Multi-valued solutions to Hessian equations

Existence of viscosity multi-valued solutions with asymptotic behavior for Hessian equations

Some Multi-valued Solutions to Monge–Ampère Equations

In this paper, we construct several types of multi-valued solutions to the Monge–Ampère equation in higher dimensions. Recently, there has been considerable interest in understanding the behavior of a metric generated by a solution of the Monge–Ampère equation at a singularity. See, for instance Loftin, Yau and Zaslow [14] and Leung [12]. To explain our results let us refer first to the theory ...

Approximation of solutions to history-valued neutral functional differential equations

K e y w o r d s N e u t r a l functional differential equation, Analytic semigroup, Mild solution, FaedoGalerkin approximation. 1. I N T R O D U C T I O N Let H be a separable Hilbert space and 0 < T, T < ec. For 0 < t < T, Ct := C([-~-, t]; H) be the Banach space of all continuous functions from l--T, t] into H endowed with the supremum norm, Ilullt := sup I lu(s) l l , u • Ct, (1.1) -r<_s<t w...

A Level Set Framework for Capturing Multi-Valued Solutions of Nonlinear First-Order Equations

We introduce a level set method for the computation of multi-valued solutions of a general class of nonlinear first-order equations in arbitrary space dimensions. The idea is to realize the solution as well as its gradient as the common zero level set of several level set functions in the jet space. A very generic level set equation for the underlying PDEs is thus derived. Specific forms of the...

Quasiconvex functions and Hessian equations

In this note we construct new examples of quasiconvex functions defined on the set Sn×n of symmetric matrices. They are built on the k-th elementary symmetric function of the eigenvalues, k = 1, 2, ..., n. The idea is motivated by Šverák’s paper [S]. The proof of our result relies on the theory of the so-called k-Hessian equations, which have been intensively studied recently, see [CNS], [T], [...