Andreas Axelsson, Stephen Keith, Alan McIntosh
We solve the Kato square root problem for second order elliptic systems in divergence form under mixed boundary conditions on Lipschitz domains. This answers a question posed by J.-L. Lions in 1962. To do this we develop a general theory of quadratic estimates and functional calculi for complex perturbations of Dirac-type operators on Lipschitz domains.
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The Kato square root problem for parabolic operators with an anti-symmetric part in BMO
本研究解决了系数可分解为强制性复部与BMO实反对称部之和的抛物算子的Kato平方根问题，且允许部分无界系数，拓展了该问题的适用范围。
The metric for matrix degenerate Kato square root operators
本文通过构造适应算子的辅助黎曼度量，证明了各向异性退化矩阵系数的Kato平方根估计，并利用链式法则和Piola变换推导了退化椭圆方程边值问题的$L^2$可解性。
We give a simplified and direct proof of the Kato square root estimate for parabolic operators with elliptic part in divergence form and coefficients possibly depending on space and time in a merely measurable way. The argument relies on the nowadays classical reduction to a quadratic estimate and a Carleson-type inequality. The precise organization of the estimates is different from earlier works. In particular, we succeed in separating space and time variables almost completely despite the non-autonomous character of the operator. Hence, we can allow for degenerate ellipticity dictated by a …
The Kato square root estimate with Robin boundary conditions
本文研究了带有Robin边界条件的Kato平方根估计，针对二阶散度形式椭圆算子在局部均匀域上的估计问题，提出了新的分析方法。
This is the text of a series of three lectures given at the CMA of the Australian National University on the recent solution of the square root problem for divergence form elliptic operators, a long-standing conjecture posed by Kato in the early 60's. In this text, the motivations for this problem and its situation are given. The ideas from harmonic analysis on the T(1) theorem and T(b) theorem for square functions are described. In particular, an apparently new formulation of a local T(b) theorem for square functions is stated. The ideas of the full proof are presented. I want to thank the CM…
We solve the Kato square root problem for divergence form operators on complete Riemannian manifolds that are embedded in Euclidean space with a bounded second fundamental form. We do this by proving local quadratic estimates for perturbations of certain first-order differential operators that act on the trivial bundle over a complete Riemannian manifold with at most exponential volume growth and on which a local Poincaré inequality holds. This is based on the framework for Dirac type operators that was introduced by Axelsson, Keith and McIntosh.
We prove the Kato conjecture for elliptic operators, $L=-\nabla\cdot\left((\mathbf A+\mathbf D)\nabla\ \right)$, with $\mathbf A$ a complex measurable bounded coercive matrix and $\mathbf D$ a measurable real-valued skew-symmetric matrix in $\mathbb{R}^n$ with entries in $BMO(\mathbb{R}^n)$;\, i.e., the domain of $\sqrt{L}\,$ is the Sobolev space $\dot H^1(\mathbb{R}^n)$ in any dimension, with the estimate $\|\sqrt{L}\, f\|_2\lesssim \| \nabla f\|_2$.
A second order approach to the Kato square root problem on open sets
本文针对开集上带混合边界条件的耦合二阶椭圆系统，利用 Ahlfors–David 正则性与定量连通性条件，以二阶方法证明 Kato 平方根性质，从而获得更短、技术更简单的证明。
We solve the Kato square root problem for general elliptic operators and systems with measurable and complex coefficients on any domain of the Euclidean space. The operators are subject to Dirichlet boundary conditions. We also allow perturbations by general complex potentials.
We consider smooth, complete Riemannian manifolds which are exponentially locally doubling. Under a uniform Ricci curvature bound and a uniform lower bound on injectivity radius, we prove a Kato square root estimate for certain coercive operators over the bundle of finite rank tensors. These results are obtained as a special case of similar estimates on smooth vector bundles satisfying a criterion which we call generalised bounded geometry. We prove this by establishing quadratic estimates for perturbations of Dirac type operators on such bundles under an appropriate set of assumptions.
Let $L$ be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces $L^{p}(R^{n};X)$ of $X$-valued functions on $R^n$. We characterize Kato's square root estimates $\|\sqrt{L}u\|_{p} \eqsim \|\nabla u\|_{p}$ and the $H^{\infty}$-functional calculus of $L$ in terms of R-boundedness properties of the resolvent of $L$, when $X$ is a Banach function lattice with the UMD property, or a noncommutative $L^{p}$ space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. I…
On the Kato problem for elliptic operators in non-divergence form
本文研究非散度型二阶椭圆算子及其规范化伴随算子的Kato平方根问题，给出适用于小BMO范数实系数情形的结果，且相关估计不依赖系数光滑性假设。
On Kato's Square Root Property for the Generalized Stokes Operator
本文证明 $\mathbb{R}^d$ 上带有界可测椭圆系数的广义 Stokes 算子具有 Kato 平方根性质，刻画其平方根定义域并得到梯度等价估计，还推出平方根对系数的全纯依赖性。
The Kato problem for weighted elliptic and parabolic operators of higher order
本文解决了系数仅可测且受Muckenhoupt $A_2$权控制的任意阶$2m$抛物算子的Kato平方根问题，并证明该方法同样适用于椭圆框架下的加权Kato问题。
We prove the Kato conjecture for degenerate elliptic operators in R^n. More precisely, we consider the divergence form operator L_w = -1/w div (wA) grad, where w is a Muckenhoupt A_2 weight and A is a complex valued n x n matrix which is bounded and uniformly elliptic. We show that if the associated semigroup satisfies Gaussian upper bounds, then the Kato square root estimate holds.
We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with $L^2$ boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms.
We study the Kato problem for degenerate divergence form operators. This was begun by Cruz-Uribe and Rios who proved that given an operator $L_w=-w^{-1}{\rm div}(A\nabla)$, where $w\in A_2$ and $A$ is a $w$-degenerate elliptic measure (i.e, $A=w\,B$ with $B$ an $n\times n$ bounded, complex-valued, uniformly elliptic matrix), then $L_w$ satisfies the weighted estimate $\|\sqrt{L_w}f\|_{L^2(w)}\approx\|\nabla f\|_{L^2(w)}$. Here we solve the $L^2$-Kato problem: under some additional conditions on the weight $w$, the following unweighted $L^2$-Kato estimates hold $$ \|L_w^{1/2}f\|_{L^2(\mathbb{R}…
We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of Riemannian-like metrics that are permitted to be of low regularity and degenerate on sets of measure zero. We also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi-Lipschitz. As a consequence, we demonstrate the invariance of the Kato square root problem under Lipschitz transformations of the space and obtain solutions to this problem on functions and forms on compact manifolds with a …
We solve the Kato square root problem for bounded measurable perturbations of subelliptic operators on connected Lie groups. The subelliptic operators are divergence form operators with complex bounded coefficients, which may have lower order terms. In this general setting we deduce inhomogeneous estimates. In case the group is nilpotent and the subelliptic operator is pure second order, then we prove stronger homogeneous estimates. Furthermore, we prove Lipschitz stability of the estimates under small perturbations of the coefficients.
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$ …
This article focuses on Lp-estimates for the square root of elliptic systems of second order in divergence form on a bounded domain. We treat complex bounded measurable coefficients and allow for mixed Dirichlet/Neumann boundary conditions on domains beyond the Lipschitz class. If there is an associated bounded semigroup on Lp0 , then we prove that the square root extends for all p $\in$ (p0, 2) to an isomorphism between a closed subspace of W1p carrying the boundary conditions and Lp. This result is sharp and extrapolates to exponents slightly above 2. As a byproduct, we obtain an optimal p-i…
We prove a local $Tb$ theorem for paraproducts acting on vector valued functions, with matrix weighted averaging operators. The condition on the weight is that its square is in the $L_2$ associated matrix $A_\infty$ class. We also introduce and use a new matrix reverse Hölder class. This result generalizes the previously known case of scalar weights from the proof of the Kato square root problem, as well as the case of diagonal weights, recently used in the study of boundary value problems for degenerate elliptic equations.
We show that, under general conditions, the operator $\bigl (-\nabla \cdot μ\nabla +1\bigr)^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W^{1,p}_D(Ω)$ and $L^p(Ω)$, for $p \in {]1,2[}$ if one presupposes that this isomorphism holds true for $p=2$. The domain $Ω$ is assumed to be bounded, the Dirichlet part $D$ of the boundary has to satisfy the well-known Ahlfors-David condition, whilst for the points from $\overline {\partial Ω\setminus D}$ the existence of bi-Lipschitzian boundary charts is required.
We study second-order divergence-form systems on half-infinite cylindrical domains with a bounded and possibly rough base, subject to homogeneous mixed boundary conditions on the lateral boundary and square integrable Dirichlet, Neumann, or regularity data on the cylinder base. Assuming that the coefficients A are close to coefficients A\_0 that are independent of the unbounded direction with respect to the modified Carleson norm of Dahlberg, we prove a priori estimates and establish well-posedness if A\_0 has a special structure. We obtain a complete characterization of weak solutions whose g…
Existence results for mixed local and nonlocal elliptic equations involving singularity and nonregular data
针对含奇异性与测度数据的混合局部-非局部椭圆方程，运用Schauder不动点定理、Marcinkiewicz空间嵌入等工具，证明弱解、极弱解与对偶解的存在性，拓展了相关算子的已有研究。
$L^p$-estimates for the square root of elliptic systems with mixed boundary conditions II
该研究针对带混合边界条件的散度型二阶复椭圆方程组平方根建立$L^p$估计，刻画了估计成立区间的上下端点，并证明外推区间在$(1,\u221e)$中相对开。
We discuss stability of square root domains for uniformly elliptic partial differential operators $L_{a,Ω,Γ} = -\nabla\cdot a \nabla$ in $L^2(Ω)$, with mixed boundary conditions on $\partial Ω$, with respect to additive perturbations. We consider open, bounded, and connected sets $Ω\in \mathbb{R}^n$, $n \in \mathbb{N} \backslash\{1\}$, that satisfy the interior corkscrew condition and prove stability of square root domains of the operator $L_{a,Ω,Γ}$ with respect to additive potential perturbations $V \in L^p(Ω) + L^{\infty}(Ω)$, $p>n/2$. Special emphasis is put on the case of uniformly ellipt…
Assuming $T_0$ to be an m-accretive operator in the complex Hilbert space $\mathcal{H}$, we use a resolvent method due to Kato to appropriately define the additive perturbation $T = T_0 + W$ and prove stability of square root domains, that is, $$ dom\big((T_0 + W)^{1/2}\big) = dom\big(T_0^{1/2}\big). $$ Moreover, assuming in addition that $dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big)$, we prove stability of square root domains in the form $$dom\big((T_0 + W)^{1/2}\big) = dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big) = dom\big(((T_0 + W)^*)^{1/2}\big), $$ which is most suitable for…
Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds embedded into the full domain. Examples are fractured and composite materials, but also wells (in geological applications), plant roots, or arteries and veins. In this manuscript, we survey the structure of mixed-dimensional PDEs in the context where the sub-manifolds are a single dimension lower than the full domain, including the important aspect of intersecti…
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