Dominant integrands growth estimates and smoothness of variational functionals in Sobolev spaces

For variational functionals in Sobolev spaces $\{W^{1,p}\}\;(1\leq p<\infty)$ we introduce a sequence of so-called dominant “growth estimates” for the gradient of appropriate order of the integrand, each of which guarantees the appropriate level of smoothness of variational functional in the $C^{1}$-smooth points of the Sobolev space. Earlier studied K-pseudopolynomial representations of the integrand are particular cases of dominant growth estimates. However, unlike the pseudopolynomial case $(p\in \mathbb{N})$, our approach enables us to consider variational problems on the complete Sobolev scale $(1\leq p<\infty)$.