Sobolev spaces are essential in functional analysis and partial differential equations.
The space
Definition of
The Sobolev space
- A function
belongs tou if:L2(−1,1)
- The first weak derivative
must also be inu′ , which means:L2(−1,1)
Weak derivatives are used to accommodate functions that might not be differentiable in the classical sense everywhere on the interval, including functions that are continuous and differentiable almost everywhere but may have points of derivative discontinuity.
Norm in
The norm in the Sobolev space
Importance and Applications
Sobolev spaces such as
Example Function for
Working with a concrete example clarifies above definitions.
Consider the function
\Verification of
First, we check if the function
Calculate the integral:
This result confirms that
Verification of
Next, compute the derivative of
Check if
Computing the integral:
Thus:
Since this integral is also finite,
Thus, the function
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