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15 papers · 1 filter
A variational theory for monotone vector fields
Nassif Ghoussoub
Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators, but also as natural extensions of gradients of convex pote…
Scattering theory for the Gross-Pitaevskii equation in three dimensions
S. Gustafson, K. Nakanishi, T. -P. Tsai
We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized ene…
Random Homogenization of Fractional Obstacle Problems
L. A. Caffarelli, A. Mellet
We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domain…
Bessel potentials and optimal Hardy and Hardy-Rellich inequalities
Nassif Ghoussoub, Amir Moradifam
We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R^n,, so that the following inequalities hold for al…
Schrodinger equations and Hamiltonian systems of PDEs with selfdual boundary conditions
Nassif Ghoussoub, Abbas Moameni
Selfdual variational calculus is further refined and used to address questions of existence of local and global solutions for various parabolic semi-linear equations, Hamiltonian s…
Random homogenization of an obstacle problem
Luis A. Caffarelli, Antoine Mellet
We study the homogenization of an obstacle problem in a perforated domain. The holes are periodically distributed but have random size and shape. The capacity of the holes is assum…