Absence of bubbling phenomena for non-convex anisotropic nearly umbilical and quasi-Einstein hypersurfaces

We prove that, for every closed (not necessarily convex) hypersurface Σ in ℝ n + 1 {\mathbb{R}^{n+1}} and every p > n {p>n} , the L p {L^{p}} -norm of the trace-free part of the anisotropic second fundamental form controls from above the W 2 , p {W^{2,p}} -closeness of Σ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime p ≤ n {p\leq n} , the lack of convexity assumptions may lead in general to bubbling phenomena. Moreover, we obtain a stability theorem for quasi-Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates in the convex setting.

Maximal Fluctuations Around the Wulff Shape for Edge-Isoperimetric Sets in $$\varvec{{\mathbb {Z}}^d}$$: A Sharp Scaling Law

We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice $${\mathbb {Z}}^d$$ Z d from the limiting Wulff shape in arbitrary dimensions. As the number n of elements diverges, we prove that the symmetric difference to the corresponding Wulff set consists of at most $$O(n^{(d-1+2^{1-d})/d})$$ O ( n ( d - 1 + 2 1 - d ) / d ) lattice points and that the exponent $$(d-1+2^{1-d})/d$$ ( d - 1 + 2 1 - d ) / d is optimal. This extends the previously found ‘$$n^{3/4}$$ n 3 / 4 laws’ for $$d=2,3$$ d = 2 , 3 to general dimensions. As a consequence we obtain optimal estimates on the rate of convergence to the limiting Wulff shape as n diverges.

On Minimizers of an anisotropic liquid drop model

We consider a variant of Gamow's liquid drop model with an anisotropic surface energy. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. We show that for smooth anisotropies, in the small nonlocality regime, minimizers converge to the Wulff shape in $C^1$-norm and quantify the rate of convergence. We also obtain a quantitative expansion of the energy of any minimizer around the energy of a Wulff shape yielding a geometric stability result. For certain crystalline surface tensions we can determine the global minimizer and obtain its exact energy expansion in terms of the nonlocality parameter.

Anisotropic liquid drop models

We introduce and study certain variants of Gamow’s liquid drop model in which an anisotropic surface energy replaces the perimeter. After existence and nonexistence results are established, the shape of minimizers is analyzed. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. In sharp contrast, Wulff shapes are the unique minimizers for certain crystalline surface tensions. We also introduce and study several related liquid drop models with anisotropic repulsion for which the Wulff shape is the minimizer in the small mass regime.

Dynamic self-diffusion behaviors of nickel adatoms on clusters with Wulff shape

The dynamic self-diffusion behaviors of a nickel (Ni) adatom on the surface of the 4033-atom clusters with Wulff shape are studied. The interfacet and intrafacet self-diffusion processes are simulated via molecular dynamics (MDs) method, and the corresponding energy barriers are calculated by using the nudged elastic band (NEB) method. Based on the research and detailed analyses on all the possible diffusion processes, we found that the interfacet self-diffusion processes and intrafacet self-diffusion processes on the (100) facets are difficult to sustain at room-temperature because of high-energy barriers. However, the Ni adatoms can diffuse the inner (111) facet of the cluster easily either from the center of the facets to the step edges or along the step edges. This means that the epitaxial growth for Wulff[Formula: see text] cannot take place at low-temperature, and all facets of the cluster will grow uniformly.

Crystallization in the hexagonal lattice for ionic dimers

We consider finite discrete systems consisting of two different atomic types and investigate ground-state configurations for configurational energies featuring two-body short-ranged particle interactions. The atomic potentials favor some reference distance between different atomic types and include repulsive terms for atoms of the same type, which are typical assumptions in models for ionic dimers. Our goal is to show a two-dimensional crystallization result. More precisely, we give conditions in order to prove that energy minimizers are connected subsets of the hexagonal lattice where the two atomic types are alternately arranged in the crystal lattice. We also provide explicit formulas for the ground-state energy. Finally, we characterize the net charge, i.e. the difference of the number of the two atomic types. Analyzing the deviation of configurations from the hexagonal Wulff shape, we prove that for ground states consisting of [Formula: see text] particles the net charge is at most of order [Formula: see text] where the scaling is sharp.

Structures and energetics of low-index stoichiometric BiPO4 surfaces

Four low-index surfaces of monazite BiPO4 in the Wulff shape are investigated.

New stability results for spheres and Wulff shapes

We prove that a closed convex hypersurface of the Euclidean space with almost constant anisotropic first and second mean curvatures in the Lp-sense is W2,p-close (up to rescaling and translations) to the Wulff shape. We also obtain characterizations of geodesic hyperspheres of space forms improving those of [10] and [11].