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### Open Access Journals

This special issue of

*Discrete and Continuous Dynamical Systems*is devoted to the latest advances and trends in the modelling and in the analysis of this family of complex phenomena. In particular, we gathered contributions from different fields of science (mathematical analysis, mathematical physics, engineering) with the intent of presenting an updated picture of current research directions, offering a new and interdisciplinary perspective in the study of these processes.

Motivated by the

*Spring School on Rate-independent Evolutions and Hysteresis Modelling*, held at the Politecnico di Milano and University of Milano on May 27-31, 2013, this special issue contains different kinds of original contributions: some of them originate from the courses held in that occasion and from the discussions they stimulated, but are here presented in a new perspective; some others instead are original contributions in related topics. All the papers are written in the clearest possible language, accessible also to students and non-experts of the field, with the intent to attract and introduce them to this topic.

Final acceptance of all the papers in this volume was made by the normal referee procedure and standard practices of AIMS journals.

We wish to thanks all the referees, who kindly agreed to devote their time and effort to read and check all the papers carefully, providing useful comments and recommendations. We are also grateful to all the authors for their great job and the high quality of their contributions. We finally wish to express our gratitude to AIMS and in particular to Prof. Alain Miranville for the opportunity to publish this special issue and for the technical support.

*energetic*sense. By recovering in this context a result by RINDLER [49,50] we prove the existence of optimal controls for a suitably large class of cost functionals and comment on their possible approximation.

We are interested in the *regularity* of local minimizers of energy integrals of the Calculus of Variations. Precisely, let $Ω $ be an open subset of $\mathbb{R}^{n}$. Let $f≤\left( {x, \xi } \right) $ be a real function defined in $Ω × \mathbb{R}^{n}$ satisfying the growth condition $|{f_{\xi x}}\left( {x, \xi } \right)| \le h\left( x \right)|\xi {{\rm{|}}^{p - 1}}$, for $x∈ Ω $ and $\xi ∈ \mathbb{R}^{n}$ with $|\xi {\rm{|}} \ge {M_0}$ for some $M_{0}≥ 0$, with $h \in L_{{\rm{loc}}}^r\left( \Omega \right) $ for some $r>n$. This growth condition is more general than those considered in the mathematical literature and allows us to handle some cases recently studied in similar contexts. We associate to $f\left( {x, \xi } \right) $ the so-called *natural* $p-$*growth conditions* on the second derivatives ${f_{\xi \xi }}\left( {x, \xi } \right)$; i.e., $\left( {p - 2} \right) - $growth for $|{f_{\xi \xi }}\left( {x, \xi } \right)| $ from above and $\left( {p - 2} \right) - $growth from below for the quadratic form $({f_{\xi \xi }}\left( {x, \xi } \right)\lambda , \lambda {\rm{ }})$; for details see either (3) or (7) below. We prove that these conditions are sufficient for the *local Lipschitz continuity* of any minimizer $u \in W_{{\rm{loc}}}^{1, p}\left( \Omega \right) $ of the energy integral $\int_\Omega {f(x, Du\left( x \right)){\mkern 1mu} dx} $.

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