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We prove that any solution of a degenerate elliptic PDE is of class C 1 provided the inverse of the equation's degeneracy law satisfies an integrability criterium, viz. σ − 1 ∈ L 1 (1 λ d λ). The proof is based upon the construction of a sequence of converging tangent hyperplanes that approximate u (x) , near x 0 , by an error of order o (| x − x 0 |). Explicit control of such hyperplanes is carried over through the construction, yielding universal estimates upon the C 1 –regularity of solutions. Among the main new ingredients required in the proof, we develop an alternative recursive algorithm for renormalization of approximating solutions. This new method is based on a technique tailored to prevent the sequence of degeneracy laws constructed through the process from being, itself, degenerate. [ABSTRACT FROM AUTHOR]

Abstract: Active scalars appear in many problems of fluid dynamics. The most common examples of active scalar equations are 2D Euler, Burgers, and 2D surface quasi-geostrophic equations. Many questions about regularity and properties of solutions of these equations remain open. We develop the idea of nonlocal maximum principle introduced in Kiselev, Nazarov and Volberg (2007) , formulating a more general criterion and providing new applications. The most interesting application is finite time regularization of weak solutions in the supercritical regime. [Copyright &y& Elsevier]

Abstract: We consider non-local linear Schrödinger-type critical systems of the type where Ω is antisymmetric potential in , v is an valued map and Ωv denotes the matrix multiplication. We show that every solution of (1) is in fact in , for every , in other words, we prove that the system (1) which is a-priori only critical in happens to have a subcritical behavior for antisymmetric potentials. As an application we obtain the regularity of weak 1/2-harmonic maps into compact sub-manifolds without boundary. [Copyright &y& Elsevier]

Abstract: In dimension , for that can be written as a sum of squares of smooth functions, we prove that a convex solution u to a subelliptic Monge–Ampère equation is itself smooth if the elementary st symmetric curvature of u is positive (the case uses an additional nondegeneracy condition on the sum of squares). Our proof uses the partial Legendre transform, Calabi''s identity for where σ is the square of the third order derivatives of u, the Campanato method Xu and Zuily use to obtain regularity for systems of sums of squares of Hörmander vector fields, and our earlier work using Guan''s subelliptic methods. [Copyright &y& Elsevier]