Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel
Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel
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DOI：
10.1016/j.aim.2021.107692
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发表时间：
2020-01
影响因子：
1.7
通讯作者：
T. Mengesha;A. Schikorra;Sasikarn Yeepo
中科院分区：
文献类型：
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作者：
T. Mengesha;A. Schikorra;Sasikarn Yeepo
本文研究了方程<Lsu，φ>：=<$Rn <$Rn K（x，y）（u（x）− u（y））（φ（x）− φ（y））的内Lp-正则性理论，也称为Calderon-Zygmund理论|x− y| n+2sdxdy =< f，φ>，<$φ∈ Cc ∞（Rn）.本文证明了对s∈（0，1），t∈[s，2 s]，p∈[2，∞），K是椭圆对称的，K（n，y）是一致Hölder连续的，只要2 s-t < 1且f∈（H 00 t，p′（Ω））n，解u属于H1 oc 2 s-t，p（Ω）.可微性和可积性的增加与K的Hölder系数无关。例如，在f∈ L l o c p的情况下，我们可以推导出解u∈ H l o c 2 s− δ，p对任何δ∈（0，s]，只要2 s− δ< 1。这一正则性结果不同于发散型方程div（K <$u）= f的经典类似结果，其中C γ-Hölder连续系数K <$u只允许在H1 + γ中有解。事实上，我们证明的正则性估计是上述非局部方程的微分稳定性效应的另一种表现，这是许多作者观察到的-只是在我们的情况下，我们没有得到“小”的可微性改进，而是一直到min {2 s− t，1}。通过与（更简单的）方程< L d i a g s，t u，φ>：=<$R n K（z，z）（− Δ）t 2 u（z）（− Δ）2 s− t 2 φ（z）d z=< g，φ>，<$φ∈ C c∞（R n）的比较，证明了只要K是Hölder连续的，并且s，t，2 s− t∈（0，1），那么“交换子”L s u− L d i a g s，t u的行为就像一个低阶算子。
We study interior L p-regularity theory, also known as Calderon-Zygmund theory, of the equation< L s u, φ>:=∫ R n∫ R n K (x, y)(u (x)− u (y))(φ (x)− φ (y))| x− y| n+ 2 s d x d y=< f, φ>,∀ φ∈ C c∞(R n). We prove that for s∈(0, 1), t∈[s, 2 s], p∈[2,∞), K an elliptic, symmetric, and K (⋅, y) is uniformly Hölder continuous, the solution u belongs to H l o c 2 s− t, p (Ω) as long as 2 s− t< 1 and f∈(H 00 t, p′(Ω))⁎. The increase in differentiability and integrability is independent of the Hölder coefficient of K. For example, in the event that f∈ L l o c p, we can deduce that the solution u∈ H l o c 2 s− δ, p for any δ∈(0, s] as long as 2 s− δ< 1. This regularity result is different from its classical analogue for divergence-form equations div (K¯∇ u)= f where a C γ-Hölder continuous coefficient K¯ only allows solutions in H 1+ γ. In fact, the regularity estimates we prove are another manifestation of the differential stability effects of nonlocal equations of the above that are observed by many authors–only that in our case we do not get a “small” differentiability improvement, but all the way up to min{2 s− t, 1}. The proof argues by comparison with the (much simpler) equation< L d i a g s, t u, φ>:=∫ R n K (z, z)(− Δ) t 2 u (z)(− Δ) 2 s− t 2 φ (z) d z=< g, φ>,∀ φ∈ C c∞(R n), and showing that as long as K is Hölder continuous and s, t, 2 s− t∈(0, 1) then the “commutator” L s u− L d i a g s, t u behaves like a lower order operator.