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arXiv · 2212.06452

Weak limit of homeomorphisms in $W^{1,n-1}$: invertibility and lower semicontinuity of energy

Anna Doležalová·Stanislav Hencl·Anastasia Molchanova
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Abstract

Let $Ω$, $Ω'\subset\mathbb{R}^n$ be bounded domains and let $f_m\colonΩ\toΩ'$ be a sequence of homeomorphisms with positive Jacobians $J_{f_m} >0$ a.e. and prescribed Dirichlet boundary data. Let all $f_m$ satisfy the Lusin (N) condition and $\sup_m \int_Ω(|Df_m|^{n-1}+A(|\text{cof} Df_m|)+ϕ(J_f))<\infty$, where $A$ and $φ$ are positive convex functions. Let $f$ be a weak limit of $f_m$ in $W^{1,n-1}$. Provided certain growth behaviour of $A$ and $φ$, we show that $f$ satisfies the (INV) condition of Conti and De Lellis, the Lusin (N) condition, and polyconvex energies are lower semicontinuous.

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Anna Doležalová, Stanislav Hencl, Anastasia Molchanova. 2023-09-11. Weak limit of homeomorphisms in $W^{1,n-1}$: invertibility and lower semicontinuity of energy. https://doi.org/10.1051/cocv%2F2024006

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