Absence of Lavrentiev gap for non-autonomous functionals with ( p , q )-growth
Antonio Esposito, Francesco Leonetti, Pier Vincenzo Petricca · Advances in Nonlinear Analysis · 2016
Abstract We consider non-autonomous functionals of the form ℱ ( u , Ω ) = ∫ Ω f ( x , D u ( x ) ) 𝑑 x {\mathcal{F}(u,\hskip-0.569055pt\Omega)\hskip-0.853583pt=\hskip-0.853583pt\int% _{\Omega}f(x,\hskip-0.569055ptDu(x))\hskip-0.569055pt\,dx} , where u : Ω → ℝ N {u\colon\kern-0.711319pt\Omega\hskip-0.569055pt\to\hskip-0.569055pt\mathbb{R}^% {N}} , Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} . We assume that f ( x , z ) {f(x,z)} grows at least as | z | p {|z|^{p}} and at most as | z | q {|z|^{q}} . Moreover, f ( x , z ) {f(x,z)} is Hölder continuous with respect to x and convex with respect to z . In this setting, we give a sufficient condition on the density f ( x ,