Absence of Lavrentiev gap for non-autonomous functionals with ( p , q )-growth

We consider non-autonomous functionals of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="script">ℱ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo rspace="1.9pt">,</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo rspace="1.6pt" stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="1.6pt">=</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:msub> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo rspace="1.9pt">,</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="3.6pt" stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {F(u,-0.569055ptΩ)-0.853583pt=-0.853583pt∫% Ωf(x,-0.569055ptDu(x))-0.569055pt\,dx} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mpadded width="-0.6pt"> <m:mi mathvariant="normal">Ω</m:mi> </m:mpadded> <m:mo rspace="1.9pt">→</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> {u-0.711319ptΩ-0.569055pt→-0.569055ptR^% {N}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> {Ωⁿ} . We assume that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {f(x,z)} grows at least as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> </m:math> {|z|ᵖ} and at most as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>q</m:mi> </m:msup> </m:math> {|z|q} . Moreover, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo>