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Archive for Rational Mechanics and Analysis

, Volume 117, Issue 4, pp 295–320 | Cite as

On planar mappings with prescribed principal strains

  • Julian Gevirtz
Article

Abstract

Let D ⊂ℝ2 be a domain and let f be a mapping of D into ℝ2 whose Jacobian matrix J f is continuous and has positive determinant. The principal strain functions M and m of f are the positive roots square of the eigenvalues of J f T J f and are assumed to be distinct at all points of D. In this case J f = T1ST2, where T1, T2 are rotations through angles φ and −θ, and S is the diagonal matrix with entries M, m. For given M, m the functions θ, φ satisfy a nonlinear hyperbolic system. By means of an argument motivated by the blow-up phenomenon for solutions of such systems we show that if ▽M, ▽m are locally Lipschitz continuous in D, and in each compact subset K of D, J f satisfies a Hölder condition with exponent α k >(√5−1)/2, then J f is, in fact, locally Lipschitz continuous in D, and, moreover, satisfies a bound of the form
$$\mathop {\lim }\limits_{{\text{ }}z \to z_0 } \sup \parallel J_f (z) - J_f (z_0 )\parallel /\left| {z - z_0 } \right| \leqq C(M, m, {\text{dist}} {\text{(}}z_0 , \partial D)).$$
We use the fact that this bound is actually sharp when M and m are constant to show that the radius of convexity of the family of mappings of the unit disk with locally Lipschitz continuous first-order partial derivatives and constant principal strains M>m is m2/M2.

Keywords

Neural Network Complex System Nonlinear Dynamics Diagonal Matrix Jacobian Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Julian Gevirtz
    • 1
  1. 1.Facultad de MatemáticasUniversidad Católica de ChileCorreo 22 SantiagoChile

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