Area Integral Estimates for the Biharmonic Operator in Lipschitz Domains

Pipher, Jill; Verchota, Gregory C.

doi:10.2307/2001830

Cited by 8 publications

(11 citation statements)

References 9 publications

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“…Such results, in special cases, have been proven in earlier works [10], [25], [5]. Here we make no restriction on the order of differentiation or the size of the (determined) systems.…”

Section: Introductionsupporting

confidence: 63%

See 1 more Smart Citation

Area integral estimates for higher order elliptic equations and systems

Dahlberg¹,

Kenig²,

Pipher³

et al. 1997

Annales De L’institut Fourier

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“…Such results, in special cases, have been proven in earlier works [10], [25], [5]. Here we make no restriction on the order of differentiation or the size of the (determined) systems.…”

Section: Introductionsupporting

confidence: 63%

“…Then we have forward using interior estimates and assuming that the square function is defined with respect to cones with a larger aperture than those used to define ^(V 771 " 1^) . (See for example [10] and then [25]. )…”

mentioning

confidence: 99%

Area integral estimates for higher order elliptic equations and systems

Dahlberg¹,

Kenig²,

Pipher³

et al. 1997

Annales De L’institut Fourier

Self Cite

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“…A duality argument as in [41] is also possible. This would use solvability of the biharmonic Dirichlet problem and the area integral estimates of [30] or [8]. As a consequence the single layer can be shown to be invertible for 1 <p< 2+~ when n--3 because of [31], and for 2(n-1)/(n+l)-c<p<2+~ when n~>4 by the recent result of Z. Shen [37].…”

Section: ( O)( (T2) ' S-~s = St2--~s and (T+)': Wa~ ---+ Wa~ Is Amentioning

confidence: 99%

The biharmonic Neumann problem in Lipschitz domains

Verchota

2005

Acta Math.

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“…(See Dahlberg, Kenig, and Verchota [4, pp. 130-133] for details in the range 1 < p < 2 in R and Pipher and Verchota [11,12] for proof of existence in the range 2 -e < p < oo in R.) This suggests possible problems on domains with sharp intruding corners. Some positive results are known for the duality property.…”

Section: Motivation and Backgroundmentioning

confidence: 99%

The Duals of Harmonic Bergman Spaces

Coffman¹,

Cohen²

1990

Proceedings of the American Mathematical Society

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Abstract.In this paper we show that for SI , a starlike Lipschitz domain, the dual of the space of harmonic functions in LP{S1) need not be the harmonic functions in Lq(Sl), where \/p + \/q = 1 . We show that, as a consequence, the harmonic Bergman projection for Í2 need not extend to a bounded operator on Lp(Sl) for all 1 < p < oo . The duality result is a partial answer to a question of Nakai and Sario [9] posed initially in the Proceedings of the London Mathematical Society in 1978. We treat the duality question as a biharmonic problem, and our result follows from the failure of uniqueness for the biharmonic Dirichlet problem in domains with sharp intruding corners. IntroductionIn this paper we give a partial answer to a question posed by Nakai and Sario concerning the duals of harmonic Bergman spaces. Their question, initially posed in the Proceedings of the London Mathematical Society [9, p. 345] was whether, for 1 < p < oo, D. c R" , the dual of the space of harmonic functions in LP(Q.) is the space of harmonic functions in Lq(Q), where l/p + l/q = 1, 1 < p, q < oo. When the answer is affirmative, we will say that the duality property holds (for the given value of p). We note that every domain has the property for p = 2, and domains which are bounded and have a sufficiently smooth boundary have the property for every p .In 1981 Nakai and Sario [10] showed that the duality property can fail for Í2 = R \{z,, z2, ... , zn}, and one of the authors subsequently discovered that duality also fails for the punctured disk.In this paper we show that there is a p between 1 and 2 and a Lipschitz domain with an intruding corner for which the duality property fails. We show that, as a consequence, there are Lipschitz domains on which the harmonic Bergman projection does not extend to a bounded operator on a full range of p's between 1 and oo .Our approach is to reduce the duality question to a biharmonic problem. We first show that the duality property is equivalent to the direct sum decomposition

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Area Integral Estimates for the Biharmonic Operator in Lipschitz Domains

Abstract: Abstract. Let DCR" be a Lipschitz domain and let « be a function biharmonic in D , i.e., AAu = 0 in D . We prove that the nontangential maximal function and the square function of the gradient of u have equivalent Lp(dp) norms, where dp e A°°(do) and da is surface measure on 3D.

Cited by 8 publications

References 9 publications

Area integral estimates for higher order elliptic equations and systems

Area integral estimates for higher order elliptic equations and systems

The biharmonic Neumann problem in Lipschitz domains

The Duals of Harmonic Bergman Spaces

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