<div>Understanding grain growth in dielectric ceramics is essential to controlling the electrical and mechanical properties necessary to produce ceramic capacitors and sensors. The effect of alloying barium titanate with strontium titanate on the equilibrium crystal shape was investigated in order to determine possible impacts on grain growth. The equilibrium crystal shape was studied through three experimental methods to identify possible changes in grain boundary energy or anisotropy with changing composition.</div><div>The first method was by imaging intergranular pores to observe faceting behavior and relative interfacial energies. Intergranular pores were reconstructed to determine the relative surface energies of the identified facets. The second method was to perform atomic force microscopy on surface facets to collect topography data. The topography data was combined with orientation data obtained by EBSD analysis from the same region, and used to calculate the normal vector of the surface facets. These datasets were plotted in a stereographic projection to study the faceting anisotropy. The third method involved collecting EBSD orientation data and images of surface faceting behavior. The surface faceting behavior of each grain was categorized by type of facet and plotted on a stereographic projection at the corresponding orientation. This allowed for the analysis of faceting transitions and the differentiation of faceted and continuous regions of the equilibrium crystal shape. The analysis of faceting behavior across compositions has implications on grain growth of the barium titanate/strontium titanate system.</div>

<p dir="ltr">We investigate the surface diffusion flow of smooth curves with anisotropic surface energy.</p><p dir="ltr">This geometric flow is the H−1-gradient flow of an energy functional. It preserves the area</p><p dir="ltr">enclosed by the evolving curve while at the same time decreases its energy. We show the</p><p dir="ltr">existence of a unique local in time solution for the flow but also the existence of a global in</p><p dir="ltr">time solution if the initial curve is close to the Wulff shape. In addition, we prove that the</p><p dir="ltr">global solution converges to the Wulff shape as t → ∞. In the current setting, the anisotropy</p><p dir="ltr">is not too strong so that the Wulff shape is given by a smooth curve. In the last section, we</p><p dir="ltr">formulate the corresponding problem when the Wulff shape exhibits corners.</p>