Neumann problem for Monge-Ampere type equations revisited.

This paper concerns a priori second order derivative estimates of solutions of the Neumann problem for the Monge-Amp\`ere type equations in bounded domains in n dimensional Euclidean space. We first establish a double normal second order derivative estimate on the boundary under an appropriate notion of domain convexity. Then, assuming a barrier condition for the linearized operator, we provide a complete proof of the global second derivative estimate for elliptic solutions, as previously studied in our earlier work. We also consider extensions to the degenerate elliptic case, in both the regular and strictly regular matrix cases.

Variable Slope Forecasting Methods and COVID-19 Risk

There are many real-world situations in which complex interacting forces are best described by a series of equations. Traditional regression approaches to these situations involve modeling and estimating each individual equation (producing estimates of “partial derivatives”) and then solving the entire system for reduced form relationships (“total derivatives”). We examine three estimation methods that produce “total derivative estimates” without having to model and estimate each separate equation. These methods produce a unique total derivative estimate for every observation, where the differences in these estimates are produced by omitted variables. A plot of these estimates over time shows how the estimated relationship has evolved over time due to omitted variables. A moving 95% confidence interval (constructed like a moving average) means that there is only a five percent chance that the next total derivative would lie outside that confidence interval if the recent variability of omitted variables does not increase. Simulations show that two of these methods produce much less error than ignoring the omitted variables problem does when the importance of omitted variables noticeably exceeds random error. In an example, the spread rate of COVID-19 is estimated for Brazil, Europe, South Africa, the UK, and the USA.

The Neumann Problem of Complex Hessian Quotient Equations

In this paper, we consider the Neumann problem of complex Hessian quotient equations $\frac{\sigma _k (\partial \bar{\partial } u)}{\sigma _l (\partial \bar{\partial } u)} = f(z)$ with $0 \leq l < k \leq n$ and establish the global $C^1$ estimates and reduce the global 2nd derivative estimate to the estimate of double normal 2nd derivatives on the boundary. In particular, we can prove the global $C^2$ estimates and the existence theorem for the Neumann problem of complex Hessian quotient equations $\frac{\sigma _n (\partial \bar{\partial } u)}{\sigma _l (\partial \bar{\partial } u)} = f(z)$ with $0 \leq l < n$ by the method of continuity.