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Titlebook: Geometric Partial Differential Equations; Antonin Chambolle,Matteo Novaga,Enrico Valdinoci Conference proceedings 2013 Scuola Normale Supe

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Existence and qualitative properties of isoperimetric sets in periodic media,media. We also describe the qualitative properties of the homogenized surface tension, also known as stable norm (or minimal action) in Weak KAM theory. In particular we investigate its strict convexity and differentiability properties.
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Homogenization with oscillatory Neumann boundary data in general domain,is oscillatory. Our result on the Neumann problem is the nonlinear version of the classical result in [4] for divergence-form operators with co-normal boundary data. We also discuss the Dirichlet boundary problem.
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On general existence results for one-dimensional singular diffusion equations with spatially inhomoen by the graph of a periodic function. We prove that the initial value problem admits a unique global-in-time viscosity solution for a general periodic continuous initial datum. The notion of a viscosity solution used here is the same as proposed by Giga, Giga and Rybka, who established a compariso
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