Lipschitz constant of function with compact support

Question

Consider the function $$f_{n}(x)=\min\{1, \frac{1}{\ln n}\ln_{+}\left(\frac{n^{2}}{|x|}\right)\}.$$ Clearly we have that $supp(f_{n})=B(0, n^{2})$ but is this function Lipschitz and if it is indeed, what is the Lipschitz constant? Can someone help?

Answer 1

In this answer How to show $\min\{f_1,f_2\}$ is Lipschitz when $f_1,f_2$ are Lipschitz? it was proved that the minimum of two lipschitz functions is lipschitz. Thus also the maximum of two lipschitz functions is lipschitz. Hence, your function is lipschitz. To compute the lipschitz constant you may use that your function is piecewise smooth and you can just compute the supremum of the derivatives (only the $\ln$ term will matter on its domain).

The function is constant except on $B(0,n^2)\setminus B(0,n)$. This means the lipschitz constant is equal to $$ \max_{\vert x \vert\in [n, n^2]} \left\vert \nabla\left( \frac{1}{\ln(n)} \ln\left( \frac{n^2}{\vert x \vert}\right) \right) \right\vert = \frac{1}{\vert \ln(n) \vert} \max_{\vert x \vert\in [n, n^2]} \left\vert\left( \nabla \ln\left( \frac{1}{\vert x \vert}\right) \right) \right\vert = \frac{1}{\vert \ln(n) \vert} \max_{\vert x \vert\in [n, n^2]} \left\vert \frac{1}{\vert x \vert} \frac{x}{\vert x \vert} \right\vert = \frac{1}{\vert \ln(n) \vert} \max_{\vert x \vert\in [n, n^2]} \frac{1}{\vert x \vert} = \frac{1}{ \vert \ln(n) \vert \cdot n}$$