Space Mappings with Bounded
                    Distortion

Reshetnyak, Yu. G.

doi:10.1090/mmono/073

Cited by 384 publications

(218 citation statements)

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“…By a wellknown result of Iwaniec and Martin [14],ũ must belong to W 1,n loc (R n ; R n ). Then a classical result of Liouville's theorem (see Reshetnyak [23]…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

A linear boundary value problem for weakly quasiregular mappings in space

Yan¹

2001

Calculus of Variations and Partial Differential Equations

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Abstract. Given a number L ≥ 1, a weakly L-quasiregular map on a domain Ω in space R n is a map u in a Sobolev spaceIn this paper, we study the problem concerning linear boundary values of weakly L-quasiregular mappings in space R n with dimension n ≥ 3. It turns out this problem depends on the power p of the underlying Sobolev space. For p not too far below the dimension n we show that a weakly quasiregular map in W 1,p (Ω; R n ) can only assume a quasiregular linear boundary value; however, for all L ≥ 1 and 1 ≤ p < nL L+1 , we prove a rather surprising existence result that every linear map can be the boundary value of a weakly L-quasiregular map in W 1,p (Ω; R n ). Classification (1991):30C65, 30C70, 35F30, 49J30 Mathematics Subject

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“…By a wellknown result of Iwaniec and Martin [14],ũ must belong to W 1,n loc (R n ; R n ). Then a classical result of Liouville's theorem (see Reshetnyak [23]…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

A linear boundary value problem for weakly quasiregular mappings in space

Yan¹

2001

Calculus of Variations and Partial Differential Equations

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“…We finally mention that the Jacobian determinant was extensively studied in the literature; see, e.g., [4,5,28,38,39,44,51,52,64,65,66,67,80,81,85,86] and the references therein.…”

Section: This Impliesmentioning

confidence: 99%

Estimates for the topological degree and related topics

Nguyên

2014

J. Fixed Point Theory Appl.

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Abstract. This is a survey paper on estimates for the topological degree and related topics which range from the characterizations of Sobolev spaces and BV functions to the Jacobian determinant and nonlocal filter problems in Image Processing. These results are obtained jointly with Bourgain and Brezis. Several open questions are mentioned.Mathematics Subject Classification. 55M25, 49J99, 46E35, 46B50.

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“…In this case, f is called a mapping with bounded distortion. In more detail, the definitions and properties of quasiregular mappings (or mappings with bounded distortion) can be found, e.g, in [8,9]. Note that the main difficulty encountered in proving a property of the mapping f satisfying a relation of the form (3) or (4) for an unbounded function Q.x/ is connected with the necessity of tracing the behavior of the right-hand side of the corresponding inequality depending on the behavior of the function Q: Clearly, these difficulties do not appear for mappings with bounded distortion.…”

Section: Preliminariesmentioning

confidence: 99%