arXiv subjects
Publications and source records attributed to Florentine Fleißner.
The mathematical theory of a novel variational approximation scheme for general second and fourth order partial differential equations \begin{equation}\label{eq: A} \partial_t u - \nabla\cdot\Big(u\nabla\frac{δϕ}{δu}(u)\Big|\nabla\frac{δϕ}{δu}(u)\Big|^{q-2}\Big) \ = \ 0, \quad\quad u\geq0, \end{equation} $q\in(1, +\infty)$, is developed.
This paper will deal with differentiability properties of the class of Hellinger-Kantorovich distances which was recently introduced on the space of finite nonnegative Radon measures.
Generalized Lambda-semiflows are an abstraction of semiflows with non-periodic solutions, for which there may be more than one solution corresponding to given initial data. A select class of solutions to generalized Lambda-semiflows is introduced. It is proved that such minimal solutions are unique corresponding to given ranges and generate all other solutions by time reparametrization. Special qualities of minimal solutions are shown. The concept of minimal solutions is applied to gradient flows in metric spaces and generalized semiflows. Generalized semiflows have been introduced by Ball.
We consider the Cauchy problem for the gradient flow \begin{equation} \label{eq:81} \tag{$\star$} u'(t)=-\nablaϕ(u(t)),\quad t\ge 0;\quad u(0)=u_0, \end{equation} generated by a continuously differentiable function $ϕ:\mathbb H \to \mathbb R$ in a Hilbert space $\mathbb H$ and study the reverse approximation of solutions to ($\star$) by the De Giorgi Minimizing Movement approach. We prove that if $\mathbb H$ has finite dimension and $ϕ$ is quadratically bounded from below (in particular if $ϕ$ is Lipschitz) then for every solution $u$ to ($\star$) (which may have an infinite number of solutions) there exist perturbations $ϕ_τ:\mathbb H \to \mathbb R \ (τ>0)$ converging to $ϕ$ in the Lipschitz norm such that $u$ can be approximated by the Minimizing Movement scheme generated by the recursive minimization of $Φ(τ,U,V):=\frac 1{2τ}|V-U|^2+ ϕ_τ(V)$: \begin{equation} \label{eq:abstract} \tag{$\star\star$} U_τ^n\in \operatorname{argmin}_{V\in \mathbb H} Φ(τ,U_τ^{n-1},V)\quad n\in\mathbb N, \quad U_τ^0:=u_0. \end{equation} We show that the piecewise constant interpolations with time step $τ> 0$ of all possible selections of solutions $(U_τ^n)_{n\in\mathbb N}$ to ($\star\star$) will converge to $u$ as $τ\downarrow 0$. This result solves a question raised by Ennio De Giorgi. We also show that even if $\mathbb H$ has infinite dimension the above approximation holds for the distinguished class of minimal solutions to ($\star$), that generate all the other solutions to ($\star$) by time reparametrization.
We present new abstract results on the interrelation between the minimizing movement scheme for gradient flows along a sequence of Gamma-converging functionals and the gradient flow motion for the corresponding limit functional, in a general metric space. We are able to allow a relaxed form of minimization in each step of the scheme, and so we present new relaxation results too.