We consider conformally invariant energies W on the group \({{\,\mathrm{GL}\,}}^{\!+}(2)\) of \(2\times 2\)-matrices with positive determinant, i.e., \(W:{{\,\mathrm{GL}\,}}^{\!+}(2)\rightarrow {\mathbb {R}}\) such that

where \({{\,\mathrm{SO}\,}}(2)\) denotes the special orthogonal group and provides an explicit formula for the (notoriously difficult to compute) quasiconvex envelope of these functions. Our results, which are based on the representation \(W(F)=h\bigl (\frac{\lambda _1}{\lambda _2}\bigr )\) of W in terms of the singular values \(\lambda _1,\lambda _2\) of F, are applied to a number of example energies in order to demonstrate the convenience of the singular-value-based expression compared to the more common representation in terms of the distortion \({\mathbb {K}}:=\frac{1}{2}\frac{\Vert F \Vert ^2}{\det F}\). Applying our results, we answer a conjecture by Adamowicz (in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni, vol 18(2), pp 163, 2007) and discuss a connection between polyconvexity and the Grötzsch free boundary value problem. Special cases of our results can also be obtained from earlier works by Astala et al. (Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, Princeton, 2008) and Yan (Trans Am Math Soc 355(12):4755–4765, 2003). Since the restricted domain of the energy functions in question poses additional difficulties with respect to the notion of quasiconvexity compared to the case of globally defined real-valued functions, we also discuss more general properties related to the \(W^{1,p}\)-quasiconvex envelope on the domain \({{\,\mathrm{GL}\,}}^{\!+}(n)\) which, in particular, ensure that a stricter version of Dacorogna’s formula is applicable to conformally invariant energies on \({{\,\mathrm{GL}\,}}^{\!+}(2)\).

我们认为，形不变能量W¯¯该组\（{{\，\ mathrm {GL} \}} ^ {\！+}（2）\）的\（2 \倍2 \）正决定-matrices，即\（W：{{\，\ mathrm {GL} \，}} ^ {\！+}（2）\ rightarrow {\ mathbb {R}} \）这样

其中\（{{\，\ mathrm {SO} \，}}（2）\）表示特殊的正交组，并为这些函数的（非常难以计算）拟凸包络提供了明确的公式。我们的研究结果，这是基于表示\（W（F）= H \ bigl（\压裂{\拉姆达_1} {\拉姆达_2} \ bigr）\）的W¯¯中的奇异值的术语\（\拉姆达_1 ，将F的\ lambda _2 \）应用于许多示例能量，以证明与基于失真\（{\ mathbb {K} }：= \ frac {1} {2} \ frac {\ Vert F \ Vert ^ 2} {\ det F} \）。应用我们的结果，我们回答了Adamowicz的一个猜想（在：Atti della Accademia Nazionale dei Lincei。Classe di Scienze Fisiche，Matematiche e Naturali。Rendiconti Lincei。Serie IX。Matematica e Applicazioni，第18（2）卷，第163页，2007年）并讨论了多凸性和格罗茨自由边界值问题之间的联系。我们的结果的特殊情况也可以从Astala等人的早期工作中获得。（椭圆偏微分方程和平面上的拟保形映射，普林斯顿大学出版社，普林斯顿，2008年）和Yan（Trans Am Math Soc 355（12）：4755–4765，2003）。由于与全局定义的实值函数相比，所讨论的能量函数的受限域在拟凸性概念上带来了更多困难，因此我们还将讨论与\（W ^ {1，p} \）-域\（{{\，\ mathrm {GL} \，}} ^ {\！+}（n）\）上的拟凸包络，尤其要确保较严格的Dacorogna公式适用于\（{{\，\ mathrm {GL} \，}} ^ {\！+}（2）\）上的保形不变能量。