Method and apparatus for minimizing convection during crystal growth from solution

A method and apparatus are disclosed for growing in a gravitational field a microscopic crystal from a solution. The solution is held in a vertical chamber which is relatively thin, the thin being generally perpendicular to the vertical. There is a substrate crystal disposed at either the upper or lower end of the chamber and the crystal grows from this substrate crystal in one direction. The temperature conditions of the solution are controlled so that, as the crystal forms, the effects of buoyant convection within the solution are minimized. This is accomplished in two different ways depending upon whether the crystal is grown from the upper or lower end of the chamber. When grown from the upper end of the chamber, the temperature of the solution is controlled so that it remains essentially isothermal so that there is essentially no heat loss from the solution. When the crystal is grown from the lower end of the chamber, the temperature of the solution is controlled so that there is a differential in temperature throughout the solution which provides a positive thermal gradient within the chamber

Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the stochastic heat equation with multiplicative noise and in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise

Existence of minimal and maximal solutions to first--order differential equations with state--dependent deviated arguments

We prove some new results on existence of solutions to first--order ordinary differential equations with deviating arguments. Delay differential equations are included in our general framework, which even allows deviations to depend on the unknown solutions. Our existence results lean on new definitions of lower and upper solutions introduced in this paper, and we show with an example that similar results with the classical definitions are false. We also introduce an example showing that the problems considered need not have the least (or the greatest) solution between given lower and upper solutions, but we can prove that they do have minimal and maximal solutions in the usual set--theoretic sense. Sufficient conditions for the existence of lower and upper solutions, with some examples of application, are provided too

Density estimates for solutions to one dimensional Backward SDE's

In this paper, we derive sufficient conditions for each component of the solution to a general backward stochastic differential equation to have a density for which upper and lower Gaussian estimates can be obtained