Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growth

Abstract

We establish local Calderón–Zygmund estimates for solutions to certain parabolic problems with irregular obstacles and nonstandard p(x,t)${p(x,t)}$-growth. More precisely, we will show that the spatial gradient Du${Du}$ of the solution to the obstacle problem is as integrable as the obstacle ψ${\psi }$, i.e. |Dψ|p(·),|∂tψ|γ1'∈L loc q⇒|Du|p(·)∈L loc q,foranyq>1,$ |D\psi |^{p(\,\cdot \,)},|\partial _t\psi |^{\gamma _1^{\prime }}\in L^q_\mathrm {loc}\Rightarrow |Du|^{p(\,\cdot \,)}\in L^q_\mathrm {loc},\quad \text{for any}~q>1, $ where γ1'=γ1γ1-1${\gamma _1^{\prime }=\frac{\gamma _1}{\gamma _1-1}}$ and γ1${\gamma _1}$ is the lower bound for p(·)${p(\,\cdot \,)}$.

Keywords

• calderón–zygmund estimates
• nonlinear parabolic obstacle problems
• variational inequality
• nonstandard growth
• localizable solution
• 35k86
• 35b45
• 35b51
• 42b20
• 49n60