论文标题
在有限弹性方面的多凸功能的各个地方
Everywhere regularity results for a polyconvex functional in finite elasticity
论文作者
论文摘要
在这里，我们为$ 2 \ times2- $尺寸可压缩的有限弹性开发了polycovex功能的规律性理论。特别是，我们考虑功能$ i（u）= \ int \limits_Ω{\ frac {\ frac {1} {2} | \ nabla u |^2+ρ（\ det \ nabla u） w^{1,2}（ω，\ mathbb {r}^2）$和$ρ：\ m athbb {r} \ rightarrow \ rightArrow \ mathbb {r} _0^+$ spooth and convex and convex and convex with $ρ（s）= 0 $ s）= 0 $ s $ s \ s \ s \ s \ les $ $ p y $ s $ 0. 状况。 我们表明的第一个结果是，每个固定点都需要在本地进行Hölder-Coninuul。其次，我们证明，如果$ \ |ρ'\ | _ {l^\ infty（\ Mathbb {r}）} <1 $ S.T.整数仍然是均匀凸的，然后所有固定点必须位于$ w_ {loc}^{2,2}。的确，我们表明，所有固定点是$ w_ {loc}^{2,2} $ and jacobian的所有固定点，其jacobian是适当的hölder-contium，属于$ c_ {loc}^loc}^{\ infty}。 $ \ |ρ'\ | _ {l^\ infty（\ Mathbb {r}）} <1 $所有固定点必须平滑。
Here we develop a regularity theory for a polyconvex functional in $2\times2-$dimensional compressible finite elasticity. In particular, we consider energy minimizers/stationary points of the functional $I(u)=\int\limits_Ω{\frac{1}{2}|\nabla u|^2+ρ(\det\nabla u)\;dx},$ where $Ω\subset\mathbb{R}^2$ is open and bounded, $u\in W^{1,2}(Ω,\mathbb{R}^2)$ and $ρ:\mathbb{R}\rightarrow\mathbb{R}_0^+$ smooth and convex with $ρ(s)=0$ for all $s\le0$ and $ρ$ becomes affine when $s$ exceeds some value $s_0>0.$ Additionally, we may impose boundary conditions. The first result we show is that every stationary point needs to be locally Hölder-continuous. Secondly, we prove that if $\|ρ'\|_{L^\infty(\mathbb{R})}<1$ s.t. the integrand is still uniformly convex, then all stationary points have to be in $W_{loc}^{2,2}.$ Next, a higher-order regularity result is shown. Indeed, we show that all stationary points that are additionally of class $W_{loc}^{2,2}$ and whose Jacobian is suitably Hölder-continuous are of class $C_{loc}^{\infty}.$ As a consequence, these results show that in the case when $\|ρ'\|_{L^\infty(\mathbb{R})}<1$ all stationary points have to be smooth.
