Regularity for non-uniformly elliptic equations

I discuss local regularity properties of solutions of certain linear and nonlinear non-uniformly elliptic equations. We start with weak solutions of the linear equation
∇ · a(x)∇u(x) = 0.
Assuming certain integrability conditions on the ellipticity contrast of the coefficient field a, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are sharp and improve upon classical results in the literature.
Moreover, we consider variational integrals with (p, q)-growth and provide local boundedness and Lipschitz regularity for minimizer under improved (and in case of local boundedness optimal) relations between p, q and the dimension.
The talk is based on joint works with P. Bella (TU Dortmund) and J. Hirsch (U Leipzig).
References
[1] P. Bella and M. Schäffner, Local Boundedness and Harnack Inequality for Solutions of Linear Nonuniformly Elliptic Equations. Comm. Pure Appl. Math. doi:10.1002/cpa.21876.
[2] P. Bella and M. Schäffner, On the regularity of minimizers for scalar integral functionals with (p, q)-growth. arXiv:1904.12279 (to appear Anal. PDE ).
[3] J. Hirsch and M. Schäffner, Growth conditions and regularity, an optimal local boundedness result. Comm. Cont. Math. https://doi.org/10.1142/S0219199720500297