# Maximum principle for Delta u >0

• loesung
In summary, the conversation discusses using the Taylor expansion formula to show that if a function u is twice continuously differentiable on a set U and satisfies a certain condition, then its maximum value cannot be achieved by any point in U. The explanation involves expanding u using Taylor's theorem and using vector calculus to show that u(x) must be positive definite. The contradiction of this assumption leads to the conclusion that \Delta u must be greater than 0. The conversation also mentions not understanding the origin of some equations and asks for clarification.
loesung
Hello!

I would like to show the following: $$u\in C^2(U) \cap C(\bar{U})$$ satisfies $$\Delta u(x)>0$$ for any $$x\in U$$, then $$\max_U u$$ cannot be achieved by any point in $$U$$. Here, $$u\in \mathbb{R}^n$$, i.e. it's not complex valued.

Apparently, one can use the Taylor expansion formula to show this. But how?

Los

Last edited:
loesung said:
Apparently, one can use the Taylor expansion formula to show this. But how?
Los

Yes. Expand upto the first order , assume the contrary & apply the mean value theorem.

I was given the following explanation, though I really don't understand what's going on with eqs. (3) and (4):

Using Taylors theorem, write

$$u(x)=u(x_0)+\nabla u(x_0)\cdot(x-x_0)+\frac{1}{2}(x-x_0)^T\nabla^2 u(x^*)(x-x_0)$$

Because we suppose a max is obtained, we have from vector calculus that

$$Du(x_0)=0, \qquad (1)$$

$$\nabla^2 u (x_0)\leq 0 \mbox{ as a matrix}, \qquad (2)$$

However, somehow it's important to note, for reasons that to me are not clear, that $$u$$ is positive definite...:

$$v^T\nabla^2u(x_0)v\leq 0 \mbox{ for all }v\in\mathbb{R}^n,\qquad (3)$$

and so $$u(x_0)= \text{tr}(D^2u(x_0))\leq 0, \qquad (4)$$

which contradicts the assumption that $$\Delta u\geq 0$$

In particular I don't understand where $$(1)$$ and $$(2)$$ come from, and supposedly $$(2)\Rightarrow (3)$$...? I would appreciate if this could be cleared up!

Los

Last edited:

## What is the maximum principle for Delta u >0?

The maximum principle for Delta u >0 is a mathematical theorem that states that if a function u satisfies a certain differential equation (specifically, the Laplace equation), then the maximum value of u can only occur on the boundary of the domain where the equation is being solved.

## How is the maximum principle for Delta u >0 used in science?

The maximum principle for Delta u >0 is used in various fields of science, including physics, engineering, and mathematics, to analyze and solve problems involving diffusion, heat conduction, and electrostatics. It is also used in numerical methods to ensure the stability and accuracy of solutions.

## What are the implications of the maximum principle for Delta u >0?

The maximum principle for Delta u >0 has several important implications. It guarantees the uniqueness of solutions to certain differential equations, and it also provides a way to find the maximum and minimum values of a function in a given domain. Additionally, it can be used to prove the existence of solutions to certain problems.

## Are there any limitations to the maximum principle for Delta u >0?

While the maximum principle for Delta u >0 is a powerful tool in solving certain differential equations, it does have some limitations. It only applies to equations that satisfy certain conditions, such as being linear and having constant coefficients. Additionally, it may not hold for non-smooth or irregular domains.

## Are there any real-world applications of the maximum principle for Delta u >0?

Yes, the maximum principle for Delta u >0 has many practical applications in the real world. It is used in engineering to analyze and design heat and mass transfer systems, in physics to study diffusion and equilibrium, and in economics to model optimal decision-making processes. It is also used in computer science and image processing to smooth out noisy data.

• Differential Equations
Replies
1
Views
883
• Differential Equations
Replies
1
Views
2K
• Differential Equations
Replies
2
Views
1K
• Math POTW for Graduate Students
Replies
1
Views
1K
• Differential Equations
Replies
2
Views
1K
• Differential Equations
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
58
Views
3K
• Differential Equations
Replies
17
Views
6K
• Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
• Differential Equations
Replies
1
Views
2K