On the boundary regularity of phase-fields for Willmore's energy

@article{Dondl2019OnTB,
  title={On the boundary regularity of phase-fields for Willmore's energy},
  author={P. Dondl and Stephan Wojtowytsch},
  journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics},
  year={2019},
  volume={149},
  pages={1017 - 1035}
}
Abstract We demonstrate that Radon measures which arise as the limit of the Modica-Mortola measures associated with phase-fields with uniformly bounded diffuse area and Willmore energy may be singular at the boundary of a domain and discuss implications for practical applications. We furthermore give partial regularity results for the phase-fields uε at the boundary in terms of boundary conditions and counterexamples without boundary conditions. 
1 Citations
Uniform regularity and convergence of phase-fields for Willmore’s energy
We investigate the convergence of phase fields for the Willmore problem away from the support of a limiting measure $$\mu $$μ. For this purpose, we introduce a suitable notion of essentially uniformExpand

References

SHOWING 1-10 OF 20 REFERENCES
Uniform regularity and convergence of phase-fields for Willmore’s energy
We investigate the convergence of phase fields for the Willmore problem away from the support of a limiting measure $$\mu $$μ. For this purpose, we introduce a suitable notion of essentially uniformExpand
Hitchhiker's guide to the fractional Sobolev spaces
This paper deals with the fractional Sobolev spaces W s;p . We analyze the relations among some of their possible denitions and their role in the trace theory. We prove continuous and compactExpand
Phase Field Models for Thin Elastic Structures with Topological Constraint
This article is concerned with the problem of minimising the Willmore energy in the class of connected surfaces with prescribed area which are confined to a small container. We propose a phase fieldExpand
Phase-field approximations of the Willmore functional and flow
We discuss in this paper phase-field approximations of the Willmore functional and the associated $${\mathrm L}^{2}$$L2-flow. After recollecting known results on the approximation of the WillmoreExpand
Local solutions to a free boundary problem for the Willmore functional
We consider a free boundary problem for the Willmore functional $$\mathcal{W}(f) = \frac{1}{4} \int _\Sigma H^2\,d\mu _f$$W(f)=14∫ΣH2dμf. Given a smooth bounded domain $$\Omega \subset {\mathbbExpand
The gradient theory of phase transitions and the minimal interface criterion
In this paper I prove some conjectures of GURTIN [15] concerning the Van der Waals-Cahn-Hilliard theory of phase transitions. Consider a fluid, under isothermal conditions and confined to a boundedExpand
Some remarks on Γ-convergence and least squares method
In the study of semicontinuity, relaxation, and Γ-convergence problems, few attention has been devoted, up to now, to questions concerning functionals arising in the study of differential equationsExpand
On a Modified Conjecture of De Giorgi
We study the Γ-convergence of functionals arising in the Van der Waals–Cahn–Hilliard theory of phase transitions. Their limit is given as the sum of the area and the Willmore functional. The problemExpand
Γ-convergence for nonlocal phase transitions
We discuss the Γ-convergence, under the appropriate scaling, of the energy functional ‖u‖Hs(Ω)2+∫ΩW(u)dx, with s∈(0,1), where ‖u‖Hs(Ω) denotes the total contribution from Ω in the Hs norm of u, andExpand
Functional Analysis, Sobolev Spaces and Partial Differential Equations
Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint.Expand
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