In this paper, we establish the global C^{2,α} and W^{2,p} regularity for the Monge-Amp`ere equation det D^2u = f subject to boundary condition Du(Ω) = Ω^∗, where Ω and Ω^∗ are bounded convex domains in the Euclidean space R^n with C^{1,1} boundaries, and f is a Ho ̈lder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications.

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We study wetting front (traveling wave) solutions to the Richards equation that describe the vertical infiltration of water through one-dimensional periodically layered unsaturated soils. We prove the existence, uniqueness, and large time asymptotic stability of the traveling wave solutions under prescribed flux boundary conditions and certain constitutive conditions. The traveling waves are connections between two steady state solutions that form near the ground surface and towards the underground water table. We found a closed form expression of the wave speed. The speed of a traveling wave is equal to the ratio of the flux difference and the difference of the spatial averages of the two steady states. We give both analytical and numerical examples showing that the wave speeds in the periodic soils can be larger or smaller than those in the homogeneous soils which have the same mean diffusivity and conductivity

A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in Rn. The nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion, including a notion of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory. In particular, it is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider. The numerous applications elucidate different interpretations of the operator and the associated governing equations. For example, a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process. Sufficient conditions on the kernel of the nonlocal operator and the notion of volume constraints are shown to lead to a well-posed problem. Volume constraints are a proxy for boundary conditions that may not be defined for a given kernel. In particular, we demonstrate for a general class of kernels that the nonlocal operator is a mapping between a volume constrained subspace of a fractional Sobolev subspace and its dual. We also demonstrate for other particular kernels that the inverse of the operator does not smooth but does correspond to diffusion. The impact of our results is that both a continuum analysis and a numerical method for the modeling of anomalous diffusion on bounded domains in Rn are provided. The analytical framework allows us to consider finite-dimensional approximations using discontinuous and continuous Galerkin methods, both of which are conforming for the nonlocal diffusion equation we consider; error and condition number estimates are derived.