Statement of the Fundamental Lemma

Oct 13, 2025 by windailys.com 66 views

Introduction

The calculus of variations is a branch of mathematics that deals with the optimization of functionals, which are functions that map a set of functions to the real numbers. In this article, we will focus on the fundamental lemma of the calculus of variations in higher dimensions, which is a crucial result in the field. The fundamental lemma states that if a function $f$ is continuous on a bounded domain $\Omega$ in $\mathbb{R}^n$ and the integral of $f$ times any continuous function $v$ that vanishes on the boundary of $\Omega$ is zero, then $f$ must be identically zero.

Statement of the Fundamental Lemma

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain and $f\in C(\overline{\Omega})$ . We want to show that

\int_\Omega f(x)v(x)\text{d}x=0\space\forall v\in C(\overline{\Omega}):v|_{\partial\Omega=0}\iff f(x)=0

Proof of the Fundamental Lemma

To prove the fundamental lemma, we will use a proof by contradiction. Assume that $f$ is not identically zero, and let $x_0$ be a point in $\Omega$ where $f(x_0)\neq 0$ . Without loss of generality, we can assume that $f(x_0)>0$ . We will show that there exists a continuous function $v$ that vanishes on the boundary of $\Omega$ and satisfies

\int_\Omega f(x)v(x)\text{d}x>0

This will contradict the assumption that the integral is zero for all such functions $v$ .

Construction of the Function $v$

Let $r>0$ be a small positive number such that the ball $B(x_0,r)$ is contained in $\Omega$ . We define the function $v$ as follows:

v(x)=\begin{cases} \frac{f(x_0)}{f(x_0)-f(x)}&\text{if }x\in B(x_0,r)\\ 0&\text{if }x\notin B(x_0,r) \end{cases}

This function is continuous on $\overline{\Omega}$ and vanishes on the boundary of $\Omega$ . We can now compute the integral of $f$ times $v$ over $\Omega$ :

\int_\Omega f(x)v(x)\text{d}x=\int_{B(x_0,r)} f(x)v(x)\text{d}x

Using the definition of $v$ , we get

\int_\Omega f(x)v(x)\text{d}x=\int_{B(x_0,r)} f(x)\frac{f(x_0)}{f(x_0)-f(x)}\text{d}x

This integral is equal to

\int_{B(x_0,r)} f(x_0)\text{d}x-f(x_0)\int_{B(x_0,r)} \frac{f(x)}{f(x_0)-f(x)}\text{d}x

The first integral is equal to $f(x_0)$ times the volume of the ball $B(x_0,r)$ . The second integral is equal to $-f(x_0)$ times the volume of the ball $B(x_0,r)$ , since the integrand is equal to $-1$ on the ball. Therefore, the integral is equal to $2f(x_0)$ times the volume of the ball $B(x_0,r)$ .

Conclusion

We have shown that if $f$ is not identically zero, then there exists a continuous function $v$ that vanishes on the boundary of $\Omega$ and satisfies

\int_\Omega f(x)v(x)\text{d}x>0

This contradicts the assumption that the integral is zero for all such functions $v$ . Therefore, we must have $f(x)=0$ for all $x\in\Omega$ .

Applications of the Fundamental Lemma

The fundamental lemma has many applications in the calculus of variations. One of the most important applications is the Euler-Lagrange equation, which is a necessary condition for a functional to have a local minimum. The Euler-Lagrange equation is a partial differential equation that is derived from the fundamental lemma.

Conclusion

In this article, we have proved the fundamental lemma of the calculus of variations in higher dimensions. This result is a crucial tool in the field of the calculus of variations, and it has many applications in physics, engineering, and economics. We have shown that if a function $f$ is continuous on a bounded domain $\Omega$ in $\mathbb{R}^n$ and the integral of $f$ times any continuous function $v$ that vanishes on the boundary of $\Omega$ is zero, then $f$ must be identically zero.

References

[1] Evans, L. C. (2010). Partial Differential Equations. American Mathematical Society.
[2] Gilbarg, D., & Trudinger, N. S. (2001). Elliptic Partial Differential Equations of Second Order. Springer.
[3] Lions, J. L. (1969). Quelques Méthodes de Résolution de Problèmes aux Limites Non Linéaires. Dunod.

Q: What is the fundamental lemma of the calculus of variations in higher dimensions?

A: The fundamental lemma of the calculus of variations in higher dimensions is a result that states that if a function $f$ is continuous on a bounded domain $\Omega$ in $\mathbb{R}^n$ and the integral of $f$ times any continuous function $v$ that vanishes on the boundary of $\Omega$ is zero, then $f$ must be identically zero.

Q: What is the significance of the fundamental lemma?

A: The fundamental lemma is a crucial result in the calculus of variations, and it has many applications in physics, engineering, and economics. It is used to derive the Euler-Lagrange equation, which is a necessary condition for a functional to have a local minimum.

Q: How is the fundamental lemma used in the calculus of variations?

A: The fundamental lemma is used to prove the existence of a minimum for a functional. It is also used to derive the Euler-Lagrange equation, which is a necessary condition for a functional to have a local minimum.

Q: What are some of the applications of the fundamental lemma?

A: The fundamental lemma has many applications in physics, engineering, and economics. Some of the applications include:

Optimization problems: The fundamental lemma is used to solve optimization problems, such as finding the minimum of a functional.
Euler-Lagrange equation: The fundamental lemma is used to derive the Euler-Lagrange equation, which is a necessary condition for a functional to have a local minimum.
Variational problems: The fundamental lemma is used to solve variational problems, such as finding the minimum of a functional subject to certain constraints.

Q: What are some of the challenges in applying the fundamental lemma?

A: Some of the challenges in applying the fundamental lemma include:

Computational complexity: The fundamental lemma can be computationally complex to apply, especially for high-dimensional problems.
Numerical instability: The fundamental lemma can be numerically unstable, especially for problems with singularities or discontinuities.
Lack of regularity: The fundamental lemma requires the function $f$ to be continuous on the domain $\Omega$ . If the function $f$ is not continuous, the fundamental lemma may not apply.

Q: What are some of the future directions in the application of the fundamental lemma?

A: Some of the future directions in the application of the fundamental lemma include:

High-dimensional problems: The fundamental lemma can be applied to high-dimensional problems, such as those in machine learning and data analysis.
Nonlinear problems: The fundamental lemma can be applied to nonlinear problems, such as those in physics and engineering.
Stochastic problems: The fundamental lemma can be applied to stochastic problems, such as those in finance and economics.

Q: What are some of the resources available for learning more about the fundamental lemma?

A: Some of the resources available for learning more about the fundamental lemma include:

Textbooks: There are many textbooks available on the calculus of variations, including those by Evans, Gilbarg, and Trudinger.
Online courses: There are many online courses available on the calculus of variations, including those on Coursera and edX.
Research papers: There are many research papers available on the calculus of variations, including those on arXiv and ResearchGate.

Q: What are some of the open problems in the application of the fundamental lemma?

A: Some of the open problems in the application of the fundamental lemma include:

High-dimensional problems: The fundamental lemma can be computationally complex to apply to high-dimensional problems.
Nonlinear problems: The fundamental lemma can be difficult to apply to nonlinear problems, especially those with singularities or discontinuities.
Stochastic problems: The fundamental lemma can be difficult to apply to stochastic problems, especially those with uncertainty or randomness.