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http://hdl.handle.net/10773/35429
Title: | A solution to Newton’s least resistance problem is uniquely defined by its singular set |
Author: | Plakhov, Alexander |
Issue Date: | Oct-2022 |
Publisher: | Springer Nature |
Abstract: | Let $u$ minimize the functional $F(u) = \int_\Omega f(\nabla u(x))\, dx$ in the class of convex functions $u : \Omega \to {\mathbb R}$ satisfying $0 \le u \le M$, where $\Omega \subset {\mathbb R}^2$ is a compact convex domain with nonempty interior and $M > 0$, and $f : {\mathbb R}^2 \to {\mathbb R}$ is a $C^2$ function, with $\{ \xi : \, \text{the smallest eigenvalue of} \, f''(\xi) \, \text{is zero} \}$ being a closed nowhere dense set in ${\mathbb R}^2$. Let epi$(u)$ denote the epigraph of $u$. Then any extremal point $(x, u(x))$ of epi$(u)$ is contained in the closure of the set of singular points of epi$(u)$. As a consequence, an optimal function $u$ is uniquely defined by the set of singular points of epi$(u)$. This result is applicable to the classical Newton's problem, where $F(u) = \int_\Omega (1 + |\nabla u(x)|^2)^{-1}\, dx$. |
Peer review: | yes |
URI: | http://hdl.handle.net/10773/35429 |
DOI: | 10.1007/s00526-022-02300-w |
ISSN: | 0944-2669 |
Publisher Version: | https://doi.org/10.1007/s00526-022-02300-w |
Appears in Collections: | CIDMA - Artigos OGTCG - Artigos |
Files in This Item:
File | Description | Size | Format | |
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2022 CalcVar.pdf | 882.72 kB | Adobe PDF |
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