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prove the following inequality for integral

Mathematics Asked by yi li on December 24, 2020

Let $eta$ be the smooth function supported in $B_1(0)$.such that $inteta = 1$

Let $u$ be a smooth function defined on an open set $Vsubset Bbb{R}^n$ that is $uin C_c^infty(V)$.

Prove the following inequality holds:

$$int_{B_1(0)}eta(y)int_0^1int_V|Du(x-epsilon ty)|dxdtdyle int_V|Du(z)|dz$$

I do it as follows extend the integral domain $Vto Bbb{R^n}$ then the inequality $int_V|Du(x-epsilon ty)dxle int_mathbb{R^n}|Du(x-epsilon ty)|dx = int_V|Du(z)|dz$.Is my proof correct?If not how to do it?

One Answer

$$ begin{aligned} int_{B(0,1)} eta(y) int_{0}^{1} int_{V}left|D u_{m}(x-epsilon t y)right| d x d t d y &=int_{B(0,1)} eta(y) int_{0}^{1} int_{V}left|D u_{m}(x)right| d x d t d y \ &=int_{V}left|D u_{m}(x)right| d x int_{B(0,1)} eta(y) d y int_{0}^{1} d t \ &=int_{V}left|D u_{m}(x)right| d x end{aligned} $$

Correct answer by GAUSS1860 on December 24, 2020

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