论文标题
$（P，Q）$的Quasiconvex功能 - 增长和放松最小化的部分规律性
Quasiconvex functionals of $(p,q)$-growth and the partial regularity of relaxed minimizers
论文作者
论文摘要
我们建立了$ \ mathrm {c}^{\ infty} $ - 部分规律性结果，用于放松的Quasiconvex函数的轻松最小化器\ begin {align*} \ MathScr {f} [f} [u;ω]：= \ int_p_f（= \int_Ωf（\ nabla u）\ nabla u） U \colonΩ\ to \ Mathbb {r}^{n}，\ end {align*}面临$ q $ -growth条件$ | f（z）| \ leq c（1+| z | |^{q}）$，$ z \ in \ mathbb in \ mathbb {r} $ f \ in \ MATHRM {C}^{\ infty}（\ Mathbb {r}^{n \ times n}）$ for基本上是最佳的指数范围$ 1 \ leq p \ leq p \ leq q <\ leq q <\ min \ min \ {\ frac {\ frac {np {np {np} {n-1} {n-1} {n-1} {根据$ f $ $ p $ - 平均强制性条件的说明，我们的结果包括PointSise $（P，Q）$ - 增长条件作为特殊情况。此外，我们直接允许签名的整体，这是自然而然地考虑到强制性注意事项的，因此是直接方法，但在研究放松问题的研究中是新颖的。 在经典$（P，Q）$ - 增长条件的特殊情况下，我们的结果扩展了以前已知的指数范围，从Schmidt的基础工作（Arch。Cation。Cation。193，311-337（2009））的非阴性整合体的最大范围为有意义的有意义的，更重要的是$ P = 1 $ p = 1 $。作为进一步的关键新颖性，我们的结果适用于规范的签名整合体类别，并且不以任何方式依赖于衡量表示表示法（Ann。HenriPoincaré，肛门。
We establish $\mathrm{C}^{\infty}$-partial regularity results for relaxed minimizers of strongly quasiconvex functionals \begin{align*} \mathscr{F}[u;Ω]:=\int_ΩF(\nabla u)\,\mathrm{d} x,\qquad u\colonΩ\to\mathbb{R}^{N}, \end{align*} subject to a $q$-growth condition $|F(z)|\leq c(1+|z|^{q})$, $z\in\mathbb{R}^{N\times n}$, and natural $p$-mean coercivity conditions on $F\in\mathrm{C}^{\infty}(\mathbb{R}^{N\times n})$ for the basically optimal exponent range $1\leq p\leq q<\min\{\frac{np}{n-1},p+1\}$. With the $p$-mean coercivity condition being stated in terms of a strong quasiconvexity condition on $F$, our results include pointwise $(p,q)$-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise $(p,q)$-growth conditions, our results extend the previously known exponent range from Schmidt's foundational work (Arch. Ration. Mech. Anal. 193, 311-337 (2009)) for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for $p=1$. As further key novelties, our results apply to the canonical class of signed integrands and do not rely in any way on measure representations à la Fonseca & Malý (Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, 309-338 (1997)).
