A Gårding-type inequality is proved for a quadratic form associated to A-quasiconvex functions. This quadratic form appears as the relative entropy in the theory of conservation laws and it is related to the Weierstrass excess function in the calculus of variations. The former provides weak-strong uniqueness results, whereas the latter has been used to provide sufficiency theorems for local minimisers. Using this new Gårding inequality we provide an extension of these results to PDE constrained problems in dynamics and statics under A-quasiconvexity assumptions. The application in statics improves existing results by proving uniqueness of L p local minimisers in the classical A = curl case.
In the context of image processing, we study a class of integral regularizers defined in terms of spatially inhomogeneous integrands that depend on general linear differential operators. Particularly, the spatial dependence is assumed to be only measurable. The setting is made rigorous by means of the theory of Radon measures and of suitable function spaces modeled on BV. We prove the lower semicontinuity of the functionals at stake and existence of minimizers for the corresponding variational problems. Then, we embed the latter into a bilevel scheme in order to automatically compute the regularization parameters. These parameters are considered to be spatially varying, thus allowing for good flexibility and preservation of details in the reconstructed image. After identifying a series of spatially inhomogeneous regularization functionals commonly used in image processing that are included in our framework, we substantiate its feasibility by performing numerical denoising examples in which the spatial dependence of the integrand is measurable. Specifically, we use Huber versions of the first and second order total variation (and their sum) with both the Huber and the regularization parameter being spatially varying. Notably, the spatially varying version of second order total variation produces high quality reconstructions when compared to regularizations of similar type, and the introduction of the low regularity spatially dependent Huber parameter leads to a further enhancement of the image details. We expect that our theoretical investigations and our numerical feasibility study will support future work on setting up schemes where general differential operators with spatially dependent coefficients will also be part of the optimization scheme.
This article studies the equations of adiabatic thermoelasticity endowed with an internal energy satisfying an appropriate quasiconvexity assumption which is associated to the symmetrisability condition for the system. A Gårding-type inequality for these quasiconvex functions is proved and used to establish a weak-strong uniqueness result for a class of dissipative measurevalued solutions.
A Gårding-type inequality is proved for a quadratic form associated to A-quasiconvex functions. This quadratic form appears as the relative entropy in the theory of conservation laws and it is related to the Weierstrass excess function in the calculus of variations. The former provides weak-strong uniqueness results, whereas the latter has been used to provide sufficiency theorems for local minimisers. Using this new Gårding inequality we provide an extension of these results to PDE constrained problems in dynamics and statics under A-quasiconvexity assumptions. The application in statics improves existing results by proving uniqueness of L p local minimisers in the classical A = curl case.
In the context of image processing, given a k-th order, homogeneous and linear differential operator with constant coefficients, we study a class of variational problems whose regularizing terms depend on the operator. Precisely, the regularizers are integrals of spatially inhomogeneous integrands with convex dependence on the differential operator applied to the image function. The setting is made rigorous by means of the theory of Radon measures and of suitable function spaces modeled on BV. We prove the lower semicontinuity of the functionals at stake and existence of minimizers for the corresponding variational problems. Then, we embed the latter into a bilevel scheme in order to automatically compute the space-dependent regularization parameters, thus allowing for good flexibility and preservation of details in the reconstructed image. We establish existence of optima for the scheme and we finally substantiate its feasibility by numerical examples in image denoising. The cases that we treat are Huber versions of the first and second order total variation with both the Huber and the regularization parameter being spatially dependent. Notably the spatially dependent version of second order total variation produces high quality reconstructions when compared to regularizations of similar type, and the introduction of the spatially dependent Huber parameter leads to a further enhancement of the image details.
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