Abstract
Some properties of the fully anisotropic Sobolev space $$W^{1,(\Phi _1,\ldots ,\Phi _N)}$$ are investigated. In particular extension and embedding theorems for functions in $$W^{1,(\Phi _1,\ldots ,\Phi _N)}$$ are deduced. Thanks to a Caccioppoli type inequality it is possible to carry out the De Giorgi procedure to prove the local boundedness of quasi-minimizers of the classical integral functional of the Calculus of Variations satisfying fully anisotropic growth conditions.
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