The Kato square root problem on locally uniform domains. (English) Zbl 1494.47021
Summary: We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in \(\mathbb{R}^d\) under two simple geometric conditions: The Dirichlet boundary part is Ahlfors-David regular and a quantitative connectivity property in the spirit of locally uniform domains holds near the Neumann boundary part. This improves upon all existing results even in the case of pure Dirichlet or Neumann boundary conditions. We also treat elliptic systems with lower order terms. As a side product, we establish new regularity results for the fractional powers of the Laplacian with boundary conditions in our geometric setup.
MSC:
| 47A60 | Functional calculus for linear operators |
| 35J47 | Second-order elliptic systems |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 26A33 | Fractional derivatives and integrals |
Keywords:
Kato square root problem; fractional Laplacian; functional calculus; locally uniform domains; Ahlfors-David regular setsReferences:
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