A Weierstrass extremal field theory for the fractional Laplacian
A Weierstrass extremal field theory for the fractional Laplacian
复制标题
分数拉普拉斯算子的 Weierstrass 极值场理论
DOI：
10.1515/acv-2022-0099
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发表时间：
2022
影响因子：
1.7
通讯作者：
Juan
中科院分区：
文献类型：
--
作者：
Xavier Cabré;Iñigo U. Erneta;Juan
摘要本文首次将变分学中Weierstrass极值场理论的一部分推广到非局部框架。我们的模型案例是分数阶拉普拉斯函数（Gagliardo-Sobolev半模）的能量泛函，当时还没有这样的理论。在极值场存在的情况下，我们建立了非线性方程的零拉格朗日量和一个校正。因此，我们的构造假定存在欧拉-拉格朗日方程的一组解，其图产生叶理。然后，叶理中每片叶子的最小值都遵循校准的存在。作为一个应用，我们证明了分数阶半线性方程的单调解是最小解。在即将到来的工作中，我们将该理论推广到一类广泛的非局部椭圆泛函，并给出了粘滞理论的应用。
Abstract In this paper, we extend, for the first time, part of the Weierstrass extremal field theory in the Calculus of Variations to a nonlocal framework. Our model case is the energy functional for the fractional Laplacian (the Gagliardo–Sobolev seminorm), for which such a theory was still unknown. We build a null-Lagrangian and a calibration for nonlinear equations involving the fractional Laplacian in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler–Lagrange equation whose graphs produce a foliation. Then the minimality of each leaf in the foliation follows from the existence of the calibration. As an application, we show that monotone solutions to fractional semilinear equations are minimizers. In a forthcoming work, we generalize the theory to a wide class of nonlocal elliptic functionals and give an application to the viscosity theory.