Growth conditions and regularity, an optimal local boundedness result

We prove local boundedness of local minimizers of scalar integral functionals integral(Omega) f(x, del u(x)) dx, Omega subset of R-n where the integrand satisfies (p, q)-growth of the form vertical bar z vertical bar(p) less than or similar to f(x, z) less than or similar to vertical bar z vertical bar(q) + 1 under the optimal relation 1/p - 1/q <= 1/n-1.

Automated summary

The local boundedness of local minimizers of scalar integral functionals is proven for integrals over subsets of Rn where the integrand satisfies (p, q)-growth conditions. The optimal relation is maintained during the proof, showing the importance of these growth conditions.