## Abstract

We consider the obstacle problem $$\{\begin{array}{l}\text{minimize}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}I\left(u\right)={\int}_{\Omega}G\left(\nabla u\right)dx\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{among functions}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u:\Omega \to R\\ \text{such}\text{\hspace{0.17em}}\text{that}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u{|}_{\partial \Omega}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\ge \Phi \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}\text{.e}\text{.}\end{array}$$ for a given function $$ and a bounded Lipschitz domain $\Omega $ in ${\mathbf{R}}^{n}$. The growth properties of the convex integrand $G$ are described in terms of a $N$-function $A:\left[0,\infty \right)\to \left[0,\infty \right)$ with $$. If $n\le 3$, we prove, under certain assumptions on $G,{C}^{1,\infty}$-partial regularity for the solution to the above obstacle problem. For the special case where $A\left(t\right)=tln\left(1+t\right)$ we obtain ${C}^{1,\alpha}$-partial regularity when $n\le 4$. One of the main features of the paper is that we do not require any power growth of $G$.

## Citation

Martin Fuchs. Li Gongbao. "Variational inequalities for energy functionals with nonstandard growth conditions." Abstr. Appl. Anal. 3 (1-2) 41 - 64, 1998. https://doi.org/10.1155/S1085337598000438

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