Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growth

• Main Author: Erhardt André
• Format: Article
• Language: English
• Published: De Gruyter 2014-02-01
• Series: Advances in Nonlinear Analysis
• Subjects: calderón–zygmund estimates; nonlinear parabolic obstacle problems; variational inequality; nonstandard growth; localizable solution; 35k86; 35b45; 35b51; 42b20; 49n60
• Online Access: https://doi.org/10.1515/anona-2013-0024

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(·)${p(\,\cdot \,)}$.