On a range of exponents for absence of Lavrentiev phenomenon for double phase functionals

Abstract

For a class of functionals having the $(p,q)$-growth, we establish an improved range of exponents $p$, $q$ for which the Lavrentiev phenomenon does not occur. The proof is based on a standard mollification argument and Young convolution inequality. Our contribution is two-fold. First, we observe that it is sufficient to regularise only bounded functions. Second, we exploit the $L^{\infty }$ bound on the function rather than the $L^p$ estimate on the gradient. Our proof does not rely on the properties of minimizers to variational problems but it is rather a consequence of the underlying Musielak-Orlicz function spaces. Moreover, our method works for unbounded boundary data, the variable exponent functionals and vectorial problems. In addition, the result seems to be optimal for $p\le d$.

arXiv comment32 pages

References in corpus (5)

• Gossez's approximation theorems in the Musielak-Orlicz-Sobolev spaces
• On the regularity of minimizers for scalar integral functionals with $(p,q)$-growth
• Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon
• Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon
• Parabolic equations in Musielak -- Orlicz spaces with discontinuous in time $N$-function

Cited by in corpus (3)

• Regularity for double phase problems at nearly linear growth
• Lipschitz bounds for nonuniformly elliptic integral functionals in the plane
• On the Lavrentiev gap for convex, vectorial integral functionals