Bounded \(H^\infty \)-calculus for a class of nonlocal operators: the bidomain operator in the \(L_q\)-setting


The bidomain operator \({{\mathbb {A}}}\), a nonlocal operator, is studied in a bounded domain \(\Omega \subset {{\mathbb {R}}}^d\) with boundary \(\partial \Omega \) of class \(C^{2-}\) within the \(L_q\)-setting for \(1<q<\infty \). Assuming a fairly general framework, it is shown that this operator is sectorial, invertible on functions with mean zero, and admits a bounded \({H}^\infty \)-calculus with \({H}^\infty \)-angle 0. In particular, it has the property of maximal \(L_p-L_q\)-regularity and the fractional power domains \({{\mathcal {D}}}({{\mathbb {A}}}^\alpha )\) are identified. Furthermore, it is shown that \(-{{\mathbb {A}}}\) generates a strongly continuous, compact, analytic and exponentially stable semigroup on \(L_{q,0}(\Omega )\) with q-independent, purely countable point spectrum \(\sigma ({{\mathbb {A}}})\).

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Hieber, M., Prüss, J. Bounded \(H^\infty \)-calculus for a class of nonlocal operators: the bidomain operator in the \(L_q\)-setting. Math. Ann. 378, 1095–1127 (2020).

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Mathematics Subject Classification

  • 35K50
  • 42B20
  • 92C35