We study three different problems in non linear PDE. The first problem relates to finding equilibrium configurations in periodic media, more precisely, given an Area-Dirichlet functional J, which is periodic under integer translations and given three planes in R[superscript d], we proof there exists at least one minimizer such that it’s positive part, negative part and zero set remain at a uniform bounded distance of each plane. The second and third problem are related to non local diffusion, in the elliptic non symmetric case and parabolic case. In both cases we are interested in proving interior regularity for solutions of the aforementioned equations. / text

This dissertation presents a design concept for shape-morphing structures that have two stable configurations. The design concept defines the methodology of transforming a planar structural shape into spatial structural shape using bistable compliant mechanisms. Bistable complaint mechanisms are used to achieve structural stable configurations. The dissertation incorporating geometrical relationships for the mechanisms that form the primary structure described in step-by-step process. This dissertation implements the design layouts for designer to creating shape-morphing structures including origami. The novel contribution of the work is classified in three models. The first model presents a methodology to induce bistability behavior into an origami reverse fold and partially spherical compliant mechanism. The second model presents the design and development of a bistable triangle-shaped compliant mechanism with motion limits and dwell behavior at the two stable configurations. This mechanism can be arrayed to create shape-morphing structures. The third model presents a design and development for a collapsible bistable compliant mechanism used for a shape morphing lamina-emergent frustum. Finally, physical prototypes of all models are presented as proof of concept.