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I would like to find a smooth approximation of the indicator function of the unit closed ball (which is a compact) of $ \mathbb{R}^N,\ N \geq 2. $ I tried to find a sequence $ \eta_m $ of the form:

$ \eta_m(x) = \left\{\begin{array}{ccc} 1,\ if\ 0 < \left|x\right| \leq 1, \\ \ \varphi_m(x),\ if\ 1 < \left|x\right| < 1 + a_m, \\ 0,\ if\ \left|x\right| \geq 1+a_m, \end{array} \right. $ where $ a_m \to 0. $ The problem is that we want that the sequence $ \eta_m $ is bounded and its derivative to be also bounded. I tried the affin function $ \varphi_m (x) = m(1 + \frac{1}{m}-\left|x\right|), $ but the boundedness of the derivative is not here. Any suggestion is welcome.

SemiMath
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  • For which norm on your functional space do you want your approximation to be? – EtienneBfx Mar 11 '22 at 09:20
  • In fact, i would like to find such a bounded sequence $ \eta_m $ such that $$ \int_{\mathbb{R}^N} \left| \nabla \eta_m\right|^p dx \to 0,\ m \to + \infty, $$ where $ p > 1 $ is fixed. – SemiMath Mar 11 '22 at 09:29

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