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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.