### A (...)-stable approximations of abstract Cauchy problems.

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Compactness in the space ${L}^{p}(0,T;B)$, $B$ being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961, 1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to several abstract time dependent problems related to evolutionary PDEs. In the present paper, the problem is examined in view of Young measure theory: exploiting...

This note addresses the Cauchy problem for the gradient flow equation in a Hilbert space $H$ $$\left\{\begin{array}{c}{u}^{\text{'}}\left(t\right)+\partial \phi \left(u\left(t\right)\right)\ni 0\phantom{\rule{1.0em}{0ex}}\text{a.e.}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{0.166667em}{0ex}}(0,T),\hfill \\ u\left(0\right)={u}_{0},\hfill \end{array}\right.$$ where $\phi :H\to (-\infty ,+\infty \phantom{\rule{0.166667em}{0ex}}]$ is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and $\partial \phi $ is (a suitable limiting version of) its subdifferential. The interest for this kind of equations is motivated by a number of examples, which show that several mathematical models describing phase transitions phenomena and leading to systems of evolutionary PDEs...

We study a variational formulation for a Stefan problem in two adjoining bodies, when the heat conductivity of one of them becomes infinitely large. We study the «concentrated capacity» model arising in the limit, and we justify it by an asymptotic analysis, which is developed in the general framework of the abstract evolution equations of monotone type.

This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space $\mathscr{H}$ $$\left\{\begin{array}{cc}{u}^{\text{'}}\left(t\right)+{\partial}_{\ell}\phi \left(u\left(t\right)\right)\ni f\left(t\right)\hfill & \mathit{\text{a.e.}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}(0,T),u\left(0\right)={u}_{0},\hfill \end{array}\right.$$ where $\phi :\mathscr{H}\to (-\infty ,+\infty ]$ is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and ${\partial}_{\ell}\phi $ is (a suitable limiting version of) its subdifferential. We will present some new existence results for the solutions of the equation by exploiting a technique, featuring some ideas from the theory of and of . Our analysis is also motivated by some models...

In the nonconvex case, solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate-independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential that is a viscous regularization of a given rate-independent dissipation...

This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations...

In this paper we summarize some of the main results of a forthcoming book on this topic, where we examine in detail the theory of curves of maximal slope in a general metric setting, following some ideas introduced in [11, 5], and study in detail the case of the Wasserstein space of probability measures. In the first part we derive new general conditions ensuring convergence of the implicit time discretization scheme to a curve of maximal slope, the uniqueness, and the error estimates. In the second...

We recall the definition of Minimizing Movements, suggested by E. De Giorgi, and we consider some applications to evolution problems. With regards to ordinary differential equations, we prove in particular a generalization of maximal slope curves theory to arbitrary metric spaces. On the other hand we present a unifying framework in which some recent conjectures about partial differential equations can be treated and solved. At the end we consider some open problems.

In the nonconvex case, solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate-independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential that is a viscous regularization...

Motivated by recent developments on calculus in metric measure spaces $(X,\mathrm{d},\mathrm{m})$, we prove a general duality principle between Fuglede’s notion [15] of $p$-modulus for families of finite Borel measures in $(X,\mathrm{d})$ and probability measures with barycenter in ${L}^{q}(X,\mathrm{m})$, with $q$ dual exponent of $p\in (1,\infty )$. We apply this general duality principle to study null sets for families of parametric and non-parametric curves in $X$. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence...

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