Variational methods for materials science, mechanics, and imaging

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LeoniDMS-1007989 The central theme of this project is the application ofmodern calculus of variations to materials science, mechanics,and imaging. The principal investigator emphasizes the followingspecific topics: - Material Voids in Elastic Solids with Anisotropic SurfaceEnergies. Here he studies how the qualitative features of theequilibrium shape of the void depend on the smoothness andconvexity (or lack thereof) of the interfacial energy. - Evolution by Surface Diffusion. The principal investigatorstudies the short-time existence and regularity of solutions of amodel of second-phase particles and material voids in a linearlyelastic solid. - Non-Local Operators for Image Recovery. Here he studiesregularity and qualitative features of solutions of a non-localmodel for image and signal processing proposed by Gilboa andOsher. He also takes up problems in water waves andmicromagnetics. The central theme of this project is the application ofmodern calculus of variations to materials science, mechanics,and imaging. When an elastic material is subject to stresses, itcan stretch and develop voids within the material; the shape ofthe voids affects other properties of the material. Theprincipal investigator studies how the shape depends on theenergy at the interface between the material and the void. Themovement of interfaces by means of surface diffusion is not wellunderstood when the interfacial energy is highly anisotropic. The investigator studies this problem, focusing on the short-timebehavior; rigorous results here improve our understanding ofvarious behaviors of materials. He also considers problems inimage processing, water waves, and micromagnetics. The resultsand methods developed in the project are incorporated in newgraduate courses, and some are included in a series of books thatthe investigator is writing in collaboration with I. Fonseca. Graduate students are involved in the project.