Optimal Existence Theorems for Nonhomogeneous Differential Inclusions

By Stefan Müller, M. Sychev. 2001

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Some Open Problems in Elasticity (2002, methodology): “For a way of verifying that a strict local minimizer in a space based on W 1,p for 1 p ∞ satisfies this requirement when W is strictly polyconvex see Ball & Marsden [42], and for the case when W is strictly quasiconvex see Evans & Gariepy [117] and Sychev [242].”
Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation (2009, methodology): “This is an obstruction for the available versions of convex integration, the ones based on Baire category [5, 10, 15] and direct constructions [9, 11, 14].”
The Euler equations as a differential inclusion (2007, methodology): “We emphasize that the concluding argument in Section 4 appeals to the – by now standard – methods for solving differential inclusions, either by appealing to the Baire category theorem [1, 2, 5, 13], or by the more explicit convex integration method [12, 17, 18 ].”