论文标题
小部分拉普拉斯的Weierstrass极端田间理论
A Weierstrass extremal field theory for the fractional Laplacian
论文作者
论文摘要
在本文中，我们首次扩展了Weierstrass极端场理论的一部分，以变化到非本地框架的计算。我们的模型案例是分数拉普拉斯（Gagliardo-Sobolev seminorm）的能量功能，对此理论仍然未知。 我们在存在极端领域的情况下为涉及分数拉普拉斯的非线性方程式建立了无效的非线性方程式。因此，我们的构造假设存在图形产生叶面的Euler-Lagrange方程的解决方案家族。然后，叶片中每个叶片的最小性遵循校准的存在。作为一种应用，我们表明分数半线性方程的单调解决方案是最小化器。 在即将到来的工作中，我们将理论推广到一系列非局部椭圆功能，并将其应用于粘度理论。
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.
