The operator square root of the Laplacian (âˆ’â–³) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to theâ€¦ (More)

in a punctured ball, B,(O) \ (0) c Râ€œ, n 2 3, with an isolated singularity at the origin. The model equation (1.1) arises in many physical contexts but its greatest interest in recent years lies inâ€¦ (More)

Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-diffusion equations with L initial data and minimal assumptions on the drift are locally Holder continuous. As anâ€¦ (More)

The problem of minimizing /[Vu|2 + q2(x)\2(v)] dx in an appropriate class of functions v is considered. Here q(x) Â¥= 0 and A2(t>) = X2 if v < 0, = X22 if v > 0. Any minimizer u is harmonic in {u Â¥=â€¦ (More)

We obtain C 1,Î± regularity estimates for nonlocal elliptic equations that are not necessarily translation invariant using compactness and perturbative methods and our previous regularity results forâ€¦ (More)

(1.1) det(D2u) = 1 in Rn must be a quadratic polynomial. For n = 2, a classical solution is either convex or concave; the result holds without the convexity hypothesis. A simpler and more analyticalâ€¦ (More)

The operator square root of the Laplacian (âˆ’â–³) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to theâ€¦ (More)