Regularity of weak solutions to rate-independent systems in one-dimension

We show that under some appropriate assumptions, every weak solution (e.g. energetic solution) to a given rate-independent system is of class SBV, or has fi�nite jumps, or is even piecewise C1. Our assumption is essentially imposed on the energy functional, but not convexity is required

BV solutions constructed by epsilon-neighborhood method

We study a certain class of weak solutions to rate-independent systems, which is constructed by using the local minimality in a small neighborhood of order $\varepsilon$ and then taking the limit $\varepsilon \to 0$. We show that the resulting solution satisfies both the weak local stability and the new energy-dissipation balance, similarly to the BV solutions constructed by vanishing viscosity introduced recently by Mielke, Rossi and Savar\'e

Another construction of BV solutions to rate-independent systems

We study one kind of weak solutions to rate-independent systems, which is constructed by using the local minimality in a small neighborhood of order ε and then taking the limit ε → 0. We show that the resulting solution satisfies both the weak local stability and the new energy-dissipation balance, similarly to the BV solutions constructed by vanishing viscosity introduced recently by Mielke, Rossi and Savare

Regularity of weak solutions to rate-independent systems in one-dimension

We show that under some appropriate assumptions, every weak solution (e.g. energetic solution) to a given rate-independent system is of class SBV, or has fi�nite jumps, or is even piecewise C1. Our assumption is essentially imposed on the energy functional, but not convexity is required

Weak solutions to rate-independent systems: Existence and regularity

Weak solutions for rate-independent systems has been considered by many authors recently. In this thesis, I shall give a careful explanation (benefits and drawback) of energetic solutions (proposed by Mielke and Theil in 1999) and BV solutions constructed by vanishing viscosity (proposed by Mielke, Rossi and Savare in 2012). In the case of convex energy functional, then classical results show that energetic solutions is unique and Lipschitz continuous. However, in the case energy functional is not convex, there is very few results about regularity of energetic solutions. In this thesis, I prove the SBV and piecewise C^1 regularity for energetic solution without requiring the convexity of energy functional. Another topic of this thesis is about another construction of BV solutions via epsilon-neighborhood method

Application of the cut-off projection to solve a backward heat conduction problem in a two-slab composite system

The main goal of this paper is applying the cut-off projection for solving one-dimensional backward heat conduction problem in a two-slab system with a perfect contact. In a constructive manner, we commence by demonstrating the Fourier-based solution that contains the drastic growth due to the high-frequency nature of the Fourier series. Such instability leads to the need of studying the projection method where the cut-off approach is derived consistently. In the theoretical framework, the first two objectives are to construct the regularized problem and prove its stability for each noise level. Our second interest is estimating the error in -norm. Another supplementary objective is computing the eigen-elements. All in all, this paper can be considered as a preliminary attempt to solve the heating/cooling of a two-slab composite system backward in time. Several numerical tests are provided to corroborate the qualitative analysis.Peer reviewe