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The Kato square root estimate with Robin boundary conditions

Sebastian Bechtel, Andreas Rosén

Abstract

We prove the Kato square root estimate for second-order divergence form elliptic operators $-div(A\nabla)$ on a bounded, locally uniform domain $D \subseteq \mathbb{R}^n$, for accretive coefficients $A \in L^\infty(D; \mathbb{C}^n)$, under the Robin boundary condition $ν\cdot A\nabla u + bu = 0$ for a (possibly unbounded) boundary conductivity $b$. In contrast to essentially all previous estimates of Kato square root operators, no first-order approach seems possible for the Robin boundary conditions.

The Kato square root estimate with Robin boundary conditions

Abstract

We prove the Kato square root estimate for second-order divergence form elliptic operators on a bounded, locally uniform domain , for accretive coefficients , under the Robin boundary condition for a (possibly unbounded) boundary conductivity . In contrast to essentially all previous estimates of Kato square root operators, no first-order approach seems possible for the Robin boundary conditions.
Paper Structure (14 sections, 20 theorems, 118 equations)

This paper contains 14 sections, 20 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Geometry and function spaces
  3. Elliptic operators with Robin boundary conditions
  4. Off-diagonal estimates
  5. A coercivity property for the Laplacian with Robin BC
  6. Reduction to a Carleson measure estimate
  7. Reduction to a quadratic estimate
  8. Smoothing of the quadratic estimate
  9. Reduction to small scales in the interior
  10. Dyadic structures and dyadic averaging operator
  11. Smoothed principal part approximation
  12. Removing the smoothing operator
  13. Full principal part approximation and reduction to a Carleson measure property
  14. The Carleson measure estimate

Key Result

Lemma 2.6

For $r \in (1,\infty)$, the spaces $\mathrm{W}^{1,r}(D)$ and $\mathrm{H}^{1,r}(D)$ coincide with equivalence of norms.

Theorems & Definitions (54)

  • Definition 2.1: Locally uniform
  • Remark 2.2
  • Definition 2.3: Interior thickness
  • Definition 2.4: $d$-set
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7: Trace operator
  • Lemma 3.1
  • proof
  • ...and 44 more