No Lavrentiev gap for some double phase integrals

Abstract

Abstract We prove the absence of the Lavrentiev gap for non-autonomous functionals <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <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 stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≔</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>,</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="4.2pt" 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:mo>,</m:mo> </m:mrow> </m:math> \mathcal{F}(u)\coloneqq\int_{\Omega}f(x,Du(x))\,dx, where 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> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {f(x,z)} is α-Hölder continuous with respect to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <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> {x\in\Omega\subset\mathbb{R}^{n}} , it satisfies the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(p,q)} -growth conditions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>z</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> <m:mo>⩽</m:mo> <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:mo>⩽</m:mo> <m:mrow> <m:mi>L</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>z</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mi>q</m:mi> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> \lvert z\rvert^{p}\leqslant f(x,z)\leqslant L(1+\lvert z\rvert^{q}), where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>&lt;</m:mo> <m:mi>q</m:mi> <m:mo>&lt;</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo>

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