Minimizers of nonlocal polyconvex energies in nonlocal hyperelasticity (Q6566022)

The authors present a theory for the existence of minimizers of energy functionals in hyperelasticity problems based on nonlocal gradients. They start recalling from their paper [the first author et al., Adv. Nonlinear Anal. 12, Article ID 20220316, 48 p. (2023; Zbl 1541.26017)] the definition of the nonlocal gradient \(D_{\delta }^{s}u\) of a function \(u\in C_{c}^{\infty }( \mathbb{R}^{n})\), as \(D_{\delta }^{s}u(x)=c_{n,s}\int_{B(x,\delta )}\frac{ u(x)-u(y)}{\left\vert x-y\right\vert }\frac{x-y}{\left\vert x-y\right\vert } \frac{w_{\delta }(x-y)}{\left\vert x-y\right\vert ^{n-1+s}}dy\), \(x\in \mathbb{R}^{n}\), where \(c_{n,s}=\frac{n-1+s}{\gamma (1-s)}\), and \(w_{\delta }:\) \(\mathbb{R}^{n}\rightarrow \lbrack 0,+\infty )\) is a radial and nonnegative cut-off function, such that \(w_{\delta }\in C_{c}^{\infty }(B(0,\delta ))\) and its radial representative is monotonically decreasing. They also recall the definition of the space \(H^{s,p,\delta }(\Omega )\), \( 1\leq p<\infty \), as the closure of \(C_{c}^{\infty }(\mathbb{R}^{n})\) under a norm which involves that of \(u\) in \(L^{p}(\Omega _{\delta })\) and that of \( D_{\delta }^{s}u\) in \(L^{p}(\Omega )\), with \(\Omega _{\delta }=\Omega +B(0,\delta )\). They recall properties of these spaces, embeddings and compact embeddings between them. They prove that a nonlocal gradient is the gradient of another function. They prove a nonlocal Piola identity, an integration by parts result for the determinant and weak continuity properties of the minors. The authors consider functionals of the form \( I(u)=\int_{\Omega }W(x,u(x),D_{\delta }^{s}u(x))dx\), where \(W:\mathbb{R} ^{n}\times \mathbb{R}^{n}\times \mathbb{R}^{n\times n}\rightarrow \mathbb{R} \cup \{\infty \}\) satisfies: \(W\) is \(\mathcal{L}^{n}\times \mathcal{B} ^{n}\times \mathcal{B}^{n\times n}\)-measurable, where \(\mathcal{L}^{n}\) denotes the Lebesgue sigma-algebra in \(\mathbb{R}^{n}\), and \(\mathcal{B}^{n}\) and \(\mathcal{B}^{n\times n}\) the Borel sigma-algebras in \(\mathbb{R}^{n}\) and in \(\mathbb{R}^{n\times n}\), respectively; \(W(x,\cdot ,\cdot )\) is lower semicontinuous for a.e. \(x\in \mathbb{R}^{n}\); For a.e. \(x,y\in \mathbb{R} ^{n}\), the function \(W(x,y,\cdot )\) is polyconvex; There exist a constant \( c>0\), an \(a\in L^{1}(\Omega )\) and a Borel function \(h:[0,\infty )\rightarrow \lbrack 0,\infty )\) such that \(\lim_{t\rightarrow \infty }h(t)/t=\infty \) and \(W(x,y,F)\geq a(x)+c\left\vert F\right\vert ^{p}+c\left\vert cofF\right\vert ^{q}+h(\left\vert \det F\right\vert )\) for a.e. \(x\in \Omega \), all \(y\in \mathbb{R}^{n}\) and all \(F\in \mathbb{R} ^{n\times n}\), where \(p\geq n-1\) \(p>1\), \(q\geq \frac{n}{n-1}\). The main result of the paper proves that if \(u_{0}\in H^{s,p,\delta }(\Omega ,\mathbb{ R}^{n})\) and \(I\) is not identically infinity in \(H_{u_{0}}^{s,p,\delta }(\Omega _{-\delta },\mathbb{R}^{n})\), there exists a minimizer of \(I\) in \( H_{u_{0}}^{s,p,\delta }(\Omega _{-\delta },\mathbb{R}^{n})\). The proof is based on the existence and properties of a minimizing sequence for \(I\), through the use of the previously proved or recalled results. The paper ends with the derivation of the Euler-Lagrange equations that minimizers of the functional \(I\) satisfy.