 Coercivity transfers strong minimality to L1 minimality

With fixed endpoint values on a finite interval, the displayed derivative coercivity makes a finite-energy strong local minimizer of a variational integral also a local minimizer in $L^1$. Otherwise choose better competitors converging in $L^1$. The upper energy bound controls their derivatives in $L^p$, and a common endpoint controls their function norms. Compact Sobolev embedding gives a uniformly convergent subsequence; its limit is the $L^1$ limit. This contradicts strong local minimality. Convexity of the integrand is unnecessary for this neighborhood-transfer argument.