Unitarity constraints on chiral perturbative amplitudes

Low lying scalar resonances emerge as a necessary part to adjust chiral perturbation theory to experimental data once unitarity constraint is taken into consideration. I review recent progress made in this direction in a model independent approach. Also I briefly review studies on the odd physical properties of these low lying scalar resonances, including in the $\gamma\gamma\to\pi^+\pi^-, \pi^0\pi^0$ processes.Comment: Talk given at: International Workshop on Effective Field Theories: from the pion to the upsilon, February 2-6 2009, Valencia, Spai

Existence and Blow-Up Behavior for Solutions of the Generalized Jang Equation

The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up behavior are given, and it is shown that blow-up solutions are not unique.Comment: 33 pages; final versio

Boundary expansions and convergence for complex Monge-Ampere equations

We study boundary expansions of solutions of complex Monge-Ampere equations and discuss the convergence of such expansions. We prove a global conver- gence result under that assumption that the entire boundary is analytic. If a portion of the boundary is assumed to be analytic, the expansions may not converge locally

Smooth Solutions to a Class of Mixed Type Monge-Ampere Equations

We prove the existence of C^{\infty} local solutions to a class of mixed type Monge-Ampere equations in the plane. More precisely, the equation changes type to finite order across two smooth curves intersecting transversely at a point. Existence of C^{\infty} global solutions to a corresponding class of linear mixed type equations is also established. These results are motivated by, and may be applied to the problem of prescribed Gaussian curvature for graphs, the isometric embedding problem for 2-dimensional Riemannian manifolds into Euclidean 3-space, and also transonic fluid flow.Comment: Calc. Var. Partial Differential Equations, to appear, 42 page