Higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions

Abstract

Abstract We here establish the higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">⟨</m:mo> <m:mrow> <m:mi mathvariant="script">A</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:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>φ</m:mi> <m:mo>-</m:mo> <m:mi>u</m:mi> </m:mrow> <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 mathvariant="italic" rspace="0pt">d</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo separator="true"> </m:mo> <m:mrow> <m:mrow> <m:mpadded width="+5pt"> <m:mtext>for all</m:mtext> </m:mpadded> <m:mo>⁢</m:mo> <m:mi>φ</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="script">K</m:mi> <m:mi>ψ</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> \int_{\Omega}\langle\mathcal{A}(x,Du),D(\varphi-u)\rangle\,dx\geq 0\quad\text{for all}\ \varphi\in\mathcal{K}_{\psi}(\Omega), where Ω is a bounded open subset of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> \mathbb{R}^{n} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ψ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \psi\in W^{1,p}(\Omega) is a fixed function called obstacle and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="script">K</m:mi> <m:mi>ψ</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>w</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>:</m:mo

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