Regularity for a Local–Nonlocal Transmission Problem
Regularity for a Local–Nonlocal Transmission Problem
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局部-非局部传输问题的规律性
DOI：
10.1007/s00205-015-0851-4
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发表时间：
2014
影响因子：
2.5
通讯作者：
D. Kriventsov
中科院分区：
文献类型：
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作者：
D. Kriventsov
我们构造并研究了一类结合局部和非局部元素的类椭圆传输问题。设$${\mathbb{R}^n}$$ Rn被光滑的超表面Γ分成两个分量。在Γ的一侧，一个函数满足一个局部二阶椭圆方程。另一方面，它满足一个低阶的非局部方程。此外，在接口Γ上施加了非局部传输条件，这具有自然的变分解释。我们根据De Giorgi的方法，推导了相应Dirichlet问题解的存在性，并证明了在温和的假设下它们是Hölder连续的。主要的困难源于Γ附近缺乏尺度不变性，我们通过推导一个特殊的能量估计来规避这个问题，该能量估计在尺度参数中是均匀的。然后，我们转向解的最优正则性和定性性质的问题，并表明（在常系数和平坦Γ的情况下）它们满足一种传输条件，其中界面两侧的“分数法向导数”的比率是固定的。然后给出了一个微扰论证，说明了如何获得具有光滑界面的变系数方程解的正则性。在整篇文章中，我们特别关注了该问题的非线性版本，该问题具有由解的奇异积分算子给出的漂移，它在拟等转动力学的背景下有一个解释。
We formulate and study an elliptic transmission-like problem combining local and nonlocal elements. Let $${\mathbb{R}^n}$$Rn be separated into two components by a smooth hypersurface Γ. On one side of Γ, a function satisfies a local second-order elliptic equation. On the other, it satisfies a nonlocal one of lower order. In addition, a nonlocal transmission condition is imposed across the interface Γ, which has a natural variational interpretation. We deduce the existence of solutions to the corresponding Dirichlet problem, and show that under mild assumptions they are Hölder continuous, following the method of De Giorgi. The principal difficulty stems from the lack of scale invariance near Γ, which we circumvent by deducing a special energy estimate which is uniform in the scaling parameter. We then turn to the question of optimal regularity and qualitative properties of solutions, and show that (in the case of constant coefficients and flat Γ) they satisfy a kind of transmission condition, where the ratio of “fractional conormal derivatives” on the two sides of the interface is fixed. A perturbative argument is then given to show how to obtain regularity for solutions to an equation with variable coefficients and smooth interface. Throughout, we pay special attention to a nonlinear version of the problem with a drift given by a singular integral operator of the solution, which has an interpretation in the context of quasigeostrophic dynamics.