Korn and Poincaré-Korn inequalities for functions with a small jump set

In this paper we prove a regularity and rigidity result for displacements in $$GSBD^p$$ G S B D p , for every $$p>1$$ p > 1 and any dimension $$n\ge 2$$ n ≥ 2 . We show that a displacement in $$GSBD^p$$ G S B D p with a small jump set coincides with a $$W^{1,p}$$ W 1 , p function, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincaré-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in $$GSBD^p$$ G S B D p .

Optimal jump set in hyperbolic conservation laws

We investigate qualitative properties of entropy solutions to hyperbolic conservation laws, and construct an entropy solution to a scalar conservation law for which the jump set is not closed, in particular, it is dense in a space-time domain. In a second part, we establish a similar result for hyperbolic systems. We give two different approaches for scalar conservation laws and hyperbolic systems in order to obtain these results. For the scalar case, the solutions are explicitly calculated.

On the blow-up of GSBV functions under suitable geometric properties of the jump set

In this paper, we investigate the fine properties of functions under suitable geometric conditions on the jump set. Precisely, given an open set {\Omega\subset\mathbb{R}^{n}} and given {p>1}, we study the blow-up of functions {u\in\mathrm{GSBV}(\Omega)}, whose jump sets belong to an appropriate class {\mathcal{J}_{p}} and whose approximate gradients are p-th power summable. In analogy with the theory of p-capacity in the context of Sobolev spaces, we prove that the blow-up of u converges up to a set of Hausdorff dimension less than or equal to {n-p}. Moreover, we are able to prove the following result which in the case of {W^{1,p}(\Omega)} functions can be stated as follows: whenever {u_{k}} strongly converges to u, then, up to subsequences, {u_{k}} pointwise converges to u except on a set whose Hausdorff dimension is at most {n-p}.

A perfect reconstruction property for PDE-constrained total-variation minimization with application in Quantitative Susceptibility Mapping

We study the recovery of piecewise constant functions of finite bounded variation (BV) from their image under a linear partial differential operator with unknown boundary conditions. It is shown that minimizing the total variation (TV) semi-norm subject to the associated PDE-constraints yields perfect reconstruction up to a global constant under a mild geometric assumption on the jump set of the function to reconstruct. The proof bases on establishing a structural result about the jump set associated with BV-solutions of the homogeneous PDE. Furthermore, we show that the geometric assumption is satisfied up to a negligible set of orthonormal transformations. The results are then applied to Quantitative Susceptibility Mapping (QSM) which can be formulated as solving a two-dimensional wave equation with unknown boundary conditions. This yields in particular that total variation regularization is able to reconstruct piecewise constant susceptibility distributions, explaining the high-quality results obtained with TV-based techniques for QSM.

Approximation of fracture energies with p-growth via piecewise affine finite elements

The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation GSBDp(Ω), p ∈ (1, ∞), their treatment is however hindered by the very low regularity of those functions and by the lack of appropriate density results. We construct here an approximation of GSBDp functions, for p ∈ (1, ∞), with functions which are Lipschitz continuous away from a jump set which is a finite union of closed subsets of C1 hypersurfaces. The strains of the approximating functions converge strongly in Lp to the strain of the target, and the area of their jump sets converge to the area of the target. The key idea is to use piecewise affine functions on a suitable grid, which is obtained via the Freudenthal partition of a cubic grid.

On the continuity of functionals defined on partitions

We characterize the continuity of prototypical functionals acting on finite Caccioppoli partitions and prove that it is equivalent to convergence of the perimeter of the jump set.

A Korn-type inequality in SBD for functions with small jump sets

We present a Korn-type inequality in a planar setting for special functions of bounded deformation. We prove that for each function in [Formula: see text] with a sufficiently small jump set the distance of the function and its derivative from an infinitesimal rigid motion can be controlled in terms of the linearized elastic strain outside of a small exceptional set of finite perimeter. Particularly, the result shows that each function in [Formula: see text] has bounded variation away from an arbitrarily small part of the domain.

Which special functions of bounded deformation have bounded variation?

Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions that are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that SBDp functions are approximately continuous -almost everywhere away from the jump set. On the negative side, we construct a function that is BD but not in BV and has distributional strain consisting only of a jump part, and one that has a distributional strain consisting of only a Cantor part.