Regularity for Fully Nonlinear Integro-differential Operators with Regularly Varying Kernels
Regularity for Fully Nonlinear Integro-differential Operators with Regularly Varying Kernels
复制标题
具有规则变化核的完全非线性积分微分算子的正则性
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Ki
中科院分区：
文献类型：
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作者：
Soojung Kim;Yong;Ki
摘要在本文中，Caffarelli和Silvestre（Comm. Pure Appl. Math. 62，597-638，2009）关于分数拉普拉斯型积分微分算子的正则性结果被推广到与零处的对称、正则变化核相关联的积分微分算子的正则性结果。特别地，我们得到了非线性积分微分方程粘性解的一致Harnack不等式和Hölder估计，其中Kσ，β满足
Kσ，β（y）<$2 −σ| y| n+σ log 2| y| 2β（2−σ）接近于0 $$K_{sigma，{2-sigma}{ 2-sigma}{|y| ^{n+sigma}}左（logfrac{2}{|y|联系我们
（八）^{η（2-sigma）} ext{near zero} $关于σ ∈（0，2）接近2（对于给定的β∈ N $eta in mathbb R$），其中当阶数σ ∈（0，2）趋于2时，正则性估计不会爆炸。
AbstractIn this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre (Comm. Pure Appl. Math. 62, 597–638, 2009) are extended to those for the integro-differential operators associated with symmetric, regularly varying kernels at zero. In particular, we obtain the uniform Harnack inequality and Hölder estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels Kσ,β satisfying
Kσ,β(y)≍2−σ|y|n+σlog2|y|2β(2−σ)near zero$$K_{sigma,eta}(y)asympfrac{ 2-sigma}{|y|^{n+sigma}}left( logfrac{2}{|y|^{2}}
ight)^{eta(2-sigma)} ext{near zero} $$ with respect to σ ∈ (0, 2) close to 2 (for a given β∈ℝ$eta in mathbb R$), where the regularity estimates do not blow up as the order σ ∈ (0, 2) tends to 2.