Wednesday, April 28, 2021 — 4:00 pm — online talk

Higher differentiability results for solutions to a class of problems

with sub-quadratic growth conditions

Abstract: The aim of this talk is to show some regularity properties of local minimizers of integral
functionals of the form

$$\mathcal F (v,\Omega)=\int_{\Omega} f (x, Dv(x)) \, dx,$$

where $\Omega \subset \mathbb R^n$ is a bounded open set, and the function $f$ satisfies $p$−growth conditions with respect to the gradient variable, for $1 < p < 2$, under suitable assumptions on the partial map $x \mapsto D_{\xi}f(x, \xi)$.

More precisely, in [1], $W^{2,p}$ regularity is proved for solutions to unconstrained problems in case the map $x \mapsto D_{\xi}f(x, \xi)$ belongs to a Sobolev space $W^{1,q}$ for $q \geq n$, and in [2], higher differentiability results are proved for solutions to obstacle problems, both in case $x \mapsto D_{\xi}f(x, \xi)$ belongs to the Sobolev space $W^{1,n}$ and to a suitable Besov-Lipschitz space $B_{\frac n \alpha, q}^\alpha$.

References:

[1] A. Gentile. Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth. Adv. Calc. Var. (2020).

[2] A. Gentile. Higher differentiability results for solutions to a class of non- autonomous obstacle problems with sub-quadratic growth conditions. Forum Math. (2021).