1. Mathematical Physics: Sobolev embedded inequality is widely used in mathematical physics, especially in quantum mechanics, fluid mechanics, electromagnetism and other fields. For example, it can be used to prove the orthogonality of some function spaces, thus simplifying the solution of the problem.
2. Partial differential equation: Sobolev embedded inequality is an important tool to study the uniqueness and existence of solutions of partial differential equations. By introducing appropriate Sobolev embedding inequality, the existence and uniqueness of solutions of some partial differential equations can be proved.
3. Functional analysis: Sobolev embedded inequality also has important applications in functional analysis. For example, it can be used to prove the compactness of some function spaces, so that the function sequence in this space converges to a certain limit.
4. Control theory: In control theory, Sobolev embedded inequality is used to study the stability and boundedness of the system. By introducing Sobolev embedding inequality, the sufficient conditions for the stability and boundedness of the system are obtained.
5. Geometric analysis: In geometric analysis, Sobolev embedded inequality is used to study the properties of measures and integrals on manifolds. For example, it can be used to prove the finiteness of some measures, so as to obtain the existence of the integral corresponding to the measure.
6. Image processing: In image processing, Sobolev embedding inequality is used to study the characteristics and structure of images. For example, it can be used to prove the stability of some image features, so as to get the effectiveness of the features in image processing.
Generally speaking, Sobolev embedded inequality is widely used in many fields and is an important tool to solve these problems.