L 3/2-estimates of vector fields with L 1 curl in a bounded domain

Xiang, Xingfei

doi:10.1007/s00526-011-0473-0

Cited by 4 publications

(2 citation statements)

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“…Theorem 10 can also be deduced from theorem 3: first theorem 10 is proved on a half-space by a reflection argument (see also [7][8][9]), then it is extended to a general domain by local charts and partition of the unity [25, remark 3.3] (see also [112]).…”

Section: Variationsmentioning

confidence: 99%

Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations

Schaftingen

2014

J. Fixed Point Theory Appl.

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To Haïm Brezis who has taught me so much and has asked me many inspiring questions; on the occasion of his 70th birthday, with admiration and gratitude Abstract. J. Bourgain and H. Brezis have obtained in 2002 some new and surprising estimates for systems of linear differential equations, dealing with the endpoint case L 1 of singular integral estimates and the critical Sobolev space W 1,n (R n ). This paper presents an overview of the results, further developments over the last ten years and challenging open problems. Mathematics Subject Classification. 35A23 26D15 46E35.Keywords. Critical Sobolev spaces, Gagliardo-Nirenberg-Sobolev inequality, canceling differential operator, overdetermined elliptic systems, underdetermined systems. Theme Limiting Hodge theory for Sobolev formsThe study of limiting estimates for systems of linear differential equations starts from the following problem: given a function g ∈ L n (R n ; R n ), find the best regularity that a vector field u : R n → R n can have such that div u = g in R n .(1.1)If n ≥ 2, equation (1.1) is strongly underdetermined. The standard way of finding a solution u consists in lifting the underdeterminacy by solving the system div u = g in R n , curl u = 0 in R n , (1.2)

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