Existence for evolutionary problems with linear growth by stability methods
The very interesting paper under review deals with $L^2$-gradient flow associated to variational integrals of the type \[ \mathbf{F}[u]=\int_\Omega f(x,Du)dx \] over a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n,$ with a given Borel measurable integrand $f:\Omega\times\mathbb{R}^{N\times n}\to [0,\infty)$ that is convex in the gradient variable, for vector-valued maps $u:\Omega\to\mathbb{R}^N.$ The integrand $f$ satisfies a linear growth condition of the form \[ \nu|\xi|\leq f(x,\xi)\leq L(1+|\xi|)\quad \forall (x,\xi)\in\Omega\times\mathbb{R}^N \] with constants $0<\nu\leq L.$ \par On a formal level, solutions of the $L^2$-gradient flow associated to the functional $\mathbf{F}$ correspond to solutions of the system of PDEs \[ \partial_tu-\text{div}\big(D_\xi f(x,Du)\big)=0\quad \text{in}\ \Omega_T=\Omega\times(0,T). \] The authors construct solutions of the Cauchy-Dirichlet problem for that system by the SOLA (Solutions Obtained as Limits of Approximations) approach. Precisely, replacing the variational functional $\mathbf{F}$ by \[ \int_\Omega f^p(x,Du)dx \] with $p>1$ whose integrand $f^p$ fulfills a standard $p$-growth and coercivity condition, the associated $L^2$-gradient flow corresponds to solutions of the parabolic system \[ \partial_tu-\text{div}\big(D_\xi f^p(x,Du)\big)=0\quad \text{in}\ \Omega_T, \] which obeys well developed existence theory because its coefficients support a standard $p$-growth and coercivity condition. Thus the solutions of systems with linear growth are obtained as limits of solutions of perturbed $p$-growth systems when $p\searrow1.$
Solutions obtained as limits of approximations
scientific article; zbMATH DE number 7040872
scientific article