Archive for Rational Mechanics and Analysis

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

On planar mappings with prescribed principal strains

  • Julian Gevirtz


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.


Neural Network Complex System Nonlinear Dynamics Diagonal Matrix Jacobian Matrix 
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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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