Gamma-convergence of Gaussian fractional perimeter

We prove the Γ-convergence of the renormalised Gaussian fractional s-perimeter to the Gaussian perimeter as s → 1 - {s\to 1^{-}} . Our definition of fractional perimeter comes from that of the fractional powers of Ornstein–Uhlenbeck operator given via Bochner subordination formula. As a typical feature of the Gaussian setting, the constant appearing in front of the Γ-limit does not depend on the dimension.

Symmetry and scaling limits for matching of implicit surfaces based on thin shell energies

In a recent paper by Iglesias, Rumpf and Scherzer (Found. Comput. Math. 18(4), 2018) a variational model for deformations matching a pair of shapes given as level set functions was proposed. Its main feature is the presence of anisotropic energies active only in a narrow band around the hypersurfaces that resemble the behavior of elastic shells. In this work we consider some extensions and further analysis of that model. First, we present a symmetric energy functional such that given two particular shapes, it assigns the same energy to any given deformation as to its inverse when the roles of the shapes are interchanged, and introduce the adequate parameter scaling to recover a surface problem when the width of the narrow band vanishes. Then, we obtain existence of minimizing deformations for the symmetric energy in classes of bi-Sobolev homeomorphisms for small enough widths, and prove a $\Gamma$-convergence result for the corresponding non-symmetric energies as the width tends to zero. Finally, numerical results on realistic shape matching applications demonstrating the effect of the symmetric energy are presented.

Geometric linearization of theories for incompressible elastic materials and applications

We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g. encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers.

Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence

We introduce two new concepts of convergence of gradient systems $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) to a limiting gradient system $$({\mathbf{Q}},{{\mathcal {E}}}_0,{{\mathcal {R}}}_0)$$ ( Q , E 0 , R 0 ) . These new concepts are called ‘EDP convergence with tilting’ and ‘contact–EDP convergence with tilting.’ Both are based on the energy-dissipation-principle (EDP) formulation of solutions of gradient systems and can be seen as refinements of the Gamma-convergence for gradient flows first introduced by Sandier and Serfaty. The two new concepts are constructed in order to avoid the ‘unnatural’ limiting gradient structures that sometimes arise as limits in EDP convergence. EDP convergence with tilting is a strengthening of EDP convergence by requiring EDP convergence for a full family of ‘tilted’ copies of $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) . It avoids unnatural limiting gradient structures, but many interesting systems are non-convergent according to this concept. Contact–EDP convergence with tilting is a relaxation of EDP convergence with tilting and still avoids unnatural limits but applies to a broader class of sequences $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) . In this paper, we define these concepts, study their properties, and connect them with classical EDP convergence. We illustrate the different concepts on a number of test problems.

On some non-local approximation of nonisotropic Griffith-type functionals$ ^\dagger $

<abstract><p>The approximation in the sense of $ \Gamma $-convergence of nonisotropic Griffith-type functionals, with $ p- $growth ($ p &gt; 1 $) in the symmetrized gradient, by means of a suitable sequence of non-local convolution type functionals defined on Sobolev spaces, is analysed.</p></abstract>

$ \Gamma $-convergence of quadratic functionals with non uniformly elliptic conductivity matrices

<p style='text-indent:20px;'>We investigate the homogenization through <inline-formula><tex-math id="M2">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence for the <inline-formula><tex-math id="M3">\begin{document}$ L^2({\Omega}) $\end{document}</tex-math></inline-formula>-weak topology of the conductivity functional with a zero-order term where the matrix-valued conductivity is assumed to be non strongly elliptic. Under proper assumptions, we show that the homogenized matrix <inline-formula><tex-math id="M4">\begin{document}$ A^\ast $\end{document}</tex-math></inline-formula> is provided by the classical homogenization formula. We also give algebraic conditions for two and three dimensional <inline-formula><tex-math id="M5">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-periodic rank-one laminates such that the homogenization result holds. For this class of laminates, an explicit expression of <inline-formula><tex-math id="M6">\begin{document}$ A^\ast $\end{document}</tex-math></inline-formula> is provided which is a generalization of the classical laminate formula. We construct a two-dimensional counter-example which shows an anomalous asymptotic behaviour of the conductivity functional.</p>