On a sub-supersolution method for the prescribed mean curvature problem

summary:The paper is about a sub-supersolution method for the prescribed mean curvature problem. We formulate the problem as a variational inequality and propose appropriate concepts of sub- and supersolutions for such inequality. Existence and enclosure results for solutions and extremal solutions between sub- and supersolutions are established

Maintenance Optimization in a Digital Twin for Industry 4.0

The advent of Internet of Things and artificial intelligence in the era of Industry 4.0 has transformed decision-making within production systems. In particular, many decisions that previously required significant human activity are now made automatically with minimal human intervention via so-called digital twins (DTs). In the context of maintenance and reliability modeling, this naturally calls for new paradigms that can be seamlessly integrated within DTs for decision-making. The input data for time to failure needed in reliability computations are directly collected from the work center in a digital setting and often do not satisfy a known distribution. A neural network (NN) is proposed here to bypass this difficulty within the DT. Further, an algorithm inspired from machine learning is employed to solve the underlying semi-Markov decision process, whose transition model is captured via the NN. Numerical studies are carried out to demonstrate the usefulness of the approach. Finally, convergence properties of the algorithm are analyzed mathematically

On some noncoercive variational inequalities containing degenerate elliptic operators

We are concerned with the solvability of variational inequalities that contain degenerate elliptic operators. By using a recession approach, we find conditions on the boundary conditions such that the inequality has at least one solution. Existence results of Landesman-Lazer type for a nonsmooth inequality and a resonance problem for a weighted p-Laplacian are discussed in detail

Sub-supersolutions in a Variational Inequality Related to a Sandpile Problem

In this paper we study a variational inequality in which the principal operator is a generalised Laplacian with fast growth at infinity and slow growth at 0. by defining appropriate sub- and super-solutions, we show the existence of solutions and extremal solutions of this inequality above the subsolutions or between the sub- and super-solution

Some Existence Results and Properties of Solutions in Quasilinear Variational Inequalities with General Growths

This paper is about the existence and some properties of solutions of the boundary value problem: {−div(a(|∇u|)∇u)=f(x,u)inΩu=0on∂Ω, with the principal term a(|∇u|)∇u having general growth. In the non-coercive case, a sub-supersolution approach is applied to get existence and enclosure results. Other properties such as compactness of solution sets and existence of extremal solutions are also derived

On Variational and Quasi-variational Inequalities with Multivalued Lower Order Terms and Convex Functionals

In this paper, we consider the existence and some qualitative properties of solutions of variational inequalities of the form: and of quasi-variational inequalities of the form: where a is a second-order elliptic operator of Leray–Lions type, F is a multivalued lower order term, J and Ju are convex functionals, and Ju also depends on u. We concentrate here in noncoercive cases and use sub-supersolution methods to study the existence and enclosure of solutions, and also the existence of extremal solutions between sub and supersolutions