The mixed problem for harmonic functions in polyhedra of ℝ³

Venouziou, Moises; Verchota, Gregory C.

doi:10.1090/pspum/079/2500502

Cited by 5 publications

(2 citation statements)

References 20 publications

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“…Several previously studied cases of the mixed problem fall under the conditions of assumptions (1.2), (1.3), and (1.4). Venouziou and Verchota [32] establish a solution to the L p -mixed problem (1.1) in polyhedral domains in R 3 . In one particular case, they are able to solve the mixed boundary value problem in the pyramid in R 3 , when Dirichlet and Neumann data are assigned to alternating faces.…”

Section: Introductionmentioning

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: Introductionmentioning

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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“…Using Harnack's inequality for positive solutions to this PDE we can connect a point in K(α) ∩ B(0, ρ) to e 1 by a chain of balls with radii ≥ c −1 = c(p, n) −1 , and then apply Harnack's inequality in successive balls to finally get the right-hand side of (2.12). The idea to use a Rellich type inequality to make estimates as above we garnered from a paper of Venouziou and Verchota in [28].…”

Section: Definition 23mentioning

confidence: 99%

Note on an eigenvalue problem for an ODE originating from a homogeneous $p$-harmonic function

Akman¹,

Lewis²,

Vogel³

2020

St. Petersburg Math. J.

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We discuss what is known about homogeneous solutions u to the p− Laplace equation, p fixed, 1 < p < ∞, when (A) u is an entire p− harmonic function on Euclidean n space, R n , or (B) u > 0 is p− harmonic in the cone, K(α) = {x = (x 1 ,. .. , x n) : x 1 > cos α |x|} ⊂ R n , n ≥ 2, with continuous boundary value zero on ∂K(α) \ {0} when α ∈ (0, π]. We also outline a proof of our new result concerning the exact value, λ = 1 − (n − 1)/p, for an eigenvalue problem in an ODE associated with u when u is pharmonic in K(π) and p > n − 1. Generalizations of this result are stated. Our result complements work of Krol'-Maz'ya for 1 < p ≤ n − 1.

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Layer Potentials for the Harmonic Mixed Problem in the Plane

Venouziou

2012

Potential Anal

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The mixed problem for harmonic functions in polyhedra of ℝ³

Cited by 5 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

Note on an eigenvalue problem for an ODE originating from a homogeneous $p$-harmonic function

Layer Potentials for the Harmonic Mixed Problem in the Plane

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