Year: 2025
Type: article
Abstract: We study injectivity for models of Nonlinear Elasticity that involve the second gradient. We assume that $$\Omega \subset {\mathbb{R}}^n$$ is a domain, $$f\in W^{2,q}(\Omega ,{\mathbb{R}}^n)$$ satisfies $$|J_f|^{-a}\in L^1$$ and that f equals a given homeomorphism on $$\partial \Omega$$ . Under suitable conditions on q and a we show that f must be a homeomorphism. As a main new tool we find an optimal condition for a and q that imply that $$\mathcal {H}^{n-1}(\{J_f=0\})=0$$ and hence $$J_f$$ cannot change sign. We further specify in dependence of q and a the maximal Hausdorff dimension d of the critical set $$\{J_f=0\}$$ . The sharpness of our conditions for d is demonstrated by constructing respective counterexamples.
Authors: D. Campbell, S. Hencl, A. Menovschikov, S. Schwarzacher
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Awards: Delicate analytical and topological tools for variational problems and modelling, G4365426473, Properties of Sobolev homeomorphisms and their limits, Geometric and Harmonic Analysis 2, Interactions of fluids and solids - On a systematic analysis theory for partial differential equations in the context of fluid-structure interactions
OpenAlex ID: W4414210571