Rellich–Kondrachov compactness

Let $\Omega\subset\mathbb R^n$ be bounded with Lipschitz boundary. If $1\le p<n$, then W¹,ᵖ embeds compactly into Lᑫ for every $1\le q<p^*=np/(n-p)$. If $p=n$ with $n\ge2$, every finite q is allowed. On a bounded interval, the endpoint W¹,¹ also embeds compactly into every finite Lᑫ, but generally not into L∞. If $n<p<\infty$, compactness holds in continuous functions with the uniform norm, and in Hölder spaces with exponent strictly below $1-n/p$. These statements mean every bounded source sequence has a target-convergent subsequence. The whole original sequence need not converge, and its limit need not have a chosen classical derivative.

A continuous embedding only bounds the target norm by the source norm. A compact embedding adds subsequential strong convergence and is therefore stronger. Reflexivity of W¹,ᵖ for $1<p<\infty$ can already provide a weak source subsequence, but weak convergence alone does not give norm convergence. Combining weak energy convergence with strong convergence of functions in a suitable Lᑫ target often permits nonlinear lower-order terms to pass to the limit. Derivative-dependent nonlinearities can require additional monotonicity, convexity, or compactness arguments; Rellich does not automatically settle them.

For a whole-space W¹,ᵖ function, the difference between a function and its translation by h has Lᵖ norm at most $|h|\|\nabla u\|_p$. Prove this for smooth functions by the fundamental theorem along segments and then use density. A bounded energy family therefore has uniformly small translations. To obtain compactness, one also needs spatial tightness: the family cannot carry its mass arbitrarily far away. For a bounded Lipschitz domain, use an extension with support in a common bounded neighborhood, so both requirements hold.

The Lᵖ compactness criterion then provides a strongly convergent subsequence in Lᵖ. Continuous Sobolev embedding supplies a higher integrability bound, and interpolation upgrades convergence to intermediate subcritical exponents. Finite measure handles smaller exponents. This proof outline identifies why boundedness and domain geometry appear: one controls escape of mass, the other supplies a uniform extension. For a zero-boundary closure on a bounded arbitrary open set, zero extension can replace the Lipschitz extension step, yielding useful compactness versions without the same boundary regularity.

High-frequency functions can prevent strong convergence in the original energy norm while disappearing in a weaker norm. For example, dividing a sine wave by its frequency makes its function L² norm tend to zero while its derivative L² norm stays fixed. Concentrating bumps instead demonstrate failure at the critical Lᵖ-star target: their supports shrink but their critical norms remain positive. Both examples are compatible with the subcritical theorem. They show that derivative order and critical scaling are concrete restrictions rather than technical qualifications to ignore.

On the whole space, translating a fixed compactly supported bump farther and farther away preserves all its Sobolev norms but can destroy strong Lᵖ compactness. Every fixed compact region eventually sees zero, yet the global norm remains unchanged. Local compactness may still hold, while global compactness fails through escape. In an existence proof, identify which obstruction a hypothesis rules out: confinement addresses translation, subcritical targets address concentration, and a weaker derivative order addresses high-frequency variation. Boundedness of one source norm alone cannot rule out every kind of loss.