论文标题
抛物线操作员的平方根
The square root of a parabolic operator
论文作者
论文摘要
令l（t）= -div（a（x，t）$ \ nabla $ x）t $ \ in $（0，$τ$）是一个均匀的椭圆运算符，在r d和$ \ partial $ = $ = $ = $ \ partial $ $ $ $ $ \ partial $ t的域上具有边界条件。定义抛物线运算符l = $ \ partial $ + l在L 2（0，$τ$，l 2（$ω$））的（lu）（t）（t）：= $ \ partial $ u（t）$ \ partial $ \ partial $ t + l（t）u（t）u（t）u（t）。对于$ω$的边界，我们假设几乎没有规律性，并假设系数A（x，t）可以在x中可测量，则可以在x和分段的c $α$中t t to $α$> 1 2。 $ \ nabla $ x u l 2（0，$τ$，l 2（$ω$）） + u H 1 2（0，$τ$，l 2（$ω$）） + $τ$ 0 u（t）2 l 2（$ω$）dt t t t 1/2。我们还证明了这一结果的p-perversions。关键字：椭圆形和抛物线算子，加藤方形根特性，最大规律性，全体形态功能演算，非自主进化方程。
Let L(t) = --div (A(x, t)$\nabla$ x) for t $\in$ (0, $τ$) be a uniformly elliptic operator with boundary conditions on a domain $Ω$ of R d and $\partial$ = $\partial$ $\partial$t. Define the parabolic operator L = $\partial$ + L on L 2 (0, $τ$, L 2 ($Ω$)) by (Lu)(t) := $\partial$u(t) $\partial$t + L(t)u(t). We assume a very little of regularity for the boundary of $Ω$ and assume that the coefficients A(x, t) are measurable in x and piecewise C $α$ in t for some $α$ > 1 2. We prove the Kato square root property for $\sqrt$ L and the estimate $\sqrt$ L u L 2 (0,$τ$,L 2 ($Ω$)) $\approx$ $\nabla$ x u L 2 (0,$τ$,L 2 ($Ω$)) + u H 1 2 (0,$τ$,L 2 ($Ω$)) + $τ$ 0 u(t) 2 L 2 ($Ω$) dt t 1/2. We also prove L p-versions of this result. Keywords: elliptic and parabolic operators, the Kato square root property, maximal regularity, the holomorphic functional calculus, non-autonomous evolution equations.
