Compact Operators - Applications

The Rellich-Kondrachov Compactness Theorem is one of the most important applications of compact operator theory to PDEs, providing compact embeddings between Sobolev spaces.

TheoremRellich-Kondrachov Theorem

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary. Then:

If $k < n/p$ , the embedding $W^{k,p}(\Omega) \hookrightarrow L^q(\Omega)$ is compact for all $q < p^* = \frac{np}{n-kp}$
If $k > n/p$ , the embedding $W^{k,p}(\Omega) \hookrightarrow C(\overline{\Omega})$ is compact
The embedding $H^1(\Omega) \hookrightarrow L^2(\Omega)$ is compact

This theorem is fundamental to the study of elliptic PDEs. It ensures that bounded sequences in Sobolev spaces have convergent subsequences in $L^p$ spaces, enabling existence proofs via compactness arguments.

ExampleApplication to Poisson's Equation

Consider $-\Delta u = f$ in $\Omega$ with $u = 0$ on $\partial\Omega$ .

The solution operator $S : L^2(\Omega) \to H^1_0(\Omega)$ defined by $Sf = u$ is bounded. Composing with the compact embedding $H^1_0(\Omega) \hookrightarrow L^2(\Omega)$ gives a compact operator $K : L^2 \to L^2$ .

This compact operator $K$ is self-adjoint, so by the spectral theorem, there exists an orthonormal basis $\{e_n\}$ of $L^2(\Omega)$ consisting of eigenfunctions: $-\Delta e_n = \lambda_n e_n$

These are precisely the Laplacian eigenvalues and eigenfunctions.

TheoremSchauder Fixed Point Theorem

Let $X$ be a Banach space, $K \subset X$ a non-empty convex compact set, and $T : K \to K$ a continuous map. Then $T$ has a fixed point.

Combined with compact operators, this provides existence for nonlinear operator equations.

ExampleNonlinear Integral Equations

Consider the equation $u(x) = \int_0^1 k(x,y) f(y, u(y)) \, dy$

If $k$ induces a compact operator and $f$ satisfies growth conditions, one can show that the right-hand side maps a ball in $C[0,1]$ compactly into itself. By Schauder's theorem, a solution exists.

Remark

Compact embeddings transform PDE problems into compact operator equations, where the powerful machinery of spectral theory and fixed point theorems becomes available. This approach has been enormously successful in proving existence and regularity of solutions to elliptic equations.

The combination of Sobolev space theory, compact embeddings, and compact operator spectral theory forms the backbone of modern PDE analysis.