完全非线性积分算子的退化结果
Degeneracy Results for Fully Nonlinear Integral Operators
摘要：
证明了完全非线性型$K(x)(t)=\int_Ωk(t,s,x(t),x(s))\,ds$的积分算子在一大类空间中表现出与叠加算子$F(x)(t)=f(t,x(t))$相似的退化现象.特别地，$K$在$L_p$中是Fréchet可微的，仅当它关于“$x(t)$“参数是仿射的。如果$K$满足局部Lipschitz或紧性条件，则类似的退化结果成立。向量函数，无限测度空间，和一个更丰富的类的函数空间比只有$L_p$考虑。作为一个副作用，在这个更一般的设置中获得叠加算子的退化断言，补充了标量函数的已知结果。作为一个特殊的例子，证明了耦合仓本振子的连续极限中的算子处处不是Fréchet可微的或局部紧的.
英文摘要：
It is shown that integral operators of the fully nonlinear type $K(x)(t)=\int_Ωk(t,s,x(t),x(s))\,ds$ exhibit similar degeneracy phenomena in a large class of spaces as superposition operators $F(x)(t)=f(t,x(t))$. In particular, $K$ is Fréchet differentiable in $L_p$ only if it is affine with respect to the "$x(t)$" argument. Similar degeneracy results hold if $K$ satisfies a local Lipschitz or compactness condition. Also vector functions, infinite measure spaces, and a much richer class of function spaces than only $L_p$ are considered. As a side result, degeneracy assertions for superposition operators are obtained in this more general setting, complementing the known results for scalar functions. As a particular example, it is shown that the operators arising in continuous limits of coupled Kuramoto oscillators fail everywhere to be Fréchet differentiability or locally compact.
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