TY - JOUR

T1 - Neohookean deformations of annuli, existence, uniqueness and radial symmetry

AU - Iwaniec, Tadeusz

AU - Onninen, Jani

N1 - Funding Information:
Iwaniec was supported by the NSF grants DMS-0301582 and DMS-0800416 and by the Academy of Finland. Onninen was supported by the NSF grant DMS-0701059 and by the Academy of Finland.

PY - 2010/9

Y1 - 2010/9

N2 - Let X = x ∈ ℝ2; r < {pipe}x{pipe} < R} and Y = {y ∈ ℝ2; r* < {pipe}y{pipe} < R* be annuli in the plane. We consider the class F(X, Y) of all orientation preserving homeomorphisms, in the Sobolev space W1,2(X, Y) which keep the boundary circles in the same order. This means that, and,. We study the Neohookean energy integral, where Φ ∈ C∞(0, ∞) is positive and strictly convex. We assume in addition that the function, and its derivative extend continuously to [0, ∞), with Ψ(0) = 0. Then we prove: Theorem 1 The minimum of energy within the class F (X, Y) is attained for a radial map h(x) = H({pipe}x{pipe}) x/{pipe}x{pipe}. The minimizer is C∞-smooth and is unique up to a rotation of the annuli. We believe that not only the result but also some novelties in the computation might gain a particular interest.

AB - Let X = x ∈ ℝ2; r < {pipe}x{pipe} < R} and Y = {y ∈ ℝ2; r* < {pipe}y{pipe} < R* be annuli in the plane. We consider the class F(X, Y) of all orientation preserving homeomorphisms, in the Sobolev space W1,2(X, Y) which keep the boundary circles in the same order. This means that, and,. We study the Neohookean energy integral, where Φ ∈ C∞(0, ∞) is positive and strictly convex. We assume in addition that the function, and its derivative extend continuously to [0, ∞), with Ψ(0) = 0. Then we prove: Theorem 1 The minimum of energy within the class F (X, Y) is attained for a radial map h(x) = H({pipe}x{pipe}) x/{pipe}x{pipe}. The minimizer is C∞-smooth and is unique up to a rotation of the annuli. We believe that not only the result but also some novelties in the computation might gain a particular interest.

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U2 - 10.1007/s00208-009-0469-7

DO - 10.1007/s00208-009-0469-7

M3 - Article

AN - SCOPUS:77953913930

SN - 0025-5831

VL - 348

SP - 35

EP - 55

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 1

ER -