We consider the minimization of an energy functional given by the sum of a crystalline perimeter and a nonlocal interaction of Riesz type, under volume constraint. We show that, in the small mass regime, if the Wulff shape of the anisotropic perimeter has certain symmetry properties, then it is the unique global minimizer of the total energy. In dimension two this applies to convex polygons which are reflection symmetric with respect to the bisectors of the angles. We further prove a rigidity result for the structure of (local) minimizers in two dimensions.

中文翻译：

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非局部各向异性等渗问题中多态的极小值

我们考虑了在体积约束下由晶体周长和Riesz型非局部相互作用之和给出的能量函数的最小化。我们表明，在小质量状态下，如果各向异性周长的Wulff形状具有某些对称性，则它是总能量的唯一全局最小化器。在第二维中，这适用于相对于角度的平分线反射对称的凸多边形。我们进一步证明了（局部）最小化器在二维结构上的刚度结果。