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The square root problem for elliptic operators. A survey. (English) Zbl 0723.47032

Functional-analytic methods for partial differential equations, Proc. Conf. Symp., Tokyo/Jap. 1989, Lect. Notes Math. 1450, 122-140 (1990).
[For the entire collection see Zbl 0707.00017.]
Consider an elliptic sesquilinear form defined on \({\mathcal V}\times {\mathcal V}\) by \[ J[u,v]=\int_{\Omega}\{\sum a_{jk}\frac{\partial u}{\partial x_ k}\frac{\overline{\partial v}}{\partial x_ j}+\sum a_ k\frac{\partial u}{\partial x_ k}\bar v+\sum \alpha_ ju\frac{\overline{\partial v}}{\partial x_ j}+au\bar v\}dx, \] where \({\mathcal V}\) is a closed linear subspace of the Sobolev space \(H^ 1(\Omega)\) which contains \(C^{\infty}_ C(\Omega)\), \(\Omega\) is an open subset of \({\mathbb{R}}^ n\), \(a_{jk}\), \(a_ k\), \(\alpha_ j\), \(a\in L_{\infty}(\Omega)\) and Re\(\sum a_{jk}(x)\zeta_ k\overline{\zeta_ j}\geq \kappa | \zeta |^ 2\) for all \(\zeta =(\zeta_ j)\in {\mathbb{C}}^ n\) and some \(\kappa >0\). Let A be the operator in \(L_ 2(\Omega)\) with largest domain \({\mathcal D}(A)\subset {\mathcal V}\) such that \(J[u,v]=(Au,v)\) for all \(u\in {\mathcal D}(A)\) and all \(v\in {\mathcal V}\). Then \(A+\lambda I\) is a maximal accretive operator in \(L_ 2(\Omega)\) for some positive number \(\lambda\) and so has a maximal accretive square root \((A+\lambda I)^{1/2}\). The problem of determining whether its domain \({\mathcal D}((A+\lambda I)^{1/2})\) is equal to \({\mathcal V}\), possibly for particular choices of \(\Omega\) and \({\mathcal V}\), has become known as the square root problem of Kato for elliptic operators. It seems to be a more difficult problem now than when posed by Kato almost 30 years ago.
Reviewer: A.McIntosh

MSC:

47B44 Linear accretive operators, dissipative operators, etc.
47F05 General theory of partial differential operators
35J25 Boundary value problems for second-order elliptic equations

Citations:

Zbl 0707.00017