Higher differentiability of solutions to a class of obstacle problems under non-standard growth conditions

We establish the higher differentiability of integer order of solutions to a class of obstacle problems assuming that the gradient of the obstacle possesses an extra integer differentiability property. 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>∫</m:mo> <m:mi>Ω</m:mi> </m:msub> <m:mrow> <m:mrow> <m:mo>〈</m:mo> <m:mrow> <m:mi>𝒜</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</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>)</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>(</m:mo> <m:mrow> <m:mi>φ</m:mi> <m:mo>-</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>〉</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:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo> </m:mo> <m:mrow> <m:mrow> <m:mtext>for all </m:mtext> <m:mo>⁢</m:mo> <m:mi>φ</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>𝒦</m:mi> <m:mi>ψ</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>Ω</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> ∫Ω(x,Du),D(φ-u)\,dx≥ 0% for all φψ(Ω). The main novelty is that the operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒜</m:mi> </m:math> {A} satisfies the so-called <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:math> {p,q} -growth conditions with p and q linked by the relation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mfrac> <m:mi>q</m:mi> <m:mi>p</m:mi> </m:mfrac> <m:mo><</m:mo> <m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mi>n</m:mi> </m:mfrac> </m:mrow> <m:mo>-</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mi>r</m:mi> </m:mfrac> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> q/p<1+1/n-1/r, for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>r</m:mi> <m:mo>></m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> {r>n} . Here <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>(</m:mo> <m:mi>Ω</m:mi>