We prove existence and uniqueness of minimizers for a family of energy functionals that arises in Elasticity and involves polyconvex integrands over a certain subset of displacement maps. This work extends previous results by Awi and Gangbo to a larger class of integrands. We are interested in Lagrangians of the form \(L(A,u)=f(A)+H(\det A)-F\cdot u \). Here the strict convexity condition on \(f \) and \(H \) have been relaxed to a convexity condition. Meanwhile, we have allowed the map \(F \) to be non-degenerate. First, we study these variational problems over displacements for which the determinant is positive. Second, we consider a limit case in which the functionals are degenerate. In that case, the set of admissible displacements reduces to that of incompressible displacements which are measure preserving maps. Finally, we establish that the minimizer over the set of incompressible maps may be obtained as a limit of minimizers corresponding to a sequence of minimization problems over general displacements provided we have enough regularity on the dual problems. We point out that these results do not rely on the direct methods of the calculus of variations.

中文翻译：

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一类带有凸积分的变分问题的极小化子的唯一性。

我们证明了在弹性中出现的能量函数族的最小化子的存在和唯一性，并且涉及位移映射的某些子集上的多凸被积。这项工作将Awi和Gangbo的先前结果扩展到了更大种类的被整数。我们对\（L（A，u）= f（A）+ H（\ det A）-F \ cdot u \）形式的Lagrangian感兴趣。在这里，\（f \）和\（H \）上的严格凸条件已被放宽为凸条件。同时，我们允许映射\（F \）不堕落。首先，我们研究关于行列式为正的位移的这些变分问题。其次，我们考虑功能退化的极限情况。在这种情况下，可允许的位移量减少到不可压缩的位移量，后者是量度图。最后，我们建立了不可压缩映射集上的最小化器，作为与一般位移上的一系列最小化问题相对应的最小化器的极限，只要我们对双重问题有足够的规律性即可。我们指出，这些结果并不依赖于变化微积分的直接方法。