Quasiconvexity and PDE constraints: analysis and applications

In this thesis we focus on questions of stability, existence and uniqueness for PDE con- strained problems in both dynamics and statics under appropriate convexity conditions. In the first part, we establish a Gårding-type inequality for quantities associated to (A, 0)- quasiconvex functions, where A is a constant-coefficient, linear differential operator with constant rank. In dynamics, we initially apply a simplified version of our derived inequality to prove weak-strong uniqueness results for conservation laws possessing involutions i.e. a differential constraint A propagated by the initial data, provided that the system is endowed with an A-quasiconvex entropy. In addition to this, combining our Gårding-type inequality with the relative entropy method, we establish a weak-strong uniqueness result for the hyperbolic system of adiabatic thermoelasticity under quasiconvexity conditions. In particular, we show that classical solutions of that system are unique within a suitable class of dissipative measure-valued solutions, provided that the internal energy is stronly (curl, 0)-quasiconvex. In statics, we investigate the so-called Weierstrass problem of finding necessary and sufficient conditions for local minimisers. More precisely, we prove an A-quasiconvexity based sufficiency theorem for local minimisers for general problems constrained by an operator A. An additional contribution of our result is that we infer uniqueness of these local minimisers and quantify the difference in energy between them and arbitrary comperitors. In the second part, in the context of image processing, we study a class of PDE constrained variational problems whose regularising terms depend on the differential operator. We prove the lower semicontinuity of the functionals in question and existence of minimisers for the corresponding variational problems. Then, we embed the latter into a bilevel scheme in order to automatically compute the space-dependent regularisation parameters, and we establish existence of optima for the scheme. We finally substantiate its feasibility by numerical examples in image denoising.