Regularity for Double Phase Variational Problems

Calderón–Zygmund estimates and non-uniformly elliptic operators

“…In the other direction, (1.12)-(1.13) sharply describe the interaction between coefficients and gradient in order to get a moderate rate of non-uniform ellipticity of the operator; see (1.5). Finally, as first emphasised by Zhikov and then expanded in [15,16,22], bounds (1.12)-(1.13) also play a crucial role to exclude the Lavrentiev phenomenon that functionals as P otherwise exhibit. Our results also extend to the vectorial case; see Theorem 1.3, while parabolic estimates are possible too [4].…”

“…Such non-standard growth conditions (in Marcellini's terminology [40]), have been attracting increasing attention, starting by the pioneering papers of Marcellini [40][41][42]. See for instance [5,13,22,36,44,47] and our previous papers [15,16] for the regularity of minima of the functional P.…”

Lipschitz estimates for systems with ellipticity conditions at infinity

“…For similar results, we refer to [26][27][28][29][30][31] and more recently [1,3,10]. A new impulse to the subject has been given by the recent articles already cited [6,7]. Everywhere Lipschitz continuity up to the boundary for either the Dirichlet or the Neumann problem has been recently considered by Cianchi and Maz'ya [5] under uniformly elliptic conditions.…”

Existence of weak solutions for elliptic systems with p,q-growth

“…A common feature is that to get regularity results p and q must be not too far apart, as examples of irregular solutions by Giaquinta [17], Marcellini [25], [27] and Hong [21] show. Notice that the rate of Hölder continuity of A(·, z) interacts with the ratio q/p precisely as in (1.5), see Esposito-Leonetti-Mingione [15] and Colombo-Mingione [8] where the same bound appears. This is suggested by an example in [15], where the minimizer of a functional with q/p > 1 + γ/n fails to be locally W 1,q -regular; see also Fonseca-Malý-Mingione [16].…”