arXiv subjects

Michael Gelantalis

Publications and source records attributed to Michael Gelantalis.

3 recordsLinked to original sources

arXiv2019

Symmetry of constrained minimizers of the Cahn-Hilliard energy on the torus

We establish sufficient conditions for a function on the torus to be equal to its Steiner symmetrization and apply the result to volume-constrained minimizers of the Cahn-Hilliard energy. We also show how two-point rearrangements can be used to establish symmetry for the Cahn-Hilliard model. In two dimensions, the Bonnesen inequality can then be applied to quantitatively estimate the sphericity of superlevel sets.

math.AP↗

arXiv2016

Existence and properties of certain critical points of the Cahn-Hilliard energy

The Cahn-Hilliard energy landscape on the torus is explored in the critical regime of large system size and mean value close to $-1$. Existence and properties of a "droplet-shaped" local energy minimizer are established. A standard mountain pass argument leads to the existence of a saddle point whose energy is equal to the energy barrier, for which a quantitative bound is deduced. In addition, finer properties of the local minimizer and appropriately defined constrained minimizers are deduced. The proofs employ the $Γ$-limit (identified in a previous work), quantitative isoperimetric inequalities, variational arguments, and Steiner symmetrization.

math.AP↗

arXiv2014

Energy barrier and $Γ$-convergence in the $d$-dimensional Cahn-Hilliard equation

We study the d-dimensional Cahn-Hilliard equation on the flat torus in a parameter regime in which the system size is large and the mean value is close---but not too close---to -1. We are particularly interested in a quantitative description of the energy landscape in the case in which the uniform state is a local but not global energy minimizer. In this setting, we derive a sharp leading order estimate of the size of the energy barrier surrounding the uniform state. A sharp interface version of the proof leads to a $Γ$-limit of the rescaled energy gap between a given function and the uniform state.

math.AP↗