# On planar mappings with prescribed principal strains

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## Abstract

Let We use the fact that this bound is actually sharp when

*D*⊂ℝ^{2}be a domain and let*f*be a mapping of*D*into ℝ^{2}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 }=*T*_{1}*ST*_{2}, where*T*_{1},*T*_{2}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$$\mathop {\lim }\limits_{{\text{ }}z \to z_0 } \sup \parallel J_f (z) - J_f (z_0 )\parallel /\left| {z - z_0 } \right| \leqq C(M, m, {\text{dist}} {\text{(}}z_0 , \partial D)).$$

*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*m*^{2}/*M*^{2}.## Keywords

Neural Network Complex System Nonlinear Dynamics Diagonal Matrix Jacobian Matrix
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© Springer-Verlag 1992