Minimizers of an Energy Functional

In summary, an energy functional is a mathematical function that describes the total energy of a physical system and its evolution over time. Minimizers of an energy functional are the values of the system's variables that represent the most stable or equilibrium state of the system. These minimizers are typically found using mathematical optimization techniques and are important for understanding and predicting the behavior of the system. Real-world applications of minimizers of an energy functional include predicting material behavior, designing energy systems, and optimizing chemical reactions.
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Let ##U## be a bounded open subset of ##\mathbb{R}^n##. Given a continuous function ##\phi : \overline{U} \to \mathbb{R}##, show that any real-valued function ##u## of class ##C^2(\overline{U})## such that ##\Delta u = \phi## in ##U## and ##u|_{\partial U} = 0## is a minimizer of the energy functional $$\mathscr{E}(u) = \int_U d^n x\, \left(\frac{1}{2}|\nabla u(x)|^2 + \phi(x) u(x)\right)$$ over the class of functions ##\Sigma = \{v\in C^2(\overline{U}) : v|_{\partial U} = 0\}##.
 
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First, as

$$
\Delta u(x) = \phi (x)
$$

we have

$$
0 = \int_U d^nx \left( - \Delta u(x) + \phi (x) \right) \eta (x)
$$

for an arbitrary function ##\eta (x)##. If we take ##\eta \in \Sigma##, integrating by parts gives

$$
0 = \int_U d^nx \left( \nabla u(x) \cdot \nabla \eta (x)+ \phi (x) \eta (x) \right)
$$

Take an arbitrary ##v \in \Sigma##. We can write down an obvious inequality

$$
0 \leq \left( \| \nabla u \| - \| \nabla v \| \right)^2
$$

where ##\| \nabla u \| := \left( \sum_{i=1}^n (\partial_i u)^2 \right)^{1/2}## (##\| \nabla u \|^2 \equiv | \nabla u |^2##). Which implies

$$
0 \leq \frac{1}{2} | \nabla u |^2 + \frac{1}{2} | \nabla v |^2 - \| \nabla u \| \| \nabla v \|
$$

or

$$
\| \nabla u \| \| \nabla v \| \leq \frac{1}{2} | \nabla u |^2 + \frac{1}{2} | \nabla v |^2 .
$$

As ##\nabla u \cdot \nabla v \leq | \nabla u \cdot \nabla v | \leq \| \nabla u \| \| \nabla v \|## (Cauchy-Schwarz) we have

$$
\nabla u \cdot \nabla v \leq \frac{1}{2} | \nabla u |^2 + \frac{1}{2} | \nabla v |^2 .
$$

From which we have

$$
\int_U d^nx \left( \nabla u \cdot \nabla v + \phi v - \frac{1}{2} | \nabla u |^2 \right) \leq \int_U d^nx \left( \frac{1}{2} | \nabla v |^2 + \phi v \right)
$$

Using ##\int_U d^nx \left( \nabla u \cdot \nabla \eta+ \phi \eta \right) = 0## in this integral gives

\begin{align*}
\int_U d^nx \left( \nabla u \cdot \nabla [\eta + v] + \phi [\eta + v] - \frac{1}{2} | \nabla u |^2 \right)
\leq \int_U d^nx \left( \frac{1}{2} | \nabla v |^2 + \phi v \right)
\end{align*}

Putting ##\eta (x) = u(x) - v(x)## we have,

$$
\int_U d^nx \left( \frac{1}{2} | \nabla u (x) |^2 + \phi (x) u (x) \right) \leq \int_U d^nx \left( \frac{1}{2} | \nabla v (x) |^2 + \phi (x) v (x) \right)
$$

which is the desired result. Obviously, this choice of ##\eta (x)## means that ##\eta \in \Sigma##.
 
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Related to Minimizers of an Energy Functional

What is an energy functional?

An energy functional is a mathematical function that describes the energy of a physical system in terms of its state or configuration. It is used in the field of physics and engineering to model and analyze various systems.

What are minimizers of an energy functional?

Minimizers of an energy functional are the states or configurations of a physical system that correspond to the lowest possible energy value. In other words, they are the states that minimize the energy functional and are considered the most stable or optimal configurations of the system.

How do minimizers of an energy functional relate to the laws of thermodynamics?

Minimizers of an energy functional are closely related to the laws of thermodynamics, specifically the second law which states that all natural processes tend towards a state of minimum energy. This means that the minimizers of an energy functional are the most likely states that a physical system will naturally evolve towards.

What are some common techniques for finding minimizers of an energy functional?

There are several techniques for finding minimizers of an energy functional, including gradient descent, variational methods, and numerical optimization algorithms. These methods involve finding the critical points of the energy functional and determining which points correspond to minimizers.

What are some real-world applications of studying minimizers of an energy functional?

Studying minimizers of an energy functional has many real-world applications, such as in material science, where it is used to understand the properties and behavior of materials. It is also used in physics, chemistry, and engineering to model and analyze various systems, such as fluids, crystals, and electromagnetic fields.

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