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We investigate the minima of functionals of the form $${\int }_{[a,b]}g\left(\dot{u}\left(s\right)\right)\mathrm{d}s$$ where $g$ is strictly convex. The admissible functions $u:[a,b]\to ℝ$ are not necessarily convex and satisfy $u\le f$ on $[a,b]$, $u\left(a\right)=f\left(a\right)$, $u\left(b\right)=f\left(b\right)$, $f$ is a fixed function on $[a,b]$. We show that the minimum is attained by $\overline{f}$, the convex envelope of $f$.

We investigate the minima of functionals of the form $${\int }_{[a,b]}g\left(\dot{u}\left(s\right)\right)\mathrm{d}s$$ where is strictly convex. The admissible functions $u:[a,b]⟶ℝ$ are not necessarily convex and satisfy $u\le f$ on , , , is a fixed function on . We show that the minimum is attained by $\overline{f}$, the convex envelope of .