40 research outputs found
Explicit and implicit non-convex sweeping processes in the space of absolutely continuous functions
We show that sweeping processes with possibly non-convex prox-regular
constraints generate a strongly continuous input-output mapping in the space of
absolutely continuous functions. Under additional smoothness assumptions on the
constraint we prove the local Lipschitz continuity of the input-output mapping.
Using the Banach contraction principle, we subsequently prove that also the
solution mapping associated with the state-dependent problem is locally
Lipschitz continuous.Comment: Changes: p. 2 line 10; p. 5 lines 1 to 6; p. 9 line -1;
Acknowledgment section; New References [3] and [23
Explicit and implicit non-convex sweeping processes in the space of absolutely continuous functions
We show that sweeping processes with possibly non-convex prox-regular constraints generate a strongly continuous input-output mapping in the space of absolutely continuous functions. Under additional smoothness assumptions on the constraint we prove the local Lipschitz continuity of the input-output mapping. Using the Banach contraction principle, we subsequently prove that also the solution mapping associated with the state-dependent problem is locally Lipschitz continuous
Variational Methods for Evolution
The meeting focused on the last advances in the applications of variational methods to evolution problems governed by partial differential equations. The talks covered a broad range of topics, including large deviation and variational principles, rate-independent evolutions and gradient flows, heat flows in metric-measure spaces, propagation of fracture, applications of optimal transport and entropy-entropy dissipation methods, phase-transitions, viscous approximation, and singular-perturbation problems
Variational Methods for Evolution: Abstracts from the workshop held December 4–10, 2011
The meeting focused on the last advances in the applications of
variational methods to evolution problems governed by partial differential
equations. The talks covered a broad range of topics, including large deviation
and variational principles, rate-independent evolutions and gradient
flows, heat flows in metric-measure spaces, propagation of fracture, applications
of optimal transport and entropy-entropy dissipation methods, phasetransitions,
viscous approximation, and singular-perturbation problems
Differential, energetic, and metric formulations for rate-independent processes
We consider different solution concepts for rate-independent systems. This includes energetic solutions in the topological setting and differentiable, local, parametrized and BV solutions in the Banach-space setting. The latter two solution concepts rely on the method of vanishing viscosity, in which solutions of the rate-independent system are defined as limits of solutions of systems with small viscosity. Finally, we also show how the theory of metric evolutionary systems can be used to define parametrized and BV solutions in metric spaces
Advances in Multiscale and Multifield Solid Material Interfaces
Interfaces play an essential role in determining the mechanical properties and the structural integrity of a wide variety of technological materials. As new manufacturing methods become available, interface engineering and architecture at multiscale length levels in multi-physics materials open up to applications with high innovation potential. This Special Issue is dedicated to recent advances in fundamental and applications of solid material interfaces
Some results on quasistatic evolution problems for unidirectional processes
The present thesis is devoted to the study of some models of quasistatic evolutions for materials, in the presence of unidirectional phenomena, such as damage and fracture. In particular, these models concern the coupling between damage and plasticity, and the growth of brittle and cohesive fractures in antiplane linearized elasticity
Analysis of gradient descents in random energies and heat baths
This thesis concerns the mathematical analysis of random gradient descent
evolutions as models for rate-independent dissipative systems under the influence
of thermal effects. The basic notions of the theory of gradient descents
(especially rate-independent evolutions) are reviewed in chapter 2.
Chapters 3 and 4 focus on the scaling regime in which the microstructure
dominates the thermal effects and comprise a rigorous justification of rateindependent
processes in smooth, convex energies as scaling limits of ratedependent
gradient descents in energies that have rapidly-oscillating random
microstructure: chapter 3 treats the one-dimensional case with quite a broad
class of random microstructures; chapter 4 treats a case in which the microstructure
is modeled by a sum of “dent functions” that are scattered in
Rn using a suitable point process. Chapters 5 and 6 focus on the opposite
scaling regime: a gradient descent system (typically a rate-independent process)
is placed in contact with a heat bath. The method used to “thermalize”
a gradient descent is an interior-point regularization of the Moreau–Yosida
incremental problem for the original gradient descent. Chapter 5 treats
the heuristics and generalities; chapter 6 treats the case of 1-homogeneous
dissipation (rate independence) and shows that the heat bath destroys the
rate independence in a controlled and deterministic way, and that the effective
dynamics are a gradient descent in the original energetic potential
but with respect to a different and non-trivial effective dissipation potential.
The appendices contain some auxiliary definitions and results, most of them
standard in the literature, that are used in the main text