Integrability of the Brouwer degree for irregular arguments

Abstract

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