Elliptic and parabolic regularity for second‐ order divergence operators with mixed boundary conditions

Haller-Dintelmann, Robert; Jonsson, Alf; Knees, Dorothee; Rehberg, Joachim

doi:10.1002/mma.3484

Cited by 34 publications

(47 citation statements)

References 62 publications

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“…We close this gap by establishing a full interpolation theory under geometric assumptions in the spirit of what has become standard for treating mixed boundary value problems [3,8,17,24]. In particular, we confirm the formula for the complex interpolation spaces that was conjectured in connection with fractional powers of divergence form operators in [3,Rem.…”

Section: Introduction and Main Resultssupporting

confidence: 65%

“…This leads to the problem of identifying such interpolation spaces in the presence of rough boundaries to the fractional Sobolev-and Bessel potential spaces, in analogy with what is long known for function spaces on R d or on smooth domains with pure Dirichet boundary condition [6,36,40]. Interpolation theory related to the spaces W 1,p D (O) has recently been studied in [3,4,8,15,19,24], but mostly with a focus on interpolating with respect to integrability. Interpolation in differentiability appears only in [4,15] for p = 2 and in [19] for general p on certain model sets.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Bechtel

Egert

2019

J Fourier Anal Appl

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A full interpolation theory for Sobolev functions with smoothness between 0 and 1 and vanishing trace on a part of the boundary of an open set is established. Geometric assumptions are of mostly measure theoretic nature and reach beyond Lipschitz regular domains. Previous results were limited to regular geometric configurations or Hilbertian Sobolev spaces. Sets with porous boundary and their characteristic multipliers on smoothness spaces play a major role in the arguments.

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Section: Introduction and Main Resultssupporting

confidence: 65%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Bechtel

Egert

2019

J Fourier Anal Appl

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“…Hence, establishing the existence of a solution to the variational problem (3.4) is equivalent to showing that the operator T R n : X p (R n ) → X ′ p ′ (R n ) is an isomorphism (see also [11,Proposition 7.2], [30,Theorem 5.6], and [53, Theorem 3.1] for the standard Stokes system).…”

Section: )mentioning

confidence: 99%

“….3],[30, Theorem 5.6],[53, Theorem 3.1]).Consequently, whenever condition (3.3) holds and for all given data (ξ, ζ) ∈ H −1 p (R n ) n × L p (R n ), there exists a unique solution (u, π) ∈ H 1 p (R n ) n × L p (R n ) of the equation T R n (u, π) = (ξ, ζ) or, equivalently, of the variational problem (3.4), satisfying inequality (3.5).Next we use Lemma 3.1 and show the well-posedness of the L ∞ -coefficient Stokes system in the space H 1 p (R n ) n × L p (R n ) for any p ∈ R(p * , n) (cf [38,. Theorem 4.2] for p = 2 with A(x) = µ(x)I, [42, Proposition 2.9] and [2, Theorem 3] for p ∈ (1, n) in the constant-coefficient case).…”

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confidence: 99%

Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier–Stokes systems with L_∞ strongly elliptic coefficient tensor

Kohr

Mikhailov

Wendland

2019

Complex Variables and Elliptic Equations

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We obtain well-posedness results in Lp-based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier-Stokes systems with L∞ strongly elliptic coefficient tensor, in complementary Lipschitz domains of R n , n ≥ 3. The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces two linear transmission problems for the anisotropic Stokes system to equivalent mixed variational formulations with data in Lp-based weighted Sobolev and Besov spaces. We show that such a mixed variational formulation is well-posed in the space H 1 p (R n ) n × Lp(R n ), n ≥ 3, for any p in an open interval containing 2. These results are used to define the Newtonian and layer potential operators for the considered anisotropic Stokes system. Various mapping properties of these operators are also obtained. The potentials are employed to show the well-posedness of some linear transmission problems, which then is combined with a fixed point theorem in order to show the well-posedness of the nonlinear transmission problem for the anisotropic Stokes and Navier-Stokes systems in Lp-based weighted Sobolev spaces, whenever the given data are small enough.for an unsteady exterior Stokes problem). The authors in [34] obtained mapping properties of the constant-coefficient Stokes and Brinkman layer potential operators in standard and weighted Sobolev spaces by exploiting results of singular integral operators (see also [35,36]).The methods of layer potential theory play also a significant role in the study of elliptic boundary problems with variable coefficients. Mitrea and Taylor [49, Theorem 7.1] used the technique of layer potentials to prove the well-posedness of the Dirichlet problem for the Stokes system in L p -spaces on arbitrary Lipschitz domains in a compact Riemannian manifold. Dindos and Mitrea [24, Theorems 5.1, 5.6, 7.1,7.3] used a boundary integral approach to show well-posedness results in Sobolev and Besov spaces for Poisson problems of Dirichlet type for the Stokes and Navier-Stokes systems with smooth coefficients in Lipschitz domains on compact Riemannian manifolds. A layer potential analysis of pseudodifferential operators of Agmon-Douglis-Nirenberg type in Lipschitz domains on compact Riemannian manifolds has been developed in [39]. The authors in [37] used a layer potential approach and a fixed point theorem to show well-posedness of transmission problems for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems with smooth coefficients in Lipschitz domains on compact Riemannian manifolds. Choi and Lee [21] proved the well-posedness in Sobolev spaces for the Dirichlet problem for the Stokes system with non-smooth coefficients in a Lipschitz domain Ω ⊂ R n (n ≥ 3) with a small Lipschitz constant when the coefficients have vanishing mean oscillations (VMO) with respect to all variables. Choi and Yang [22] established existence and pointwise bound of the fundamental solution for the Stokes system with measurable coefficients in the spac...

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“…The existence of p > 2 was proved in the work of Meyers. 12 See the work of Haller-Dintelmann et al 22 for advanced recent progresses on the topic. Lemma 7.…”

Section: The Hoelder Exponent In Three Dimensionsmentioning

confidence: 99%

Regularity of second derivatives in elliptic transmission problems near an interior regular multiple line of contact

Druet

2018

Math Methods in App Sciences

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We consider second-order elliptic transmission problems in 3D in which several subdomains intersect at a closed line of contact. We prove that weak solutions possess second-order generalized derivatives up to the contact line. Moreover, we quantify the index of integrability and the Hölder exponent by means of explicit formula known in the literature for two-dimensional problems. For instance, the integrability of the gradient to a power larger than the space dimension d = 3 can be expected if the oscillations of the diffusion coefficient are moderate, that is, for far larger a range than what a theory of small perturbations would allow, or if there are at most three involved materials. KEYWORDSbest Hölder exponent, elliptic transmission problem, interior multiple contact line, W 2,p regularity theory Math Meth Appl Sci. 2018;41:6457-6479.wileyonlinelibrary.com/journal/mma

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Elliptic and parabolic regularity for second‐ order divergence operators with mixed boundary conditions

Cited by 34 publications

References 62 publications

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

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

Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier–Stokes systems with L_∞ strongly elliptic coefficient tensor

Regularity of second derivatives in elliptic transmission problems near an interior regular multiple line of contact

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Elliptic and parabolic regularity for second‐ order divergence operators with mixed boundary conditions

Cited by 34 publications

References 62 publications

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

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

Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier–Stokes systems with L∞ strongly elliptic coefficient tensor

Regularity of second derivatives in elliptic transmission problems near an interior regular multiple line of contact

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Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier–Stokes systems with L_∞ strongly elliptic coefficient tensor