论文标题
P-Laplacian PDE的弱解决方案的持有人连续性具有VMO系数
Holder continuity of weak solutions of p-Laplacian PDE's with VMO coefficients
论文作者
论文摘要
我们考虑$ p $ -laplacian pde \ begin {equation} \ nabla} \ nabla \ cdot \ cdot \ cdot \ big（a（x）e（x）| du | du | du | du |， $ x \inΩ\ subseteq \ mathbb {r}^{n} $，其中$ω$是打开和界限的。更一般地，我们考虑椭圆系统的解决方案\ begin {equation} \ nabla \ cdot \ left（a（x）g'\ big（a（x）| du | \ big）\ frac {du} {| du | |} {| du |} {| du |} \ right） \ begin {equination} \int_Ωg\ big（a（x）| du | \ big）\ dx。意味着它可能是不连续的。如果不假设$ x \ mapsto a（x）$具有任何弱的可区分性，我们表明每个$ 0 <α<1 $ in \ mathscr {c} _ {c} _ {\ text {loc}}}^{0，α}（ω）$。实际上，前面的结果是一个更普遍的结果的推论，该推论适用于功能\ begin {equation} \int_Ωf\ big（x，x，u，u，du \ big）\ dx \ dx \ dx \ notag \ end {equination {equination {qore}，如果$ f $仅是渐近的convex。
We consider solutions $u\in W^{1,p}\big(Ω;\mathbb{R}^{N}\big)$ of the $p$-Laplacian PDE \begin{equation} \nabla\cdot\big(a(x)|Du|^{p-2}Du\big)=0,\notag \end{equation} for $x\inΩ\subseteq\mathbb{R}^{n}$, where $Ω$ is open and bounded. More generally, we consider solutions of the elliptic system \begin{equation} \nabla\cdot\left(a(x)g'\big(a(x)|Du|\big)\frac{Du}{|Du|}\right)=0\text{, }x\inΩ\notag \end{equation} as well as minimizers of the functional \begin{equation} \int_Ωg\big(a(x)|Du|\big)\ dx.\notag \end{equation} In each case, the coefficient map $a\ : \ Ω\rightarrow\mathbb{R}$ is only assumed to be of class $VMO(Ω)\cap L^{\infty}(Ω)$, which means that it may be discontinuous. Without assuming that $x\mapsto a(x)$ has any weak differentiability, we show that $u\in\mathscr{C}_{\text{loc}}^{0,α}(Ω)$ for each $0<α<1$. The preceding results are, in fact, a corollary of a much more general result, which applies to the functional \begin{equation} \int_Ωf\big(x,u,Du\big)\ dx\notag \end{equation} in case $f$ is only asymptotically convex.
