Layer potentials and Hodge decompositions in two dimensional Lipschitz domains

Mitrea, Dorina

doi:10.1007/s002080100266

Cited by 17 publications

(17 citation statements)

References 20 publications

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“…We now turn to establishing uniqueness of solutions to the mixed problem. We rely on the results of the previous section and uniqueness of the regularity problem due to Dahlberg and Kenig [9] (also, see the work of D. Mitrea [23,Corollary 4.2] for the result in two dimensions). More specifically, we prove that if u solves (1.1) with zero data and (∇u) * ∈ L 1 (∂Ω), then u also solves the regularity problem with zero data and hence u = 0.…”

Section: Uniquenessmentioning

confidence: 99%

The mixed problem in Lipschitz domains with general decompositions of the boundary

Taylor¹,

Ott²,

Brown³

2012

Trans. Amer. Math. Soc.

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This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain Ω ⊂ R n , n ≥ 2, with boundary that is decomposed as ∂Ω = D ∪ N , D and N disjoint. We let Λ denote the boundary of D (relative to ∂Ω) and impose conditions on the dimension and shape of Λ and the sets N and D. Under these geometric criteria, we show that there exists p 0 > 1 depending on the domain Ω such that for p in the interval (1, p 0 ), the mixed problem with Neumann data in the space L p (N ) and Dirichlet data in the Sobolev space W 1,p (D) has a unique solution with the non-tangential maximal function of the gradient of the solution in L p (∂Ω). We also obtain results for p = 1 when the Dirichlet and Neumann data comes from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces. 1 2 J.L. TAYLOR, K.A. OTT, AND R.M. BROWN data [9]. The mixed boundary value problem in Lipschitz domains appears as an open problem in Kenig's CBMS lecture notes [16, Problem 3.2.15]. There is a large literature on boundary value problems in polyhedral domains and we do not attempt to summarize this work here. See the work of Bȃcuţȃet. al. [1] for recent results for the mixed problem in polyhedral domains and additional references.Under mild restrictions on the boundary data we can use energy estimates to show that there exists a solution of the mixed problem with ∇u in L 2 of the domain. Our goal in this paper is to obtain more regularity of the solution and, in particular, to show that ∇u lies in L p (∂Ω). Brown [2] showed that the solution satisfies ∇u ∈ L 2 (∂Ω) when the data f N is in L 2 (N ) and f D is in the Sobolev space W 1,2 (D) for a certain class of Lipschitz domains. Roughly speaking, his results hold when the Dirichlet and Neumann portions of the boundary meet at an angle strictly less than π. In this same class of domains, Sykes and Brown [31] obtain L p results for 1 < p < 2 and I. Mitrea and M. Mitrea [24] establish well-posedness in an essentially optimal range of function spaces. Lanzani, Capogna and Brown [20] establish L p results in two dimensional graph domains when the data comes from weighted L 2 -spaces and the Lipschitz constant is less than one. The aforementioned results rely on a variant of the Rellich identity. The Rellich identity cannot be used in the same way in general Lipschitz domains because it produces estimates in L 2 , and even in smooth domains simple examples show that we cannot expect to have solutions with gradient in L 2 (∂Ω).Ott and Brown [26] establish conditions on Ω, N , and D which ensure uniqueness of solutions of the L p -mixed problem and they also establish conditions on Ω, N , D and f N and f D which ensure that solutions to the L p -mixed problem exist. All of this work is done under an additional geometric assumption on the boundary of D. More specifically, the authors address solvability of the mixed problem for the Laplacian in bounded Lipschitz domains under the assumption that the boundary between D and N (relative to ∂Ω) is l...

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Section: Uniquenessmentioning

confidence: 99%

The mixed problem in Lipschitz domains with general decompositions of the boundary

Taylor¹,

Ott²,

Brown³

2012

Trans. Amer. Math. Soc.

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“…is an isomorphism for 2 3 ppp1 (this is implicit in [23]). Going further, given the dilation invariant nature of the problem, there is no loss of generality in assuming that the atom a is supported in a surface ball of radius one.…”

Section: The Two-dimensional Casementioning

confidence: 97%

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

Escauriaza

Mitrea

2004

Journal of Functional Analysis

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We prove the well-posedness of the transmission problem for the Laplacian across a Lipschitz interface, with optimal non-tangential maximal function estimates, for data in Lebesgue and Hardy spaces on the boundary. As a corollary, we show that the spectral radius of the (adjoint) harmonic double layer potential K Ã in L p 0 ð@OÞ is less than 1 2 ; whenever O is a bounded convex domain and 1opp2: r

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“…Then (17) see, e.g., [6], [5]. As for (18), we employ the following twodimensional identity proven in [4]:…”

Section: Theorem 32mentioning

confidence: 99%

Counterexamples to the well-posedness of 𝐿^{𝑝} transmission boundary value problems for the Laplacian

Mitrea

Ott

2007

Proc. Amer. Math. Soc.

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Abstract. In this note we show that the well-posedness range p ∈ (1, 2] for L p transmission boundary value problems for the Laplacian in the class of Lipschitz domains established by Escauriaza and Mitrea (2004) is sharp. Our approach relies on Mellin transform techniques for singular integrals naturally associated with the transmission problems and on a careful analysis of the L p spectra of such singular integrals.

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Layer potentials and Hodge decompositions in two dimensional Lipschitz domains

Cited by 17 publications

References 20 publications

The mixed problem in Lipschitz domains with general decompositions of the boundary

The mixed problem in Lipschitz domains with general decompositions of the boundary

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

Counterexamples to the well-posedness of 𝐿^{𝑝} transmission boundary value problems for the Laplacian

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