We consider non-autonomous functionals of the form ℱ ( u , Ω ) = ∫ Ω f ( x , D u ( x ) ) ? x {mathcal{F}(u,hskip-0.569055ptOmega)hskip-0.853583pt=hskip-0.853583ptint _{Omega}f(x,hskip-0.569055ptDu(x))hskip-0.569055pt,dx} , where u : Ω → ℝ N {ucolonkern-0.711319ptOmegahskip-0.569055pttohskip-0.569055ptmathbb{R}^ {N}} , Ω ⊂ ℝ n {Omegasubsetmathbb{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 , z ) {f(x,z)} that ensures the absence of a Lavrentiev gap.
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