Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients

Carbonaro, Antonella; Dragičević, Oliver

doi:10.4171/jems/984

Cited by 28 publications

(128 citation statements)

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“…Strongly elliptic operators with mixed boundary conditions are considered in [6,7]. All results in [1] for C 0 -semigroups are for strongly elliptic operators. The main emphasis in this paper is to consider degenerate elliptic operators.…”

Section: T D Domentioning

confidence: 99%

“…Ouhabaz in [15,Theorem 3.9] proved that (1) is true for generators of sub-Markovian semigroups. It is interesting to note that [15,Theorem 3.9] gives the same constant K in (1) as in [14].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we will prove the sectorial estimate (1) for degenerate elliptic secondorder differential operators with bounded complex-valued coefficients. The results are generalizations of [14].…”

Section: Introductionmentioning

confidence: 99%

“…Remark 1.3. The conditions, conclusion and some implications can be rephrased in the recently introduced terminology of Carbonaro and Dragičević [1]. For every p ∈ (1, ∞) and bounded…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, if Δ 2 (C) > 0 (the operator is strongly elliptic) then [1, Theorem 1.3 (a ⇒ b)] together with the Lumer-Phillips theorem establishes that (8) implies (1), where K = tan(π/2 − φ + θ) which coincides with (5)…”

Section: Introductionmentioning

confidence: 99%

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On sectoriality of degenerate elliptic operators

Duc¹

2021

Proceedings of the Edinburgh Mathematical Society

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Let $c_{kl} \in W^{1,\infty }(\Omega , \mathbb{C})$ for all $k,l \in \{1, \ldots , d\};$ and $\Omega \subset \mathbb{R}^{d}$ be open with uniformly $C^{2}$ boundary. We consider the divergence form operator $A_p = - \sum \nolimits _{k,l=1}^{d} \partial _l (c_{kl} \partial _k)$ in $L_p(\Omega )$ when the coefficient matrix satisfies $(C(x) \xi , \xi ) \in \Sigma _\theta$ for all $x \in \Omega$ and $\xi \in \mathbb{C}^{d}$ , where $\Sigma _\theta$ be the sector with vertex 0 and semi-angle $\theta$ in the complex plane. We show that a sectorial estimate holds for $A_p$ for all $p$ in a suitable range. We then apply these estimates to prove that the closure of $-A_p$ generates a holomorphic semigroup under further assumptions on the coefficients. The contractivity and consistency properties of these holomorphic semigroups are also considered.

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Section: T D Domentioning

confidence: 99%