Differential inequalities and flow-invariance via linear functionals

AbstractWe consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type $W(F)=\frac{\mu }{2} \hspace{0.07em} \frac{\lVert F \rVert ^{2}}{\det F}+f(\det F)$ W ( F ) = μ 2 ∥ F ∥ 2 det F + f ( det F ) ; such an energy is rank-one convex if and only if the function $f$ f is convex.

Download Full-text

On the critical behavior for inhomogeneous parabolic differential inequalities in the half-space

Applied Mathematics Letters ◽

10.1016/j.aml.2021.107099 ◽

2021 ◽

Vol 117 ◽

pp. 107099

Author(s):

Ravi P. Agarwal ◽

Mohamed Jleli ◽

Bessem Samet

Keyword(s):

Half Space ◽

Critical Behavior ◽

Differential Inequalities

Download Full-text

Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Mathematics ◽

10.3390/math9161866 ◽

2021 ◽

Vol 9 (16) ◽

pp. 1866

Author(s):

Mohamed Jleli ◽

Bessem Samet ◽

Calogero Vetro

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

A Priori Estimates ◽

Sufficient Conditions ◽

Global Solutions ◽

Higher Order ◽

Differential Inequalities ◽

Structure Of Solutions ◽

Nonlinear Capacity ◽

Fractional Differential

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

Download Full-text