论文标题
在莫里的猜想的背景下，用于研究准蔬菜的数值方法
Numerical approaches for investigating quasiconvexity in the context of Morrey's conjecture
论文作者
论文摘要
确定给定功能是否为Quasiconvex通常是一项艰巨的任务。在这里，我们讨论了许多数值方法，这些方法可用于搜索给定函数$ w $的Quasiconvexity的反例。我们将使用平面各向同性级别凸函数\ [[ w _ {\ rm magic}^+（f）= \ frac {λ_{\ rm max}}} {λ_{λ_{\ rm min}} - \ log \ log \ frac {λ_ {\ rm max}}} {\ rm max}}} f = \ frac {λ_ {\ rm max}}} {λ_{\ rm min}}}+2 \logλ_ {\ rm min} \ ,, \ ,, \]其中$λ_ {\ rm max} \ rm max} \geqλ_ {在先前的贡献中，我们已经表明，此功能的准综合性意味着所有等级 - 一凸的同性恋平面能量$ w：\ operatorname {gl}^+（2）\ rightarrow \ rightarrow \ rightarrow \ mathbb {r} $具有附加性的dolumetric-ischoric split form and Ischoric split formeconcoric split s eff in form of norm of n form conf inf。 w（f）= w _ {\ rm iso}（f）+w _ {\ rm vol}（\ det f）= \ widetilde w _ {\ rm iso} \ bigG（\ frac {f} {f} \]带有凹面的体积部分。因此，关于莫里的公开问题，这个示例特别感兴趣，即排名是否意味着在平面案例中的icalvexity。
Deciding whether a given function is quasiconvex is generally a difficult task. Here, we discuss a number of numerical approaches that can be used in the search for a counterexample to the quasiconvexity of a given function $W$. We will demonstrate these methods using the planar isotropic rank-one convex function \[ W_{\rm magic}^+(F)=\frac{λ_{\rm max}}{λ_{\rm min}}-\log\frac{λ_{\rm max}}{λ_{\rm min}}+\log\det F=\frac{λ_{\rm max}}{λ_{\rm min}}+2\logλ_{\rm min}\,, \] where $λ_{\rm max}\geqλ_{\rm min}$ are the singular values of $F$, as our main example. In a previous contribution, we have shown that quasiconvexity of this function would imply quasiconvexity for all rank-one convex isotropic planar energies $W:\operatorname{GL}^+(2)\rightarrow\mathbb{R}$ with an additive volumetric-isochoric split of the form \[ W(F)=W_{\rm iso}(F)+W_{\rm vol}(\det F)=\widetilde W_{\rm iso}\bigg(\frac{F}{\sqrt{\det F}}\bigg)+W_{\rm vol}(\det F) \] with a concave volumetric part. This example is therefore of particular interest with regard to Morrey's open question whether or not rank-one convexity implies quasiconvexity in the planar case.
