The competition in the pinning of a directed polymer by a columnar pin and a background of random point impurities is investigated systematically using the renormalization group method. With the aid of the mapping to the noisy-Burgers’ equation and the use of the mode-coupling method, the directed polymer is shown to be marginally localized to an arbitrary weak columnar pin in 1+1 dimensions. T...

We present a new approach to proving the existence of semi-classical bound states of the non-linear Schrödinger equation which are concentrated near a finite set of non-degenerate critical points of the potential function. The method is based on considering a system of non-linear elliptic equations. The positivity of the solutions is considered. It is shown how the same method yields “multi-bum...

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampère type. These are: the problem of locally prescribed Gaussian curvature for surfaces in R3, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove a general local existence result for a large class of degenerate Monge-Ampère...

In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction ü = −γu̇+m(‖∇u‖)∆u− δ|u|u+ f, which is known as degenerate if m(·) ≥ 0, and non-degenerate if m(·) ≥ m0 > 0. We would like to point out that, to the author’s knowledge, exponential decay for this type of equations has been studied just for the special cases of α. Our aim ...