A space of functions \(\mathcal{K}\) is said to be convex if, whenever \(u \in \mathcal{K}\) and \(v \in \mathcal{K},(1-\lambda) u+\lambda v \in \mathcal{K}\) for all \(0 \leq \lambda \leq 1 .\) Show that the space \(\mathcal{K}\) of all piecewise continuously differentiable functions \(v(x)(-1 \leq x \leq 1)\) satisfying \(v \geq f\) and \(v(\pm 1)=0\) is convex. (These functions are called test functions.) The obstacle problem may also be formulated as: find the function \(u\) that minimises the energy $$E[v]=\int_{-1}^{1} \frac{1}{2}\left(v^{\prime}\right)^{2} d x$$ over all \(v \in \mathcal{K}\). (This is the usual energy minimisation but with the constraint incorporated.) If \(u\) is the minimiser, and \(v\) is any test function, use the fact that \(E[(1-\lambda) u+\lambda v]-E[u] \geq 0\) for all \(\lambda\) to show that $$\int_{-1}^{1} u^{\prime}(v-u)^{\prime} d x \geq 0$$ This is the variational inequality for the obstacle problem.

When dealing with constraints in optimization problems, especially in physics and mathematics, we encounter the concept of a variational inequality. In simplified terms, a variational inequality involves finding a function that satisfies certain constraints and, at the same time, minimizes or maximizes an integral that represents some form of energy or cost.

In our exercise, the variational inequality stems from the need to find a function within the convex space \( \mathcal{K} \) that minimizes the energy functional \( E[v] \). This energy functional is a measure of the 'effort' or 'work' that the function represents over an interval. When we attempt to minimize \( E[v] \) but under a restriction (the obstacle), we are essentially solving the variational inequality. This inequality states that for the minimizer \( u \), and any test function \( v \) in \( \mathcal{K} \), the integral \( \int_{-1}^{1} u'(v-u)' dx \) must be non-negative.

This inequality guarantees that \( u \) is indeed the minimizer because any direction we move away from \( u \) (within the space of allowed directions that \( v \) represents) does not decrease the energy - an essential sign we've found a function that provides the 'least resistance' or 'lowest energy' while respecting our constraints.

The energy minimization problem is a cornerstone in various fields such as physics, engineering, and mathematics. It seeks the state or function that has the least possible energy, conforming to a natural tendency observed in many physical systems.

Our problem considers an energy function \( E[v] \), which is an integral that computes a value representing the 'strength' or 'vigor' of the function \( v \) over a specified interval. This particular form, \( \int_{-1}^{1} \frac{1}{2}(v')^2 dx \), captures the concept of 'strain' or 'tension' in a physical system. This strain is often visualized as the potential energy stored within a spring or the effort exerted to deform an elastic material.

By identifying the function that minimizes this energy, we're essentially finding the most 'relaxed' state of the system—the position that the system will naturally gravitate towards if left to its own devices. This idea is incredibly powerful for modeling natural phenomena and engineering problems, where systems prefer to settle in states of minimal energy.

Piecewise continuous differentiability is a property of functions that may not be smooth everywhere but can be broken into sections, each of which is smooth within its domain. A function is piecewise continuously differentiable if, on each segment of its domain, it has a derivative that does not exhibit any jumps or discontinuities.

In practical terms, for a function to be piecewise continuously differentiable in the context of our space \( \mathcal{K} \), it must have segments within the domain \( [-1, 1] \) where it behaves nicely—think of a path that's generally easy to walk, with occasional clear steps up or down. The essential point for our functions \( u \) and \( v \) is that they are 'well-behaved' in segments and can be 'stitched' together to form a function that remains controlled and predictable, despite not being entirely smooth over the entire interval. This property is crucial for the mathematics of optimization and physics simulations, as it allows for flexibility in modeling real-life scenarios that may not be perfectly smooth but are still understandable and manageable.

The obstacle problem is a particular type of optimization problem where a function must minimize energy while also avoiding crossing a certain threshold, which can be thought of as an 'obstacle'. It is a fascinating problem that interweaves calculus, physics, and geometry.

In our scenario, the functions in the space \( \mathcal{K} \) are not only striving to minimize energy, as represented by the energy functional \( E[v] \) but also must satisfy the condition \( v(x) \geq f(x) \), representing the 'obstacle' they cannot dip below.

This restriction can be envisioned as stretching a membrane over a set of points (the obstacle) without tearing it and without allowing it to dip below these points. The goal is to achieve the least amount of tension in the membrane while complying with these conditions. The obstacle problem is deeply connected to variational inequalities since the function that 'solves' the obstacle problem is the same function that satisfies the associated variational inequality.