Coercivity of integral functionals with non-everywhere superlinear lagrangians

Abstract

AbstractWe consider the classical non-autonomous variational problem\text{minimize }\biggl{\{}F(v)=\int_{a}^{b}f(x,v(x),v^{\prime}(x))\,\mathrm{d}% x:v\in\Omega\biggr{\}},where {\Omega:=\{v\in W^{1,1}(a,b),\,v(a)=A,\,v(b)=B,\,v(x)\in I\}}, when the lagrangian f has non-everywhere superlinear growth, in the sense that it can vanish at some {x_{0}\in[a,b]}, or {s_{0}\in I}. We prove some sufficient conditions ensuring the coercivity of the functional F. As a consequence, when f is convex with respect to the last variable, the existence of the minimum can be immediately derived.