## Partial regularity under anisotropic $$(p,q)$$ growth conditions.(English)Zbl 0807.49010

Let $$\Omega$$ be an open subset of $$\mathbb{R}^ n$$, $$f: \mathbb{R}^{n\cdot N}\to \mathbb{R}$$ be a function of class $$C^ 2$$ satisfying the inequality $$|\xi|^ p\leq f(\xi)\leq C(1+ |\xi|^ q)$$ for any $$\xi\in \mathbb{R}^{n\cdot N}$$, and set $F(u)= \int_ \Omega f(Du)dx$ for any $$u\in W^{1,p}(\Omega,\mathbb{R}^ N)$$.
The regularity of minimizers of integral functionals of the kind (1) under anisotropic $$(p,q)$$ growth conditions of $$f$$ is studied. In the particular case of $F(u)= \int_ \Omega\left[| Du|^ p+ \sum^ k_{\alpha=1} | D_ \alpha u|^{p_ \alpha}\right] dx,\tag{2}$ where $$1\leq k\leq n$$ and $$2\leq p< p_ \alpha$$ for $$\alpha= 1,\dots,k$$, the following result is proved.
Theorem. If $$u\in W^{1,p}(\Omega,\mathbb{R}^ N)$$ is minimizer of (2), with $$D_ \alpha u\in L^{p_ \alpha}(\Omega)$$ for $$\alpha= 1,\dots,k$$, and if $$\max\{p_ \alpha: \alpha= 1,\dots,k\}<\bar p^*$$, where $$\bar p^*= n\bar p/(n- \bar p)$$, $$\bar p^{-1}= n^{-1}\left[(n- k) p^{- 1}+ \sum^ k_{\alpha=1} p^{-1}_ \alpha\right]$$, then $$Du$$ is Hölder continuous in an open set $$\Omega_ 0$$ such that $$\text{meas}(\Omega\backslash\Omega_ 0)= 0$$.

### MSC:

 49J40 Variational inequalities 35J50 Variational methods for elliptic systems
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