有界域中的微分算子和正则半群

设Ω Rn是一个有界区域.如果P(ξ)是一个2m次实系数椭圆多项式,利用Sobolev嵌入定理和正则半群的内插定理,证明了当k≥n/2m|1/2-1/p|(1＜p＜∞)时 p(具有Dirichlet边界条件)在Lp(Ω)中,当k＞n/4m(k∈N0)时 1在L1(Ω)中, ∞在L∞(Ω)中, 0在C0(Ω)中和 c在C( )中生成一个(1-△p)-km-正则群.结果表明在有界区域中偏微分算子比在Rn中具有更好的正则性.

Differential Operators and Regularized Semigroups on Bounded Domain

Let Ω Rn be a bounded open set. When P(ξ) is a real coefficient elliptic polyno-mial of order 2m , by using the Sobolev' s imbedding theorem and an interpolation extensiontheorem of regularized semigroup, we show that P (with Dirichlet boundary conditions)generates a (1 - Δp)-km -regularized group on Lp(Ω) (1 ＜p＜ ∞) if k≥n/2m|1/2-1/p| and so does 1 on L1(Ω) , ∞ on L∞(Ω) , 0 onC0(Ω) , and c on C( ) if k＞ n/4m(k∈N0).Our results show that on a bounded domain, partial differential operators are more regular than on Rn .