Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L data and applications to nonlinear elasticity

The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces

“…Firstly, the Jacobian plays an important role in Continuum Mechanics, specially in nonlinear elasticity, see e.g. [5,16,20,27]. This is not surprising since, for a Sobolev map, positivity of the Jacobian implies a weak form of local invertibility [21].…”

Energy minimisers with prescribed Jacobian

“…where f is such that W (x, f (x)) ≡ min Although there is a good existence theory for (1.2) whenever there is some α > 0 such that u 0 is a C 1,α -diffeomorphism and f ∈ C 0,α [19,21,52,58], little is known when f is just an L p function. We refer the reader to [29,38,47,50,55], as well as our our recent works [34,35] for results in this direction; counter-examples in the p = ∞ case were obtained in [13,51].…”

A Least-Squares Method for the Solution of the Non-smooth Prescribed Jacobian Equation

“…The, strongly underdertemined, problem consists in finding a vector field satisfying instead of (together with appropriate Dirichlet boundary conditions). Theoretical results of the prescribed Jacobian inequality are given, e.g., in [ 34 ], and to the best of our knowledge there are no proposed numerical methods to solve this type of inequality. It will be addressed by, e.g., including interior-point methods [ 33 ].…”