Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 80, pp. 1-30.
Title: Gradient regularity for non-autonomous functionals with Dini or non-Dini
continuous coefficients
Authors: Paolo Baroni (Univ. of Parma, Parma, Italy)
Alessandra Coscia (Univ. of Parma, Parma, Italy)
Abstract:
We prove C1 regularity for local vectorial minimizers of the non-autonomous functional
$$
w\in W^{1,1}_{\rm loc}(\Omega;\mathbb{R}^N)\mapsto \int_{\Omega}b(x)\big[|Dw|^p
+a(x)|Dw|^p\log(e+|Dw|)\big] \,dx\,,
$$
with Ω open subset of Rn, n≥2 , p>1,
0≤a(.)≤ ||a||L∞(Ω)<∞, and
0<ν≤b(.)≤ L. The result is valid provided that the function a(.)
is log-Dini continuous and that the coefficient b(.) is Dini continuous or
it is weakly differentiable and its gradient locally belongs to the Lorentz space
Ln,1(Ω;Rn).
Submitted November 1, 2022. Published November 23, 2022.
Math Subject Classifications: 35J15, 35J60, 35J99.
Key Words: Non-autonomous functionals; gradient continuity; Dini continuous coefficients.