Gradient estimates for Monge–Ampère type equations on compact almost Hermitian manifolds with boundary

We investigate Monge–Ampère type fully nonlinear equations on compact almost Hermitian manifolds with boundary and show a priori gradient estimates for a smooth solution of these equations.

Positive solutions for (p, q)-equations with convection and a sign-changing reaction

We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction which is dependent on the gradient. We look for positive solutions and we do not assume that the reaction is nonnegative. Using a mixture of variational and topological methods (the "frozen variable" technique), we prove the existence of a positive smooth solution.

Propiedades del soporte de soluciones de una clase de ecuaciones de evolución no lineales en dos dimensiones

In this work we consider equations of the form ∂tu + P(D)u + u^{l}∂xu = 0, where P(D) is a two-dimensional differential operator, and l ∈ N. We prove that if u is a sufficiently smooth solution of the equation, such that suppu(0), suppu(T) ⊂ [−B, B] × [−B, B] for some B > 0, then there exists R0>0 such that suppu(t) ⊂ [-R_0,R_0]×[-R_0,R_0] for every t ∈ [0, T].

Boundary conditions in topological AdS4/CFT3

We revisit the construction in four-dimensional gauged Spin(4) supergravity of the holographic duals to topologically twisted three-dimensional $$ \mathcal{N} $$ N = 4 field theories. Our focus in this paper is to highlight some subtleties related to preserving supersymmetry in AdS/CFT, namely the inclusion of finite counterterms and the necessity of a Legendre transformation to find the dual to the field theory generating functional. Studying the geometry of these supergravity solutions, we conclude that the gravitational free energy is indeed independent from the metric of the boundary, and it vanishes for any smooth solution.