Conclusion of the proof
  9. Proof of Theorem \ref{['theorem:ACH-Dirichet']}
  10. The isoperimetric problem
  11. Gamma convergence
  12. Photography map
  13. Barycenter map
  14. Conclusion of the proof
  15. Generic non-degeneracy
  16. ...and 4 more sections

Key Result

Theorem A

Assume that asu_nondegenerate, asu_coercive and asu_subcritical hold. Then, there exists $m^*>0$ such that for all $m \in (0,m^*)$ there exists $\varepsilon_{m},c_{m}>0$ such that for any $\varepsilon \in (0,\varepsilon_{m})$ the Neumann problem eq:PDE_Neumann has at least $\mathrm{cat}(\partial M)$

Figures (3)

  • Figure 1: The filled regions represent an arbitrary almost minimizer of the relative isoperimetric problem (left) and the isoperimetric problem (right). In the former case, most of the volume is concentrated inside a ball (dotted line) a half-ball centered at the boundary. In the latter case, concentration occurs around some point of $M$. However, some of the volume might be away from these balls, but it is always a small portion of the volume which can be controlled. In particular, half-balls and balls are almost minimizers of the relative isoperimetric and isoperimetric problems, repectively.
  • Figure 2: Depiction of the map $\mathcal{P}_{\varepsilon,m}$ introduced in Definition \ref{['def:photography']}. On the left, we see $u_{0,p}^m$, the indicator function of the set $E_{p,m}$, which is the set enclosed by the semi-sphere $\Sigma_{p,m}$ and $\partial M$. The map $\mathcal{P}_{\varepsilon,m}$ returns $u^m_{\varepsilon,m}$, depicted on the right, which is the smooth approximation of $u_{0,p}^m$ given by the $\Gamma$-convergence result (Theorem \ref{['theorem:G-convergence']}).
  • Figure 3: The map introduced in Definition \ref{['def:photography-Dirichlet']} turns the indicator function $u_{0,\mathcal{L}(p)}^m$ into the smooth approximation $u_{\varepsilon,\mathcal{L}(p)}^m$ given by $\Gamma$-convergence (Proposition \ref{['prop:Gamma-conv-Dirichlet']}). The functions are supported on the balls $B(\mathcal{L}(p),r_{\mathcal{L}(p),m})$, whose center $\mathcal{L}(p)$ is obtained by sending a point $p$ away from the boundary through the boundary layer map $\mathcal{L}$.

Theorems & Definitions (70)

  • Theorem A
  • Theorem B
  • Remark 1.1
  • Remark 1.2
  • Example 1.3
  • Example 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • Theorem C
  • ...and 60 more