半空间上一个积分方程解的正则性

Regularity of Solutions to an Integral Equation on Half Space

考虑了半空间Rn+上一个包含Bessel位势的积分方程:u(x)=∫Rn+{gα(x-y)-gα(x-y)}uβ(y)dy,x∈Rn+,其中α>0,β>1,x是x关于超平面xn=0的对称点,gα(x)是Bessel核.首先利用结合压缩算子的正则提升方法得到积分方程的解的L∞估计.然后借助已被广泛使用的联合压缩算子和收缩算子的正则提升方法,证明积分方程的解是Lipschitz连续的.

In this article, we consider the following integral equation involving Bessel potentials on a half space R＋n：u（x）=∫Rn＋{gα（x-y）-gα（x-y）}uβ（y）dy,x∈Rn＋,where α〉0, β〉1, x- is the reflection point of x about xn =0, and g. （x） denotes the Bessel kernel. We first apply the regularity lifting by contracting operators to boost the positive solutions for integral equation from Lq to L∞. Then, we further use the regularity lifting by combination of contracting and shrinking operators, which has been extensively employed by many authors, to show the Lipschitz continuity.