Let $\Omega$ be a bounded open set in $\mathbb{R}^d$ where $d \geq 1$ is a positive integer, with Lipschitz boundary. Let $k,l$ be non-negative integers and $1 \leq p < \infty $ then if $k > l$ and $\frac{k-l}{d} > \frac{1}{p} - \frac{1}{q}$ then the Sobolev embedding $$ W^{k,p}(\Omega) \subset W^{l,q}(\Omega) $$ is compact. As an example, for $k = 1+s, p=2$ and $l = 2,$ we have
$$ W^{1+s,2}(\Omega) \subset W^{2,q}(\Omega), \quad s > 2 -2/q. $$

My question is what does this mean for solving PDEs? and what does it mean for other choices of $k$ as in $k= 2+s,$ i.e., $$ W^{2+s,2}(\Omega) \subset W^{2,q}(\Omega), \quad s > 1 -2/q. $$ and $k= 3+s,$ i.e., $$ W^{3+s,2}(\Omega) \subset W^{2,q}(\Omega), \quad s > -2/q. $$


Note: this is more like a long comment with some example without proofs of application about the Rellich-Kondrakov theorem, it is not intended as complete answer.

I report two example of the use of the Rellich-Kondrakov theorem related to the PDE word.

Start with note that in particular the Rellich-Kondrakov theorem says that the embedding $W^{l,p}(\Omega) \longrightarrow L^p(\Omega)$ is compact.

Example 1: Helmholtz equation

The Helmholtz equation is an eigenvalue problem with this form: $$ \left\{ \begin{aligned} - \Delta u= \lambda u \quad \text{in} \quad\Omega\\ u=0 \quad \text{in} \quad \partial\Omega \end{aligned} \right. $$

The value $\lambda$ is the eigenvalue of the $-\Delta$ operator with the boundary conditions. Considering the 2 dimensional case this problem is related to the membrane vibration ($\lambda_n=$ harmonic) and the minimum eigenvalue, $\lambda_1$, is the fundamental tone.

Now you can obtain the weak formulation: $$ \left\{ \begin{aligned} \int_{\Omega} \nabla u \nabla \varphi = \lambda \int_{\Omega} u \varphi \, dx \quad \forall \varphi \in W^{1,2}_0(\Omega) \\ u \in W^{1,2}_0(\Omega) \setminus \{0\} \end{aligned} \right. $$

You can demonstrate that, under some hypothesis on $\Omega$, the set $$ \{ \lambda \in \mathbb{R} \, : \text{weak formulation problem has a solution } u \neq 0 \} $$ has a positive minimum. In this proof is used the Rellich-Kondrakov theorem to obtain a subsequence the converge to a function in $L^2$.

Example 2 Poisson Problem

With the Sobolev's theory is possible show that the weak Poisson problem is a well posed in Hadamard sense, i.e. exist a unique and stable solution. For this proof is necessary the Poincaré inequality.

Now according with wikipedia

The Rellich–Kondrachov theorem may be used to prove the Poincaré inequality

I think this can be see as an indirect use of the theorem Rellich-Kondrakov.


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