有界域中的微分算子和正则半群(英文)

Differential Operators and Regularized Semigroups on Bounded Domain

设Ω Rn是一个有界区域 .如果 P(ξ)是一个 2 m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≥ n2 m| 12 - 1p| (1<p <∞ )时 Ap (具有 Dirichlet边界条件 )在 L p (Ω)中 ,当k >n4m( k∈ N0 )时 A1在 L1(Ω)中 ,A∞ 在 L∞ (Ω)中 ,A0 在 C0 (Ω )中和 AC在 C(Ω)中生成一个 ( 1-Δp) - 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 .

Let ΩR n be a bounded open set. When P(ξ) is a real coefficient elliptic polynomial of order 2m, by using the Sobolev's imbedding theorem and an interpolation extension theorem of regularized semigroup, we show that A p(with Dirichlet boundary conditions) generates a (1-Δ p) -km regularized group on L p(Ω)(1<p<∞) if k≥n2m|12-1p| and so does A 1 on L 1(Ω), A ∞ on L ∞ (Ω), A 0 on C 0(Ω) and A C on C() if >n4m(k∈N 0).Our results show that on a bounded domain, partial differential operators are more regular than on R n.