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Approximation of Lipschitz functions preserving boundary values. (arXiv:1810.04205v1 [math.FA])
来源于:arXiv
Given an open subset $\Omega$ of a Banach space and a Lipschitz function
$u_0: \overline{\Omega} \to \mathbb{R},$ we study whether it is possible to
approximate $u_0$ uniformly on $\Omega$ by $C^k$-smooth Lipschitz functions
which coincide with $u_0$ on the boundary $\partial \Omega$ of $\Omega$ and
have the same Lipschitz constant as $u_0.$ As a consequence, we show that every
$1$-Lipschitz function $u_0: \overline{\Omega} \to \mathbb{R},$ defined on the
closure $\overline{\Omega}$ of an open subset $\Omega$ of a finite dimensional
normed space of dimension $n \geq 2$, and such that the Lipschitz constant of
the restriction of $u_0$ to the boundary of $\Omega$ is less than $1$, can be
uniformly approximated by differentiable $1$-Lipschitz functions $w$ which
coincide with $u_0$ on $\partial \Omega$ and satisfy the equation $\| D w\|_*
=1$ almost everywhere on $\Omega.$ This result does not hold in general without
assumption on the restriction of $u_0$ to the boundary of $\Omega$. 查看全文>>