CEU Electronic Theses and Dissertations, 2018
| Author | Trinh, Hai Thi |
|---|---|
| Title | On some Applications of Convexity and Differential Equations |
| Summary | We investigate some applications of convexity and differential equations to study on the planar $L_p$-Minkowski problem for $0< p<1$ and the minimum time function, in particular. We first establish necessary and sufficient conditions for the existence of solutions to the asymmetric $L_p$- Minkowski problem in $\R^2$ for $0 < p < 1$, which amounts to solve a Monge-Amp\`ere type differential equation on $\mathbb{S}^1$ in the regular case. In addition, we investigate the $\varphi$-convexity of the epigraph of the minimum time function $T$ associated with a nonlinear control system with a general closed target under the condition that the sublevel sets of $T$ are $\varphi_0$-convex for some appropriate nonnegative constant $\varphi_0$, where $\varphi$ is a continuous function which can be computed explicitly. This property of $T$ is proved based on some suitable sensitivity relation results. We also provide some sufficient conditions for convexity of sublevel sets of $T$. Furthermore, we provide an invariant result for the set of non-Lipschitz points of the minimum time function. |
| Supervisor | Böröczky, Károly; Nguyen, Luong Van |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2018/trinh_hai.pdf |
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