Integrability of the Brouwer degree for irregular arguments

Olbermann, Heiner

doi:10.1016/j.anihpc.2016.07.002

Geodesic

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Integrability of the Brouwer degree for irregular arguments

Olbermann, Heiner

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 933-959

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Résumé

We prove that the Brouwer degree $\deg (u, U, \cdot)$ for a function $u \in C^{0, α} (U; R^{n})$ is in $L^{p} (R^{n})$ if $1 \leq p < \frac{n α}{d}$ , where $U \subset R^{n}$ is open and bounded and d is the box dimension of ∂U. This is supplemented by a theorem showing that $u_{j} \to u$ in $C^{0, α} (U; R^{n})$ implies $\deg (u_{j}, U, \cdot) \to \deg (u, U, \cdot)$ in $L^{p} (R^{n})$ for the parameter regime $1 \leq p < \frac{n α}{d}$ , while there exist convergent sequences $u_{j} \to u$ in $C^{0, α} (U; R^{n})$ such that ${‖ \deg (u_{j}, U, \cdot) ‖}_{L^{p}} \to \infty$ for the opposite regime $p > \frac{n α}{d}$ .

MR Zbl

DOI : 10.1016/j.anihpc.2016.07.002

Classification : 26B10, 55M25
Keywords: Brouwer degree, Distributional Jacobian determinant, Hölder functions

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     author = {Olbermann, Heiner},
     title = {Integrability of the {Brouwer} degree for irregular arguments},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {933--959},
     publisher = {Elsevier},
     volume = {34},
     number = {4},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.07.002},
     mrnumber = {3661865},
     zbl = {1366.26018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.07.002/}
}

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Olbermann, Heiner. Integrability of the Brouwer degree for irregular arguments. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 933-959. doi: 10.1016/j.anihpc.2016.07.002

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