A soup of regularity estimates with general growth ingredients
A talk in the BI.discrete series by
Bianca Stroffolini from Naples
| Abstract: | I will present some partial regularity results for minimizers of discontinuous quasiconvex integrals with general growth. In [GSS] we study the partial regularity of minimizers of the integral functional $F({\textbf u}) \colon =\int_{\Omega} f(x,{\textbf u},D{\textbf u})\,d x,$ where $\Omega\subseteq\mathbb{R}^{n}$ is {an open} bounded set and $\textbf u : \Omega\rightarrow\mathbb{R}^{N}$, with $n$, $N\ge 2$ -- i.e., we consider vectorial minimizers of $F$. The growth conditions we impose on $f=f(x,{\textbf u},{\textbf P})$ are quite general, being as they permit "general growth conditions" with respect to the gradient variable. This allows us to treat in a unified way the degenerate (when $p>2$) or singular (when $p<2$) behaviour. We assume with respect to $x$ a weak VMO condition, uniformly in $(\textbf u,\textbf P)$, and continuity with respect to $\textbf u$. Our main result proves that a minimizer of functional is locally Hölder continuous for any Hölder exponent $0<\alpha<1$ -- i.e., if $\textbf u$ is a minimizer of functional, then $\textbf u\in C_{\text{loc}}^{0,\alpha}\left(\Omega_0,\mathbb{R}^{N}\right)$, where $\Omega_0\subset\Omega$ is an open set of full measure. For the partial Hölder regularity on boundary points for quasiconvex functionals with general growth in the gradient and VMO in the $x$-variable, we assume that the functional is non-degenerate. We prove that if the boundary and the boundary datum are of class $C^1$ then the minimizer $\textbf u$ of the functional is locally Hölder continuous for every Hölder exponent $\alpha\in(0,1)$ at any boundary point that is Lebesgue type, in some sense, with respect to $D\textbf u$, [OSS]. If time permits, I'll present regularity estimates for nonlinear elliptic systems in divergence form with double-phase growth, modeling double-phase non-Newtonian fluids in the stationary case., [SS]. [GSS] C.Goodrich, G.Scilla, B.Stroffolini (corresponding), "Partial regularity for minimizers of discontinuous quasiconvex integrals with general growth", Proceedings of the Royal Society of Edinburgh Section A: Mathematics, First View, 2021, pp. 1 - 42. [OSS] J.Ok, G.Scilla, B.Stroffolini, "Boundary Partial Hölder Regularity for Minimizers of Discontinuous Quasiconvex Integrals with VMO Coefficients and General Growth, Communications on Pure and Applied Analysis, Volume 21, Issue 12: 4173-4214 (2022), Doi: 10.3934/cpaa.2022140. [SS] G.Scilla, B.Stroffolini, "Partial regularity for steady double phase fluids", to appear in Mathematics in Engineering. Within the CRC this talk is associated to the project(s): A7 |