Dan Tiba
This paper is devoted to the memory of the renowned French mathematician H. Brezis, who passed away on July 7-th, 2024. He had some special connections with Romania and his work is a source of inspiration. He was a honorary member of the Romanian Academy.
Abstract. The level set method has outstanding applications in free boundary problems or in optimal design (geometric optimization). In the computations, a crucial question is to obtain parameterizations of the involved implicitly defined curves, surfaces. The general solution has a local character. This note is devoted to the construction of almost global parameterizations, that we introduce here. First, a general geometric description is indicated, of the region generated by our parameterization technique, on the implicitly defined surfaces and hypersurfaces. Then, we formulate necessary and/or sufficient conditions for the global character of the defined parameterizations, valid in many cases of interest. The essential points in our methodology are Hamiltonian systems and the Poincaré – Bendixson theory and we also underline the constructive characteristics of our approach. To obtain global or almost global parametrizations has strong consequences to global optimization algorithms in nonlinear programming and to extending Hamiltonian techniques in shape/topology optimization, beyond dimension two (including in the computational applications).
Keywords: (local, regional, global) parameterizations, arbitrary dimen- sion, Hamiltonian systems.
MSC: 65D17, 65K10, 34D20.
DOI 10.56082/annalsarscimath.2025.1.239
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics on Its Application, Volume 17 no 1, 2025
