Differential inclusions and Young measures involving prescribed Jacobians

Abstract

In elasticity theory, one naturally requires that the Jacobian determinant of the deformation is positive or even a-priori prescribed (e.g. for incompressibility). However, such strongly non-linear and non-convex constraints are difficult to deal with in mathematical models. This short note, which is based on joint work with K. Koumatos and E. Wiedemann, presents various recent results on how this constraint can be manipulated in subcritical Sobolev spaces, where the integrability exponent is less than the dimension. In particular, we give a characterization theorem for Young measures under this side constraint. This is in the spirit of the celebrated Kinderlehrer-Pedregal Theorem and based on convex integration and "geometry" in matrix space. Finally, applications to approximation in Sobolev spaces and to the failure of lower semicontinuity for certain integral functionals with "realistic" growth conditions are given. phone +44 (0) 2476 5 -28329 det ∇u = r > 0 a.e.orwhere by J + 1 and J − 2 we denote the positive part of J 1 and the negative part of J 2 , respectively. These assumptions are rather natural (the integrability condition is only relevant in very special cases). Notice that the lower and upper bound can also be non-active for certain x, for example if J 1 (x) = −∞, there is no lower bound at x.A question that is related to the well-known Dacorogna-Moser Theory [8] then is as follows:Question Q3. Given g ∈ W 1−1/p,p (∂ Ω; R d ) (the trace space to W 1,p (Ω; R d ), that is the space of all boundary values of W 1,p (Ω; R d )-functions), does there exist u ∈ W 1,p (Ω; R d ) with u| ∂ Ω = g and det ∇u > 0 ?The same question can also be asked with the even stricter requirement det ∇u = 1 instead of mere positivity of the Jacobian. This expresses incompressibility and is relevant in the theory of fluids. Notice that here we have no "compatibility" assumption on g. Again, one can show with a similar argument to before that this question is unsolvable if p ≥ d. However, for p < d we get a positive answer, this is proved in [9].

Approximation by mappings with singular Hessian minors

“…in the space W 1,p . Rindler [29], Koumatos, Rindler, and Wiedemann [19], [20] proved the weak dense-…”

“…in the space W 1,p . Rindler [29], Koumatos, Rindler, and Wiedemann [19], [20] proved the weak denseness of {f ∈ W 1,p : 1 < p < n, J 1 ≤ det Df ≤ J 2 a.e.} in the space W 1,p for any prescribed measurable function…”