Partial Regularity for Local Minimizers of Variational Integrals With Lower-Order Terms

We consider functionals of the form $$\mathcal{F}(u):=\int_\Omega\!F(x,u,\nabla u)\,\mathrm{d} x,$$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded. The integrand $F\colon\Omega\times\mathbb{R}^N\times\mathbb{R}^{N\times n}\to\mathbb{R}$ assumed to satisfy classical assumptions a power $p$-growth corresponding strong quasiconvexity. In addition, $F$ H\"older continuous with exponent $2\beta\in(0,1)$ in its first two variables uniformly respect third variable, bounded below by quasiconvex function depending only on $z\in\mathbb{R}^{N\times n}$. establish that local minimizers $\mathcal{F}$ are class $\mathrm{C}^{1,\beta}$ an subset $\Omega_0\subseteq\Omega$ $\mathcal{L}^n(\Omega\setminus\Omega_0)=0$. This partial regularity also holds for certain weak at which second variation strongly positive satisfying $\mathrm{BMO}$-smallness condition. extends result Kristensen Taheri (2003) case depends $u$. Furthermore, we provide direct strategy this result, contrast blow-up argument used homogeneous integrands.