Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem

Favini, Angelo; Goldstein, Gisèle Ruiz; Goldstein, Jerome A.; Obrecht, Enrico; Romanelli, Silvia

doi:10.1002/mana.200910086

Cited by 39 publications

(42 citation statements)

References 33 publications

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“…In the same papers [11,12], under C ∞ regularity assumptions on ∂Ω, on all the coecients of A and on β, γ, we also showed that the closure of the realization of A in X p , 1 < p < ∞, is m-dissipative and generates an analytic semigroup having sector of analyticity depending on the moduli of ellipticity of A. Moreover, its domain can be explicitly described.…”

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confidence: 64%

“…Existence and analyticity results for the semigroup associated with A remain true also for more general elliptic versions of A, equipped with the corresponding (GW BC). See [11]. It is very important to point out that there is a precise physical derivation of (GWBC), as it was shown in details by G.R.…”

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confidence: 90%

“…In [11,12] the authors assumed that A is a uniformly elliptic operator in divergence form of general type. Here we refer to A given as in (4) and to the assumptions on the coecients in (GGWBC) as before.…”

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confidence: 99%

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The Contribution of Angelo Favini in Twenty Years of Joint Research (1996 - 2016)

Romanelli¹

2017

Bulletin of the SUSU. MMP

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During his long activity in research, Angelo Favini has given several relevant contributions in many elds of mathematics. In our long collaboration, testied by more than thirty joint papers, I highly appreciated his deep competence, his brilliant intuitions, his extraordinary knowledge of the contemporary literature, in addition to a special acumen in nding possible critical points to be claried, or better solved, during the preparation of the papers.Here follows a short survey of the main problems studied in our joint papers over the last twenty years, often in collaboration with other outstanding mathematicians.(1) At the beginning of 1990s there was a long standing open problem concerning the existence of analytic semigroups generated by second order elliptic dierential operators degenerating at the boundary. It is well-known that the existence of a (C 0 ) semigroup guarantees the wellposedness of the abstract Cauchy problem associated with the evolution equation governed by the generator, provided that the initial datum is in its domain. The analyticity of the semigroup provides, in addition, the best possible regularity of the solution, even by starting from the initial datum in the ambient space. As already known in Probability, or in Approximation Theory, in the space of continuous functions C(Ω) (here Ω is a bounded open subset of R N with suciently smooth boundary ∂Ω), a possibly degenerate on the boundary, second order elliptic operator A can be naturally equipped with the so-called Wentzell boundary condition Au = 0, introduced in the one-dimensional case by W. Feller in his pioneer work [7] (see also [6]), and in the multidimensional case by A.D. Wentzell [38]. For Wentzell boundary conditions in dierent spaces see e.g. [33,37]. Observe that, in an evolution equation u t = Au, replacing Au by u t in Wentzell boundary condition reveals that, under suitable regularity assumptions on the elements of the domain of A, a solution u of the equation with Wentzell boundary condition remains constant with respect to t along the boundary. An easy example of degenerate elliptic second order dierential operator on the space C[0, 1] equipped with Wentzell boundary conditions is A j u = x j (1 − x) j u ′′ , j > 0, well-known also for governing some evolution problems in genetics. In [1] the existence of analytic semigroups generated by A j was proved for j ≥ 2 on C[0, 1] by using dierent types of boundary conditions, including Wentzell's ones. After that, much attention was paid to this problem and relevant contributions by A. Favini et al. appeared in dierent papers, where, in addition to the space of continuous functions, as in [27,30,31], also L p spaces, with or without weights, and Sobolev spaces were considered, see e.g. [2,20,28,29]. Results concerning the wave equation with Wentzell boundary conditions in C[0, 1] were also obtained in [16].(2) In the real-valued space C(Ω) other interesting problems concern the existence of Feller semigroups generated by second-order elliptic dierential operators degenerati...

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The Contribution of Angelo Favini in Twenty Years of Joint Research (1996 - 2016)

Romanelli¹

2017

Bulletin of the SUSU. MMP

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“…This is an extremely tough problem in general. For some related complementary results, see [6]. Chapter 6 deals with partial differential operators of order higher than two.…”

Section: Theorem 2 (Hille-yosida Generation Theorem) An Operator a Imentioning

confidence: 99%

Book Review: Semi-bounded differential operators, contractive semigroups, and beyond

Goldstein¹

2017

Bull. Amer. Math. Soc.

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“…We have become aware that Favini et al [5] have recently obtained the result stated in Theorem 1.2 with a different approach.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Analyticity on L 1 of the heat semigroup with Wentzell boundary conditions

Warma

2009

Arch. Math.

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Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem

Cited by 39 publications

References 33 publications

The Contribution of Angelo Favini in Twenty Years of Joint Research (1996 - 2016)

The Contribution of Angelo Favini in Twenty Years of Joint Research (1996 - 2016)

Book Review: Semi-bounded differential operators, contractive semigroups, and beyond

Analyticity on L 1 of the heat semigroup with Wentzell boundary conditions

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