Removable Singularities for Lu=Ψ(u) and Orlicz Capacities
Removable Singularities for Lu=Ψ(u) and Orlicz Capacities
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DOI：
10.1006/jfan.1999.3480
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发表时间：
2000-02
影响因子：
1.7
通讯作者：
S. E. Kuznetsov
中科院分区：
文献类型：
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作者：
S. E. Kuznetsov
设L是Rd中的二阶椭圆型微分算子，α>1. Baras和Pierre在1984年证明了对于Lu=uα，Γ是可去的当且仅当它的贝塞尔容量Cap 2，α′（Γ）=0。我们将这个结果推广到一般方程Lu= Lu（u），其中Lu（u）是满足Δ2和Δ 2条件的增凸函数.也就是说，我们证明了对于Lu=Ψ（u），Γ是可去的当且仅当它的Orlicz容度为零，即积分<$B <$（<$Γ| x−y| 2−d ν（dy））对于任何集中在Γ上的测度ν都等于0或∞，其中B代表任何包含Γ的球。
Suppose L is a second order elliptic differential operator in Rd and let α>1. Baras and Pierre have proved in 1984 that Γ is removable for Lu=uα if and only if its Bessel capacity Cap2, α′(Γ)=0. We extend this result to a general equation Lu=Ψ(u) where Ψ(u) is an increasing convex function subject to Δ2 and ∇2 conditions. Namely, we prove that Γ is removable for Lu=Ψ(u) if and only if its Orlicz capacity is zero, that is, the integral ∫B dx Ψ(∫Γ |x−y|2−d ν(dy)) is equal to 0 or ∞ for every measure ν concentrated on Γ, where B stands for any ball containing Γ.