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Phase transitions and minimal interfaces on manifolds with conical singularities

Daniel Grieser, Sina Held, Hannes Uecker, Boris Vertman

Abstract

Using $Γ$-convergence, we study the Cahn-Hilliard problem with interface width parameter $\varepsilon > 0$ for phase transitions on manifolds with conical singularities. We prove that minimizers of the corresponding energy functional exist and converge, as $\varepsilon \to 0$, to a function that takes only two values with an interface along a hypersurface that has minimal area among those satisfying a volume constraint. In a numerical example, we use continuation and bifurcation methods to study families of critical points at small $\varepsilon > 0$ on 2D elliptical cones, parameterized by height and ellipticity of the base. Some of these critical points are minimizers with interfaces crossing the cone tip. On the other hand, we prove that interfaces which are minimizers of the perimeter functional, corresponding to $\varepsilon = 0$, never pass through the cone tip for general cones with angle less than $2π$. Thus tip minimizers for finite $\varepsilon > 0$ must become saddles as $\varepsilon \to 0$, and we numerically identify the associated bifurcation, finding a delicate interplay of $\varepsilon > 0$ and the cone parameters in our example.

Phase transitions and minimal interfaces on manifolds with conical singularities

Abstract

Using -convergence, we study the Cahn-Hilliard problem with interface width parameter for phase transitions on manifolds with conical singularities. We prove that minimizers of the corresponding energy functional exist and converge, as , to a function that takes only two values with an interface along a hypersurface that has minimal area among those satisfying a volume constraint. In a numerical example, we use continuation and bifurcation methods to study families of critical points at small on 2D elliptical cones, parameterized by height and ellipticity of the base. Some of these critical points are minimizers with interfaces crossing the cone tip. On the other hand, we prove that interfaces which are minimizers of the perimeter functional, corresponding to , never pass through the cone tip for general cones with angle less than . Thus tip minimizers for finite must become saddles as , and we numerically identify the associated bifurcation, finding a delicate interplay of and the cone parameters in our example.
Paper Structure (27 sections, 12 theorems, 64 equations, 13 figures)

This paper contains 27 sections, 12 theorems, 64 equations, 13 figures.

Table of Contents

  1. Introduction and statement of the main results
  2. The Cahn-Hilliard problem
  3. Relation to minimal hypersurfaces
  4. Main results: a) Convergence of minimizers
  5. Main results: b) Minimizers on conical surfaces
  6. Main results: c) Numerical study of minimizers
  7. Some related problems
  8. Structure of the paper
  9. Convergence of minimizers on singular spaces
  10. Existence and regularity of solutions
  11. Self-adjoint extensions of the Laplacian
  12. Existence of minimizers
  13. Convergence
  14. Minimizers on cones in 2D: interfaces
  15. Minimizers on cones: Numerical illustrations
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $\Omega\subset{\mathbb R}^d$ be a bounded domain with Lipschitz boundary. Let $(u_\varepsilon)$ be a sequence of minimizers of $E_{\varepsilon}$ with $\varepsilon \to 0$, subject to $q(u_\varepsilon)=0$ with $|m|<1$. Then there exists a subsequence of $(u_\varepsilon)$ that converges in $L^1(\Om

Figures (13)

  • Figure 1: Three basic types T1, T2 and T3 of interfaces on a cone of height $h=2$, ellipticity $a=1.05$ (almost circular); $\varepsilon=0.1$, energy $E=E_\varepsilon$ as given, approximating the interface length. The tip interface T1 is a saddle point here (but for fixed small $\varepsilon>0$ becomes a global minimizer on a sufficiently elliptic cone); T2 (winding) is the global minimizer, and T3 (roughly horizontal) is a local minimizer; see §\ref{['numsec']} for details.
  • Figure 2: Illustration of $V_n, E'_n, G'_n$ and $G"_n$.
  • Figure 3: Illustration of curve $\gamma$.
  • Figure 4: Illustration of $E_n$.
  • Figure 5: (a) $l_j(h)$. (b) Sketch for computing $l_2$. The pacman shape arises from cutting open the cone along a radius, and flattening it into the plane.
  • ...and 8 more figures

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 1.4
  • Conjecture 1.5
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 28 more