= Coercivity transfers strong minimality to L1 minimality
{title2=$f(x,u,v)\geq c|v|^p-C,\quad p>1$}
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.
Back to article page
