论文标题
$ p $ -laplacian类型方程的全球二阶Sobolev规律性，有界域中的系数可变
A global second order Sobolev regularity for $p$-Laplacian type equations with variable coefficients in bounded domains
论文作者
论文摘要
令$ω\子集r^n $为有$ n \ ge2 $的有限凸域。假设$ a $是均匀的椭圆形，当$ n \ ge 3 $或$ w^{1，q} $时属于$ w^{1，n} $，对于某些$ q> 2 $，当$ n = 2 $。对于$ 1 <p <\ infty $，我们建立了全球二阶规则性估计$$ \ | d [| du |^{p-2} du] \ | f \ | _ {l^2（ω）} $$对于不均匀$ p $ -laplace type equation \ begin \ begin {equation} - \ Mathrm {div} \ big（\ langle a du，du \ rangle ^{\ frac {p-2} 2} a du \ big）= f \ quad \ quad \ rm {in} \ω\ω\ mbox {用dirichlet/neumann/neumann/neumann/neumant/neumant/neumant/neumann $ 0 $ -BOUND and in Lips and ylim and ylim and ylim and ylim and ylip and and and and and and and andecation} equience and andecation。边界弱二阶可区分并满足某些较小的假设的域。
Let $Ω\subset R^n$ be a bounded convex domain with $n\ge2$. Suppose that $A$ is uniformly elliptic and belongs to $W^{1,n}$ when $n\ge 3$ or $W^{1,q}$ for some $q>2$ when $n=2$. For $1<p<\infty$, we build up a global second order regularity estimate $$\|D[|Du|^{p-2} Du]\|_{L^2(Ω)}+\|D[ |\sqrt{A}Du|^{p-2} A Du]\|_{L^2(Ω)} \le C \|f\|_{L^2(Ω)} $$ for inhomogeneous $p$-Laplace type equation \begin{equation} -\mathrm{div}\big(\langle A Du,Du\rangle ^{\frac{p-2}2} A Du\big)=f \quad\rm{in }\ Ω\mbox{ with Dirichlet/Neumann $0$-boundary.} \end{equation} Similar result was also built up for certain bounded Lipschitz domain whose boundary is weakly second order differentiable and satisfies some smallness assumptions.
