Abstract:
A fundamental object studied in the modern calculus of variations is a double phase functional F(u), i.e. a functional whose growth (p or q) depends on a point of the domain. For such functionals, we establish an improved range of exponents p, q (compared with [2]) such that the so-called Lavrentiev phenomenon does not occur [1]. 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 regularize only bounded functions. Second, we exploit L^{\infty} bound on the function rather than the L^p estimate on the gradient. Our proof does not use estimates on minimizers of the functional as in [3] - the result is rather a simple consequence of functional analytic properties of Musielak-Orlicz spaces. Moreover, our method works for unbounded boundary data contrary to [3], the variable exponent functionals and vectorial problems. This is a joint work with Piotr Gwiazda (IM PAN, Poland) and Miroslav Bulíček (Charles University, Czech Republic).

[1] M. Bulíček, P. Gwiazda, J. Skrzeczkowski, An improved range of exponents for absence of Lavrentiev phenomenon for double phase functionals. In preparation.
[2] M. Colombo and G. Mingione. Regularity for double phase variational problems. Arch. Ration. Mech. Anal.,215(2):443–496, 2015.
[3] M. Colombo and G. Mingione. Bounded minimisers of double phase variational integrals. Arch. Ration. Mech.Anal., 218(1):219–273, 2015.