How Does the Derivation of Harmonic Function Derivatives Work in Evans' PDE?

  • A
  • Thread starter Shackleford
  • Start date
  • Tags
    Derivatives
In summary: This is because when $|\alpha| = 1$, the term $C_k$ becomes $C_1 = \frac{(2^{n+1}nk)^k}{\alpha(n)} = \frac{2^{n+1}n}{\alpha(n)}$.
  • #1
Shackleford
1,656
2
This is from Evans PDE page 29. Assume u is harmonic.

18.
$$ |D^\alpha u(x_0)| \leq \frac{C_k}{r^{n+k}} \|u\|_{L^1(B(x_0,r))} $$

19.
$$ C_0 = \frac{1}{\alpha(n)}, \qquad C_k = \frac{(2^{n+1}nk)^k}{\alpha(n)}, \qquad (k=1,...).
$$

20. \fint is the average integral.
$$
\begin{gather*}
\begin{split}
|u_{x_i}(x_0)| & = \left| \fint_{B(x_0,r/2)} u_{x_i} \; dx\right| \\
& = \left| \frac{2^n}{\alpha(n)r^n}\int_{B(x_0,\frac{r}{2})} u \nu_i \; dS \right| \quad \text{(by Gauss-Green)}\\
& \leq \frac{2n}{r} \|u\|_{L^\infty(\partial B(x_0,\frac{r}{2}))}.
\end{split}
\end{gather*}
$$

$$
\text{If } x \in \partial B(x_0,r/2)), \text{then } B(x,r/2) \subset B(x_0,r) \subset U, \text{ and so}
$$

21.
$$
|u(x)| \leq \frac{1}{\alpha(n)} \left(\frac{2}{r}\right)^n \|u\|_{L^1(B(x_0,r))} \\
$$
by (18), (19) for k = 0. Combining the inequalities above, we deduce

$$ |D^\alpha u(x_0)| \leq \frac{2^{n+1}n}{\alpha(n)} \frac{1}{r^n} \|u\|_{L^1(B(x_0,r))}$$

if |alpha| = 1. Verifies for k = 1.

Questions:

Am I following the derivation correctly?

For k = 0:

$$
\begin{align*}
|D^0 u(x_0)| = &|u(x_0)| \leq \frac{C_0}{r^{n+0}} \|u\|_{L^1(B(x_0,r))} = \frac{1}{\alpha(n)r^{n}} \|u\|_{L^1(B(x_0,r))} \leq \frac{1}{\alpha(n)r^{n}} \cdot 2^n\|u\|_{L^1(B(x_0,r))} \\
& = \frac{1}{\alpha(n)} \cdot \left( \frac{2}{r} \right)^n \|u\|_{L^1(B(x_0,r))}.
\end{align*}
$$

Because the ball is contained in the other, we can state
$$ u(x) \leq \frac{1}{\alpha(n)} \cdot \left( \frac{2}{r} \right)^n \|u\|_{L^1(B(x_0,r))}.$$

I'm not sure how he arrived at the deduction. I thought maybe he's thinking of (20) as this $$|u_{x_i}(x_0)| = |Du(x_0)| $$

and then comparing it to what I'm calling (21) to arrive at the formula when |alpha|=1.
 
Last edited:
Physics news on Phys.org
  • #2


Yes, you are following the derivation correctly. For k = 0, the equation becomes:

$$
|D^0 u(x_0)| = |u(x_0)| \leq \frac{C_0}{r^{n+0}} \|u\|_{L^1(B(x_0,r))} = \frac{1}{\alpha(n)r^{n}} \|u\|_{L^1(B(x_0,r))} \leq \frac{1}{\alpha(n)} \cdot \left( \frac{2}{r} \right)^n \|u\|_{L^1(B(x_0,r))}.
$$

This is because when k = 0, the term $C_k$ becomes $C_0 = \frac{1}{\alpha(n)}$.

The deduction is arrived at by comparing the inequality in (20) to the inequality in (21). In (20), we have the average integral of $u_{x_i}$ over the ball $B(x_0,r/2)$, while in (21) we have the average integral of $u$ over the same ball. By comparing these two inequalities, we can see that they are essentially the same, except for the factor of $2^n$ in front of the integral in (20). This is because when taking the average integral of $u_{x_i}$, we also need to take into account the factor of $2^n$ from the Jacobian of the transformation from $x_i$ to $x$.

Therefore, by comparing these two inequalities, we can deduce that:

$$
|D^\alpha u(x_0)| \leq \frac{2^{n+1}n}{\alpha(n)} \frac{1}{r^n} \|u\|_{L^1(B(x_0,r))}
$$

if $|\alpha| = 1$.
 

FAQ: How Does the Derivation of Harmonic Function Derivatives Work in Evans' PDE?

1. What are estimates on derivatives?

Estimates on derivatives refer to mathematical calculations or approximations of the rate of change of a function at a specific point. In other words, it is an estimation of how much a function is changing at a given point.

2. Why are estimates on derivatives important?

Estimates on derivatives are important because they provide valuable information about the behavior of a function. They can help us understand the slope of a curve, the direction of change, and the rate at which the function is changing. This information is crucial in many scientific fields, including physics, economics, and engineering.

3. How are estimates on derivatives calculated?

Estimates on derivatives can be calculated using various methods, such as the limit definition of a derivative, the power rule, or the chain rule. These methods involve using algebraic equations and calculus principles to find an approximation of the derivative at a specific point.

4. What are the applications of estimates on derivatives?

Estimates on derivatives have numerous applications in various fields of science and mathematics. They are used in physics to calculate velocity and acceleration, in economics to determine the marginal cost and revenue of a product, and in engineering to analyze the behavior of systems and design optimal solutions.

5. Can estimates on derivatives be accurate?

Estimates on derivatives are not always exact, but they can be very accurate depending on the method used and the precision of the calculations. With more advanced mathematical techniques and computing power, it is possible to achieve very precise estimates on derivatives. However, it is important to keep in mind that they are still approximations and may not always reflect the exact behavior of a function.

Back
Top