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Given $f$ and $d$ sufficiently integrable functions defined on a bounded domain $\Omega$, consider: $$\min_{y \in L^2(\Omega),\\ y \leq f \text{ a.e.}} \lVert y-d \rVert_{L^2(\Omega)}^2$$

My question is, is it possible to give a formula for the minimiser $y^*$ explicitly? I know it's the projection of $d$ onto the set $K=\{ v \in L^2 : v \leq f \text{ a.e.}\}$ but I want an explicit formula.

I thought maybe $y^* = \frac{d}{|d|}f$ but even if this is right this needs positivity of $f$ to make sense. So I don't know.

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how about this: $y^*(x)=d(x)$ if $d(x)\leq f(x)$ and $y^*(x)=f(x)$ otherwise (i.e. $d(x)>f(x)$).

that is $y^*=\min(f,d)$ (a.e. pointwise)

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    Thanks. Do you know what the formula would be if everything (the projection, the norms, etc) was in $H^1_0$ instead of $L^2$? – 2017-01-09
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    in that case things start to get reeeeally complicated, at least i do not see any easy solution. – 2017-01-09