Interpolation Theory for Sobolev Functions with Partially Vanishing Trace on Irregular Open Sets

Bechtel, Sebastian; Egert, Moritz

doi:10.1007/s00041-019-09681-1

Cited by 21 publications

(31 citation statements)

References 40 publications

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“…But this is true by Lemma 7.3. Hence, we can indeed apply the case above to complete the proof of (7). The rest of the argument from above applies verbatim.…”

Section: Proof Of Theorem 11 (I)mentioning

confidence: 86%

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$L^p$ bounds for square roots of elliptic systems on open sets

Bechtel¹

2022

Preprint

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We show L p -estimates for square roots of second order elliptic systems L in divergence form on open sets in R d subject to mixed boundary conditions. The underlying set is supposed to be locally uniform near the Neumann boundary part, and the Dirichlet boundary part is Ahlfors-David regular. The lower endpoint for the interval where such estimates are available is characterized by p-boundedness properties of the semigroup generated by −L, and the upper endpoint by extrapolation properties of the Lax-Milgram isomorphism. Our range is optimal, and upper and lower endpoints are sharp if they do not coincide with 1 or ∞.

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“…But this is true by Lemma 7.3. Hence, we can indeed apply the case above to complete the proof of (7). The rest of the argument from above applies verbatim.…”

Section: Proof Of Theorem 11 (I)mentioning

confidence: 86%

“…That the first scale is an interpolation scale was discussed in the proof of Theorem 1.1 (i) (note that the arguments in-there work both for the real and complex interpolation method). Finally, use duality to transfer the interpolation properties of the first scale to the last scale, see also [7,Prop. 5.2].…”

Section: Endpoint Casesmentioning

confidence: 99%

$L^p$ bounds for square roots of elliptic systems on open sets

Bechtel¹

2022

Preprint

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“…Here, our main Theorem 1 comes into play: If we are able to show that u ∈ (H 1+θ D (Ω), H θ −1 D (Ω)) 1/ p, p implies that A(u) is a suitable multiplier on H ε (Ω) 2 for some θ < ε < 1 2 , then the theorem shows that D θ = H 1+θ D (Ω) indeed does the job. Indeed, using the recent results in [4] which hold true under our general assumptions on Ω, one may show by interpolation techniques that…”

Section: Application: a Fracture Modelmentioning

confidence: 94%

Higher regularity for solutions to elliptic systems in divergence form subject to mixed boundary conditions

Haller-Dintelmann

Meinlschmidt

Wollner

2018

Annali di Matematica

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This work combines results from operator and interpolation theory to show that elliptic systems in divergence form admit maximal elliptic regularity on the Bessel potential scale H s−1 D (Ω) for s > 0 sufficiently small, if the coefficient in the main part satisfies a certain multiplier property on the spaces H s (Ω). Ellipticity is enforced by assuming a Gårding inequality, and the result is established for spaces incorporating mixed boundary conditions with very low regularity requirements for the underlying spatial set. To illustrate the applicability of our results, two examples are provided. Firstly, a phase-field damage model is given as a practical application where higher differentiability results are obtained as a consequence to our findings. These are necessary to show an improved numerical approximation rate. Secondly, it is shown how the maximal elliptic regularity result can be used in the context of quasilinear parabolic equations incorporating quadratic gradient terms.

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“…• It is known that maximal parabolic regularity is preserved under (real and complex) interpolation, see [35,Lemma 5.3]. Using the interpolation results from [7], this shows that the minus generator of the consistent C 0 -semigroup on the (dual of) Bessel potential spaces with non-integer differentiability index between −1 and 0 also satisfies maximal parabolic regularity. This considerably generalizes the results in [35,Thm.…”

mentioning

confidence: 92%

On the Lp-theory of C0-semigroups associated with second-order elliptic operators with complex singular coefficients

Elst

Liskevich

Sobol

et al. 2017

Proceedings of the London Mathematical Society

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We study L p -theory of second-order elliptic divergence type operators with complex measurable coefficients. The major aspect is that we allow complex coefficients in the main part of the operator, too. We investigate generation of analytic C 0 -semigroups under very general conditions on the coefficients, related to the notion of form-boundedness.We determine an interval J in the L p -scale, not necessarily containing p = 2, in which one obtains a consistent family of quasi-contractive semigroups. This interval is close to optimal, as shown by several examples. In the case of uniform ellipticity we construct a family of semigroups in an extended range of L p -spaces, and we prove p-independence of the analyticity sector and of the spectrum of the generators.MSC 2010: 35J15, 47F05, 47B44, 47D06

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Interpolation Theory for Sobolev Functions with Partially Vanishing Trace on Irregular Open Sets

Cited by 21 publications

References 40 publications

$L^p$ bounds for square roots of elliptic systems on open sets

$L^p$ bounds for square roots of elliptic systems on open sets

Higher regularity for solutions to elliptic systems in divergence form subject to mixed boundary conditions

On the Lp-theory of C0-semigroups associated with second-order elliptic operators with complex singular coefficients

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