Maximal regularity for local minimizers of non-autonomous functionals

Abstract

We establish local $C^{1,\alpha }$-regularity for some $\alpha \in (0,1)$ and $C^{\alpha }$-regularity for any $\alpha \in (0,1)$ of local minimizers of the functional \[ v\ \mapsto\ \int_Ωϕ(x,|Dv|)\,dx, \] where $\varphi$ satisfies a $(p,q)$-growth condition. Establishing such a regularity theory with sharp, general conditions has been an open problem since the 1980s. In contrast to previous results, we formulate the continuity requirement on $\varphi$ in terms of a single condition for the map $(x,t)\mapsto \varphi (x,t)$, rather than separately in the $x$- and $t$-directions. Thus we can obtain regularity results for functionals without assuming that the gap $\frac{q}{p}$ between the upper and lower growth bounds is close to $1$. Moreover, for $\varphi (x,t)$ with particular structure, including $p$-, Orlicz-, $p(x)$- and double phase-growth, our single condition implies known, essentially optimal, regularity conditions. Hence, we handle regularity theory for the above functional in a universal way.

References in corpus (4)

• Regularity for minimizers for functionals of double phase with variable exponents
• Double phase problems with variable growth
• The weak Harnack inequality for unbounded supersolutions of equations with generalized Orlicz growth
• Quasilinear elliptic equations in $\RN$ via variational methods and Orlicz-Sobolev embeddings

Cited by in corpus (23)

• Regularity for double phase problems at nearly linear growth
• The weak Harnack inequality for unbounded supersolutions of equations with generalized Orlicz growth
• Double phase image restoration
• Regularity theory for non-autonomous partial differential equations without Uhlenbeck structure
• Regularity theory for non-autonomous problems with a priori assumptions
• A fundamental condition for harmonic analysis in anisotropic generalized Orlicz spaces
• Regularity for multi-phase problems at nearly linear growth
• Regularity for Orlicz phase problems
• A revised condition for harmonic analysis in generalized Orlicz spaces on unbounded domains
• Higher integrability for parabolic systems with Orlicz growth
• Lipschitz bounds for nonuniformly elliptic integral functionals in the plane
• Partial regularity for degenerate parabolic systems with general growth via caloric approximations
• Bounded variation spaces with generalized Orlicz growth related to image denoising
• Extension in generalized Orlicz spaces
• New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data
• Strong solutions of the double phase parabolic equations with variable growth
• Convergence of generalized Orlicz norms with lower growth rate tending to infinity
• Sufficient condition for boundedness of maximal operator on weighted generalized Orlicz spaces
• Continuity at a boundary point of solutions to quasilinear elliptic equations with generalized Orlicz growth and non-logarithmic conditions
• Local and global $C^{1,\beta }$-regularity for uniformly elliptic quasilinear equations of $p$-Laplace and Orlicz-Laplace type
• Geometric Gradient Estimates For Fully Nonlinear Models With Non-Homogeneous Degeneracy and Applications
• Bloch estimates in non-doubling generalized Orlicz spaces
• Gradient estimates for parabolic problems with Orlicz growth and discontinuous coefficients