The Rellich-Kondrachov Compactness Theorem and the Poincaré Inequalities

Intuition Beginner

Imagine a family of functions that all live in a fixed box and all obey the same speed limit on their slope. They can wiggle, they can shift around, but they cannot be too tall and they cannot change too fast. The compactness theorem of this unit says that such a family is, in a strong sense, crowded: you can always pull out a sequence from it that settles down and converges. No matter how the functions in the family scatter, the slope budget plus the fixed box prevent them from running off to infinity or oscillating forever, so some sequence must bunch up around a single limiting function.

Why is this worth a theorem? Because most existence proofs in physics and geometry work by setting up a sequence that gets closer and closer to solving a problem, then extracting a limit and showing the limit is the answer. The danger is that the sequence might not have a limit at all: it could spread its energy thinner and thinner, or push a bump out toward the edge and lose it. Controlling the slope inside a bounded region is exactly the ticket that forbids these escapes and guarantees a usable limit.

The companion inequalities of this unit are the slope-budget bookkeeping made precise. The Poincaré inequality says that if a function is pinned to zero on the boundary of a bounded region, then its overall size is controlled entirely by the size of its slope: a function that starts at zero on the edge and never changes fast cannot grow large in the middle. The slope alone caps the height.

There is a wrinkle when the function is not pinned at the boundary. A constant function has zero slope but is not zero, so slope alone cannot control size. The fix is to measure size after subtracting the average value: the spread of a function around its own average is controlled by its slope. This is the Poincaré-Wirtinger inequality, and it says a function with a small slope budget cannot stray far from its mean.

A picture for all three: think of a taut sheet clamped along the rim of a frame. If the clamp holds the edge at height zero and the sheet cannot crease too sharply, the whole sheet stays close to zero. If instead the sheet floats free, it can sit at any height, but its bumps above and below its own resting level are still limited by how sharply it is allowed to crease. And a whole family of such sheets, all clamped in the same frame with the same crease limit, has the crowding property: some sequence of them must converge.

Visual Beginner

The single picture to hold is a bounded box holding a crowded family of slope-limited functions, with the two Poincaré statements drawn alongside.

Read the three panels left to right. The left panel is the compactness statement: a whole family of functions sharing one height cap and one slope limit inside a fixed box is so crowded that you can always extract a converging sequence. The thing that prevents escape is the combination of the box, which stops bumps from drifting away, and the slope limit, which stops endless oscillation.

The middle panel is the Poincaré inequality. The curve is clamped to zero at the edges of the frame, so it cannot float up; its only way to gain height is to climb away from the baseline, which costs slope. The shaded area, a stand-in for the total size of the function, is therefore bounded by how much slope the function spends.

The right panel is the Poincaré-Wirtinger inequality, the version for functions that are not clamped. Now the curve can float to any height, so we measure its size not from the baseline but from its own average, the dashed line. The bumps above and below that average line are again paid for in slope, so the spread around the average is capped by the slope budget.

Worked example Beginner

We test the Poincaré idea on the simplest clamped function and watch the slope-controls-size bookkeeping work out. Take the region to be the interval from zero to one on the number line, and the function $u(x)=x(1-x)$, a single hump that is zero at both ends, which is exactly the clamped-at-the-boundary condition.

Step 1. Confirm the clamp. At the left end, $u(0)=0\times 1=0$. At the right end, $u(1)=1\times 0=0$. The function is pinned to zero on the boundary of the interval, so the Poincaré inequality applies.

Step 2. Measure the size of the function. The peak is at the middle, where $u(1/2)=(1/2)(1/2)=1/4$. So the largest the function ever gets is one quarter. Its size is modest, capped at $0.25$.

Step 3. Measure the size of the slope. The slope is $u^{\prime }(x)=1-2x$. At the left end it is $1-0=1$; at the right end it is $1-2=-1$. So the slope ranges between minus one and plus one, and its largest magnitude is one.

Step 4. Compare the two. The size of the function, one quarter, is smaller than the size of the slope, one. The Poincaré inequality is the promise that this direction of comparison always holds for clamped functions on this interval: the function's size is at most a fixed constant times the slope's size. Here a constant of one already works comfortably, with room to spare.

Step 5. See why the clamp matters. Replace the hump by the constant function $u(x)=5$. Its slope is zero everywhere, but its size is five, not zero. A constant times zero can never reach five, so no Poincaré inequality can hold for it. The escape is allowed precisely because the constant is not clamped to zero at the boundary; remove the clamp and slope stops controlling size.

What this tells us: when a function is held at zero on the edge of a bounded region, its slope budget alone caps how big it can get, and the constant in the cap depends only on the region. The constant-function failure in the last step is the warning that the clamp, or in the unclamped version the subtraction of the average, is what makes the inequality possible.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $n\ge 1$, $1\le p<\infty$, and $\Omega \subseteq {\mathbb{R}}^n$ is open and bounded. We write $W^{1,p}(\Omega )$ for the Sobolev space with norm $\Vert u{\Vert }_{W^{1,p}(\Omega )}=(\Vert u{\Vert }_{L^p(\Omega )}^p+\Vert Du{\Vert }_{L^p(\Omega )}^p{)}^{1/p}$, and $W_0^{1,p}(\Omega )$ for the closure of $C_c^{\infty }(\Omega )$ in this norm; both, together with the $L^p$ machinery, Hölder's inequality, and Minkowski's inequality, are taken as available 02.07.06. The critical Sobolev exponent $p^{\ast }=np/(n-p)$ for $1\le p<n$ is from 02.16.01, and the bounded extension operator $E:W^{1,p}(\Omega )\to W^{1,p}({\mathbb{R}}^n)$ on a bounded $C^1$ domain is from 02.16.02. For $u\in L^1(\Omega )$ and a measurable $A\subseteq \Omega$ with $|A|>0$ we write $(u{)}_A={\text{fint}}_{A}udx=\frac{1}{|A|}{\int }_{A}udx$ for the average. For $h\in {\mathbb{R}}^n$ the translation operator is $({\tau }_{h}u)(x)=u(x+h)$.

Definition (compact embedding). A continuous embedding $X\hookrightarrow Y$ of Banach spaces is compact, written $X\hookrightarrow \hookrightarrow Y$, if the inclusion map sends bounded subsets of $X$ to precompact subsets of $Y$: every sequence bounded in the $X$-norm has a subsequence converging in the $Y$-norm. Equivalently, the inclusion is a compact operator in the sense of 02.11.05.

Definition (Fréchet-Kolmogorov / Kolmogorov-Riesz precompactness). A subset $\mathcal{F}\subseteq L^p({\mathbb{R}}^n)$, $1\le p<\infty$, is precompact (has compact closure) if and only if three conditions hold:

• (boundedness) ${\sup }_{f\in \mathcal{F}}\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}<\infty$;
• (uniform equicontinuity of translation) ${\sup }_{f\in \mathcal{F}}\Vert {\tau }_{h}f-f{\Vert }_{L^p({\mathbb{R}}^n)}\to 0$ as $h\to 0$;
• (uniform decay at infinity) ${\sup }_{f\in \mathcal{F}}\Vert f{\Vert }_{L^p({\mathbb{R}}^n\setminus B(0,R))}\to 0$ as $R\to \infty$.

This is the $L^p$ analogue of the Arzelà-Ascoli theorem: equicontinuity of translation replaces equicontinuity of values, and uniform decay at infinity replaces a uniform domain bound ^{[Kolmogorov 1931] [Riesz 1933] [Hanche-Olsen-Holden 2010]}.

Definition (Poincaré inequality). $\Omega$ admits a Poincaré inequality at exponent $p$ if there is a constant $C=C(n,p,\Omega )$ with $$\Vert u{\Vert }_{L^p(\Omega )}\le C\Vert Du{\Vert }_{L^p(\Omega )}\text{for all }u\in W_0^{1,p}(\Omega ).$$ The smallest admissible $C$ is the reciprocal of the square root of the first Dirichlet eigenvalue of $-\Delta$ when $p=2$; the inequality fails without the zero-boundary restriction, since constants are excluded only by it.

Definition (Poincaré-Wirtinger inequality). A bounded connected open $\Omega$ with $C^1$ boundary (more generally an extension domain) admits a Poincaré-Wirtinger inequality at exponent $p$ if there is a constant $C=C(n,p,\Omega )$ with $$\Vert u-(u{)}_{\Omega }{\Vert }_{L^p(\Omega )}\le C\Vert Du{\Vert }_{L^p(\Omega )}\text{for all }u\in W^{1,p}(\Omega ).$$ Connectedness is essential: on a disconnected $\Omega$ a function constant on each piece has zero gradient but nonzero variance around the global mean, defeating the bound.

Counterexamples to common slips Intermediate+

• Compactness is strictly subcritical. The embedding $W^{1,p}(\Omega )\hookrightarrow L^q(\Omega )$ is compact for $1\le q<p^{\ast }$ but only bounded, never compact, at $q=p^{\ast }$. The dilating bubble $u_{\epsilon }(x)={\epsilon }^{-(n-p)/p}\varphi (x/\epsilon )$ stays bounded in $W^{1,p}$ yet has no $L^{p^{\ast }}$-convergent subsequence: its mass concentrates at a point. Compactness is exactly the gap between the bound and the critical exponent.

• Poincaré needs the boundary condition; Poincaré-Wirtinger needs the mean subtraction. On $W^{1,p}(\Omega )$ without either device the constant $u\equiv 1$ has $Du=0$ but positive $L^p$ norm, so neither inequality can hold with the bare gradient on the right. The zero trace (for Poincaré) and the subtraction of $(u{)}_{\Omega }$ (for Poincaré-Wirtinger) are the two distinct ways to quotient out the constants.

• Boundedness of $\Omega$ is essential. On an unbounded $\Omega$ the Poincaré inequality fails: rescaling a fixed bump $u_{\lambda }(x)=u(x/\lambda )$ makes $\Vert u_{\lambda }{\Vert }_{L^p}/\Vert D{u}_{\lambda }{\Vert }_{L^p}=\lambda \Vert u{\Vert }_{L^p}/\Vert Du{\Vert }_{L^p}\to \infty$ as $\lambda \to \infty$. A finite diameter is what gives the constant; on a slab bounded in one direction the inequality survives, using only the bounded direction.

• Connectedness is essential for Poincaré-Wirtinger. On $\Omega =B(0,1)\cup B(10,1)$ (two disjoint balls) the function equal to $0$ on the first ball and $1$ on the second has $Du=0$ but $\Vert u-(u{)}_{\Omega }{\Vert }_{L^p}>0$. No Poincaré-Wirtinger constant can hold; connectedness forbids exactly this locally-constant-but-globally-varying escape.

Key theorem with proof Intermediate+

Theorem (Rellich-Kondrachov). Let $\Omega \subseteq {\mathbb{R}}^n$ be bounded with $C^1$ boundary and $1\le p<n$. Then for every $q$ with $1\le q<p^{\ast }=np/(n-p)$, the embedding $$W^{1,p}(\Omega )\hookrightarrow \hookrightarrow L^q(\Omega )$$ is compact: every sequence bounded in $W^{1,p}(\Omega )$ has a subsequence converging strongly in $L^q(\Omega )$. (For $p\ge n$ the embedding $W^{1,p}(\Omega )\hookrightarrow \hookrightarrow L^q(\Omega )$ is compact for every $q\in [1,\infty )$.) ^{[Rellich 1930] [Kondrachov 1945] [Evans 2010 §5.7]}.

Proof. Let $(u_m)$ be bounded in $W^{1,p}(\Omega )$, say $\Vert u_m{\Vert }_{W^{1,p}(\Omega )}\le M$.

Step 1 (extend and cut off). Apply the extension operator $E$ of 02.16.02: the functions ${\bar{u}}_m=E{u}_m$ are supported in a fixed bounded open set $V\supset \supset \Omega$ and satisfy $\Vert {\bar{u}}_m{\Vert }_{W^{1,p}({\mathbb{R}}^n)}\le CM$, with ${\bar{u}}_m=u_m$ on $\Omega$. It suffices to show $({\bar{u}}_m)$ is precompact in $L^q(V)$; restriction to $\Omega$ then gives the conclusion. Because $V$ is bounded the uniform-decay-at-infinity condition is automatic, so by the Fréchet-Kolmogorov criterion it remains to establish the uniform translation estimate.

Step 2 (translation estimate for smooth functions). First suppose $v\in C_c^{\infty }({\mathbb{R}}^n)$. For $h\in {\mathbb{R}}^n$, write $v(x+h)-v(x)={\int }_0^{1}Dv(x+th)\cdot hdt$, so by Minkowski's integral inequality and a change of variables $$\Vert {\tau }_{h}v-v{\Vert }_{L^1({\mathbb{R}}^n)}\le |h|{\int }_0^1\Vert Dv(\cdot +th){\Vert }_{L^1}dt=|h|\Vert Dv{\Vert }_{L^1({\mathbb{R}}^n)}.$$ By density of $C_c^{\infty }$ in $W^{1,1}({\mathbb{R}}^n)$ and in $W^{1,p}({\mathbb{R}}^n)$, the same estimate holds for every ${\bar{u}}_m$: $$\Vert {\tau }_h{\bar{u}}_m-{\bar{u}}_m{\Vert }_{L^1({\mathbb{R}}^n)}\le |h|\Vert D{\bar{u}}_m{\Vert }_{L^1(V^{\prime })}\le |h||V^{\prime }{|}^{1/p^{\prime }}\Vert D{\bar{u}}_m{\Vert }_{L^p}\le C|h|M,$$ where $V^{\prime }$ is a fixed bounded neighbourhood containing all the translates and Hölder's inequality converts the $L^1$ gradient norm to the $L^p$ one on the finite-measure set $V^{\prime }$.

Step 3 (upgrade the translation estimate to $L^q$ by interpolation). Fix $q$ with $1\le q<p^{\ast }$. The Sobolev embedding 02.16.01 gives $\Vert {\bar{u}}_m{\Vert }_{L^{p^{\ast }}({\mathbb{R}}^n)}\le C\Vert {\bar{u}}_m{\Vert }_{W^{1,p}}\le CM$, hence $\Vert {\tau }_h{\bar{u}}_m-{\bar{u}}_m{\Vert }_{L^{p^{\ast }}}\le 2CM$ as well. Choose $\theta \in (0,1]$ with $\frac{1}{q}=\theta +\frac{1-\theta }{p^{\ast }}$ (possible exactly because $q<p^{\ast }$, so $q\ge 1$ interpolates between $L^1$ and $L^{p^{\ast }}$). The interpolation inequality for $L^q$ norms gives $$\Vert {\tau }_h{\bar{u}}_m-{\bar{u}}_m{\Vert }_{L^q}\le \Vert {\tau }_h{\bar{u}}_m-{\bar{u}}_m{\Vert }_{L^1}^{\theta }\Vert {\tau }_h{\bar{u}}_m-{\bar{u}}_m{\Vert }_{L^{p^{\ast }}}^{1-\theta }\le (C|h|M{)}^{\theta }(2CM{)}^{1-\theta }=C^{\prime }M|h{|}^{\theta }.$$ The right side is independent of $m$ and tends to zero as $h\to 0$: the family $({\bar{u}}_m)$ has uniformly equicontinuous translation in $L^q$.

Step 4 (apply Fréchet-Kolmogorov and extract). The family $({\bar{u}}_m)$ is bounded in $L^q({\mathbb{R}}^n)$ (by the $L^{p^{\ast }}$ bound and finite measure of $V$), supported in the fixed bounded set $V$ (uniform decay), and uniformly equicontinuous under translation in $L^q$ (Step 3). By the Fréchet-Kolmogorov compactness criterion the family is precompact in $L^q({\mathbb{R}}^n)$, so $({\bar{u}}_m)$ has a subsequence converging in $L^q$. Restricting that subsequence to $\Omega$ gives a subsequence of $(u_m)$ converging in $L^q(\Omega )$. $\square$

Bridge. The compactness is exactly the strict-subcriticality gap of the Sobolev embedding 02.16.01 turned into a compact operator 02.11.05: at $q=p^{\ast }$ the embedding is only bounded, and the dilation symmetry that fixes $p^{\ast }$ also produces the concentrating bubbles that destroy compactness, so the foundational reason the theorem stops short of $p^{\ast }$ is the same scaling invariance that forced $p^{\ast }$ in the first place. The engine is the translation estimate $\Vert {\tau }_{h}u-u{\Vert }_{L^1}\le |h|\Vert Du{\Vert }_{L^1}$, which is the integral form of the fundamental theorem of calculus and is dual to the trace estimate of 02.16.02 — there one integrated the normal derivative to the boundary, here one integrates the gradient along the translation vector. Putting these together, the central insight is that a bounded slope budget converts a bounded family into a precompact one, and this is exactly what the direct method of the calculus of variations needs: it builds toward the existence of weak solutions in 02.16.04, where a minimizing sequence's weak $W^{1,p}$ limit is upgraded to a strong $L^q$ limit precisely by Rellich-Kondrachov, and it appears again in the spectral theory of the Laplacian, whose discrete spectrum is the compactness of this same embedding read through 02.11.05.

Exercises Intermediate+

Advanced results Master

The compactness theorem and the Poincaré inequalities organize a larger structure: the sharp range of compact embeddings across the Sobolev scale, the spectral identification of the optimal constants, the failure profile at the critical exponent, the role of the embedding in Fredholm theory and the direct method, and the geometric content of the Poincaré constant. Each refines the translation-estimate and quotient-out-the-constants arguments of the Intermediate tier.

Theorem 1 (full Rellich-Kondrachov scale; Rellich 1930, Kondrachov 1945). Let $\Omega$ be bounded with $C^1$ boundary, $1\le p<\infty$, $k\in \mathbb{N}$. If $kp<n$, the embedding $W^{k,p}(\Omega )\hookrightarrow \hookrightarrow L^q(\Omega )$ is compact for $1\le q<p_k^{\ast }$, where $\frac{1}{p_k^{\ast }}=\frac{1}{p}-\frac{k}{n}$; if $kp=n$ it is compact into $L^q$ for all $q<\infty$; if $kp>n$ it is compact into $C^{0,\gamma }(\bar{\Omega })$ for $0\le \gamma <k-n/p$, and into $C^{m,\gamma }$ for the appropriate $m$ when $k-n/p>1$ ^{[Rellich 1930] [Kondrachov 1945] [Adams-Fournier 2003]}. The strictness $q<p_k^{\ast }$ is sharp: at the endpoint $q=p_k^{\ast }$ the embedding is bounded (the Sobolev embedding of 02.16.01) but loses compactness to concentration. The compact embeddings are exactly the bounded ones at a strictly smaller integrability index, the difference between the two being the scaling-invariant endpoint.

Theorem 2 (spectral form; Poincaré 1890, Courant-Fischer). For $p=2$ and bounded $\Omega$, the optimal Poincaré constant on $H_0^1(\Omega )$ is ${\lambda }_1(\Omega {)}^{-1/2}$, where $${\lambda }_1(\Omega )=\underset{u\in H_0^1(\Omega )\setminus \{0\}}{\min }\frac{\Vert Du{\Vert }_{L^2}^2}{\Vert u{\Vert }_{L^2}^2}$$ is the first Dirichlet eigenvalue of $-\Delta$, with the minimizing $u$ the first Dirichlet eigenfunction ^{[Poincaré 1890]}. The optimal Poincaré-Wirtinger constant is ${\mu }_2(\Omega {)}^{-1/2}$, where ${\mu }_2$ is the first nonzero Neumann eigenvalue (the spectral gap above the constant Neumann eigenfunction). That the minimum is attained — rather than merely an infimum — is precisely Rellich-Kondrachov: a minimizing sequence is bounded in $H_0^1$, so a subsequence converges strongly in $L^2$, and lower semicontinuity of the Dirichlet energy promotes the weak $H^1$ limit to a genuine minimizer. The discreteness of the entire Dirichlet spectrum follows by applying 02.11.05 to the compact resolvent $(-\Delta {)}^{-1}$.

Theorem 3 (Rellich-Kondrachov and the Fredholm alternative). The compactness of $W^{1,2}(\Omega )\hookrightarrow \hookrightarrow L^2(\Omega )$ makes the solution operator of a uniformly elliptic problem $Lu=f$ a compact perturbation of the identity on $L^2(\Omega )$: writing $L=-\Delta +(L+\Delta )$ and inverting the leading part, the lower-order terms factor through the compact embedding, so $(I-K)$ with $K$ compact governs solvability. The Fredholm alternative of 02.11.05 then applies: either $Lu=f$ has a unique weak solution for every $f$, or the homogeneous problem $Lu=0$ has a finite-dimensional solution space and $Lu=f$ is solvable exactly when $f$ is orthogonal to the cokernel. Compactness is the single structural input converting an infinite-dimensional boundary-value problem into the linear algebra of finite-rank obstructions.

Theorem 4 (Maz'ya capacitary and measure-theoretic characterizations). The Poincaré and compactness phenomena persist far beyond $C^1$ domains. Maz'ya's theory characterizes the domains and measures for which $W^{1,p}(\Omega )\hookrightarrow L^q(\Omega ,\mu )$ is bounded or compact in terms of isocapacitary and isoperimetric inequalities: boundedness corresponds to a capacity-volume inequality $\mu (K)\le C{\mathrm{cap}}_p(K{)}^{q/p}$ uniformly over compact $K$, and compactness to the same with the constant tending to zero on small sets ^{[Maz'ya 2011]}. For $p=1$ the relevant inequality is isoperimetric, tying the Poincaré constant to the Cheeger constant $h(\Omega )={\inf }_A\frac{|\partial A\cap \Omega |}{|A|}$ via ${\lambda }_1\ge h^2/4$ (Cheeger's inequality). Domains with sufficiently sharp outward cusps fail the extension property of 02.16.02 and can lose both the compact embedding and the Poincaré inequality.

Theorem 5 (concentration-compactness; the failure at $p^{\ast }$ made quantitative). At the critical exponent the loss of compactness is not arbitrary but structured: a $W^{1,p}$-bounded sequence with $u_m\rightharpoonup u$ weakly fails to converge strongly in $L^{p^{\ast }}$ only through a countable sum of concentrating bubbles, $|D{u}_m{|}^p\rightharpoonup |Du{|}^p+{\sum }_j{\nu }_j{\delta }_{x_j}$ and $|u_m{|}^{p^{\ast }}\rightharpoonup |u{|}^{p^{\ast }}+{\sum }_j{\mu }_j{\delta }_{x_j}$ as measures, with the masses linked by the Sobolev inequality ${\mu }_j^{p/p^{\ast }}\le S^{-1}{\nu }_j$. This is the concentration-compactness principle: strong $L^{p^{\ast }}$ convergence holds if and only if no mass escapes into Dirac bubbles, and the bubbles are dilates of the Aubin-Talenti extremals of 02.16.01. It restores a usable substitute for compactness in critical variational problems, where Rellich-Kondrachov alone is unavailable.

Synthesis. The compactness theorem is the foundational reason the direct method of the calculus of variations produces actual minimizers rather than mere infima, and the whole structure is generated by the same single principle as the Sobolev inequalities of 02.16.01: a derivative integrated along a segment controls the function, here in the translation form $\Vert {\tau }_{h}u-u{\Vert }_{L^1}\le |h|\Vert Du{\Vert }_{L^1}$ that feeds the Fréchet-Kolmogorov criterion. This translation estimate is dual to the normal-direction integration of the trace theorem 02.16.02, one estimate run along the translation vector and the other run inward to the boundary. The compact embedding is exactly the strictly-subcritical Sobolev embedding read as a compact operator through 02.11.05; the Poincaré inequality is this same control with the constants quotiented out by the zero-boundary condition, and the Poincaré-Wirtinger inequality quotients them out instead by subtracting the mean — putting these together, both optimal constants are eigenvalues, the Dirichlet ${\lambda }_1$ and the Neumann ${\mu }_2$. The central insight at this stage is that the attainment of those eigenvalues, rather than their being mere infima, is itself the compactness theorem applied to a minimizing sequence: this is exactly the upgrade from weak to strong convergence that the direct method demands.

The central insight is that a bounded slope budget on a bounded region is precisely a precompactness certificate, and this is exactly why the theorem stops at $p^{\ast }$: the dilation symmetry that forces the critical exponent also manufactures the concentrating bubbles of Theorem 5 that no slope budget can prevent. From Poincaré's 1890 inequality through Rellich's 1930 mean-convergence lemma and Kondrachov's 1945 extension across the subcritical range to the Maz'ya capacitary theory and the concentration-compactness method, the subject is one continuous refinement of a single fundamental-theorem-of-calculus estimate, generalised until it became the compactness engine of elliptic Fredholm theory and the spectral theory of the Laplacian, and it appears again in 02.16.04 as the device that turns a minimizing sequence into a weak solution.

Full proof set Master

Proposition 1 (Poincaré inequality on a bounded domain, general constant). Let $\Omega \subseteq {\mathbb{R}}^n$ be bounded with diameter $d$. Then $\Vert u{\Vert }_{L^p(\Omega )}\le d\Vert Du{\Vert }_{L^p(\Omega )}$ for all $u\in W_0^{1,p}(\Omega )$, $1\le p<\infty$.

Proof. By density it suffices to treat $u\in C_c^{\infty }(\Omega )$, extended by zero to ${\mathbb{R}}^n$. Enclose $\Omega$ in a slab $\{a<x_n<a+d\}$ of width $d$ (possible since $\mathrm{diam}\Omega =d$). For each fixed $x^{\prime }=(x_1,\dots ,x_{n-1})$, the function $t\mapsto u(x^{\prime },t)$ vanishes for $t\le a$, so $u(x^{\prime },x_n)={\int }_a^{x_n}{D}_{n}u(x^{\prime },t)dt$. By Hölder with conjugate exponents $p,p^{\prime }$, $$|u(x^{\prime },x_n){|}^p\le (x_n-a{)}^{p/p^{\prime }}{\int }_a^{a+d}|D_{n}u(x^{\prime },t){|}^{p}dt\le d^{p-1}{\int }_a^{a+d}|D_{n}u(x^{\prime },t){|}^{p}dt.$$ Integrate in $x_n$ over the slab width $d$: ${\int }_a^{a+d}|u(x^{\prime },x_n){|}^{p}d{x}_n\le d^p{\int }_a^{a+d}|D_{n}u(x^{\prime },t){|}^{p}dt$. Integrating over $x^{\prime }\in {\mathbb{R}}^{n-1}$ and using $|D_{n}u|\le |Du|$ yields $\Vert u{\Vert }_{L^p(\Omega )}^p\le d^p\Vert Du{\Vert }_{L^p(\Omega )}^p$, i.e. $\Vert u{\Vert }_{L^p}\le d\Vert Du{\Vert }_{L^p}$. The constant uses only the slab width, so boundedness in a single direction suffices. $\square$

Proposition 2 (Poincaré-Wirtinger inequality). Let $\Omega$ be bounded, connected, with $C^1$ boundary, $1\le p<\infty$. There is $C=C(n,p,\Omega )$ with $\Vert u-(u{)}_{\Omega }{\Vert }_{L^p(\Omega )}\le C\Vert Du{\Vert }_{L^p(\Omega )}$ for all $u\in W^{1,p}(\Omega )$.

Proof. Argue by contradiction using Rellich-Kondrachov. If the inequality fails for every $C$, choose $u_k\in W^{1,p}(\Omega )$ with $\Vert u_k-(u_k{)}_{\Omega }{\Vert }_{L^p}>k\Vert D{u}_k{\Vert }_{L^p}$. Normalize $v_k=(u_k-(u_k{)}_{\Omega })/\Vert u_k-(u_k{)}_{\Omega }{\Vert }_{L^p}$, so $(v_k{)}_{\Omega }=0$, $\Vert v_k{\Vert }_{L^p}=1$, and $\Vert D{v}_k{\Vert }_{L^p}<1/k$. Then $\Vert v_k{\Vert }_{W^{1,p}}^p=1+\Vert D{v}_k{\Vert }_{L^p}^p\le 2$ for $k\ge 1$, so $(v_k)$ is bounded in $W^{1,p}(\Omega )$. By the Key Theorem (Rellich-Kondrachov, taking $q=p<p^{\ast }$, valid since $\Omega$ is a $C^1$ extension domain) a subsequence converges strongly in $L^p(\Omega )$ to some $v$ with $\Vert v{\Vert }_{L^p}=1$ and $(v{)}_{\Omega }=\lim (v_k{)}_{\Omega }=0$. For every $\phi \in C_c^{\infty }(\Omega )$ and each $i$, ${\int }_{\Omega }v{D}_i\phi dx={\lim }_k{\int }_{\Omega }{v}_k{D}_i\phi dx=-{\lim }_k{\int }_{\Omega }(D_i{v}_k)\phi dx=0$, the last equality since $\Vert D{v}_k{\Vert }_{L^p}\to 0$ and $\phi$ is bounded. Thus $Dv=0$ weakly. On the connected open set $\Omega$ a weak gradient that vanishes forces $v$ constant a.e.; with $(v{)}_{\Omega }=0$ this constant is $0$, contradicting $\Vert v{\Vert }_{L^p}=1$. The inequality therefore holds for some finite $C$. $\square$

Proposition 3 (translation estimate in $L^p$). For $u\in W^{1,p}({\mathbb{R}}^n)$, $1\le p<\infty$, and $h\in {\mathbb{R}}^n$, $\Vert {\tau }_{h}u-u{\Vert }_{L^p({\mathbb{R}}^n)}\le |h|\Vert Du{\Vert }_{L^p({\mathbb{R}}^n)}$.

Proof. For $u\in C_c^{\infty }({\mathbb{R}}^n)$, $u(x+h)-u(x)={\int }_0^{1}Du(x+th)\cdot hdt$. By Minkowski's integral inequality, viewing the difference as an integral over $t$ of the $L^p$-valued map $x\mapsto Du(x+th)\cdot h$, $$\Vert {\tau }_{h}u-u{\Vert }_{L^p}\le {\int }_0^1\Vert Du(\cdot +th)\cdot h{\Vert }_{L^p}dt\le |h|{\int }_0^1\Vert Du(\cdot +th){\Vert }_{L^p}dt=|h|\Vert Du{\Vert }_{L^p},$$ using $|Du(\cdot +th)\cdot h|\le |h||Du(\cdot +th)|$ pointwise and translation invariance of the $L^p$ norm. For general $u\in W^{1,p}$, approximate by $u_m\in C_c^{\infty }$ with $u_m\to u$, $D{u}_m\to Du$ in $L^p$; translation is an $L^p$-isometry, so both sides converge and the estimate persists. $\square$

Proposition 4 (discreteness of the Dirichlet spectrum via compactness). Let $\Omega$ be bounded with $C^1$ boundary. The operator $-\Delta$ on $\Omega$ with zero Dirichlet boundary data has compact resolvent, hence a discrete spectrum $0<{\lambda }_1\le {\lambda }_2\le \cdots \to \infty$ with an $L^2(\Omega )$-orthonormal basis of eigenfunctions.

Proof. For $f\in L^2(\Omega )$ the weak problem ${\int }_{\Omega }Du\cdot Dv={\int }_{\Omega }fv$ for all $v\in H_0^1(\Omega )$ has, by the Poincaré inequality of Proposition 1 (which makes $\Vert Du{\Vert }_{L^2}$ an equivalent norm on $H_0^1$) and the Riesz representation theorem 02.11.05, a unique solution $u=:Gf\in H_0^1(\Omega )$ with $\Vert u{\Vert }_{H_0^1}\le C\Vert f{\Vert }_{L^2}$. Thus $G:L^2(\Omega )\to H_0^1(\Omega )$ is bounded. Composing with the compact inclusion $H_0^1(\Omega )\hookrightarrow \hookrightarrow L^2(\Omega )$ from the Key Theorem ($q=2<2^{\ast }$), the operator $G:L^2(\Omega )\to L^2(\Omega )$ is compact. It is also self-adjoint and positive (from the symmetric, coercive bilinear form). The spectral theorem for compact self-adjoint operators 02.11.05 gives an orthonormal eigenbasis $\{{\varphi }_k\}$ with eigenvalues ${\sigma }_k\downarrow 0$; setting ${\lambda }_k=1/{\sigma }_k$ gives $-\Delta {\varphi }_k={\lambda }_k{\varphi }_k$ with ${\lambda }_k\uparrow \infty$, and ${\lambda }_1>0$ by the Poincaré inequality. $\square$

Connections Master

• The compact embedding is the strictly-subcritical Sobolev embedding of 02.16.01 read as a compact operator: the boundedness $W^{1,p}(\Omega )\hookrightarrow L^{p^{\ast }}(\Omega )$ proved there is upgraded here to compactness for every $q<p^{\ast }$, and the concentration phenomenon that obstructs compactness at $q=p^{\ast }$ is exactly the Aubin-Talenti bubbling of that unit's sharp-constant theory. This unit owns the compactness; 02.16.01 owns the boundedness and the sharp constant.

• The extension operator $E:W^{1,p}(\Omega )\to W^{1,p}({\mathbb{R}}^n)$ of 02.16.02 is the indispensable first step of the compactness proof: it transfers the bounded sequence to a fixed bounded set in ${\mathbb{R}}^n$ where the Fréchet-Kolmogorov criterion can be run, and it is the device whose failure on cuspidal domains makes both the compact embedding and the Poincaré inequality fail there. The trace operator of that unit also supplies the boundary control that the Poincaré inequality complements in the interior.

• The abstract framework — compact operators, the Fredholm alternative, and the spectral theorem for compact self-adjoint operators — is supplied by 02.11.05; this unit provides the single concrete compact operator (the Sobolev embedding) that makes elliptic boundary-value problems Fredholm and gives $-\Delta$ a discrete spectrum, so the abstract theory of 02.11.05 acquires its principal application here.

• The $L^p$ apparatus on which the translation estimate and the interpolation step rest — Hölder's inequality, Minkowski's integral inequality, and the interpolation of $L^q$ norms between $L^1$ and $L^{p^{\ast }}$ — is developed in 02.07.06; the mollification argument behind Fréchet-Kolmogorov is the same approximate-identity machinery used there.

• The compactness theorem is the existence engine for weak solutions of elliptic boundary-value problems in 02.16.04: the direct method minimizes an energy over $W^{1,p}$, and Rellich-Kondrachov is exactly what upgrades a minimizing sequence's weak limit to a strong $L^q$ limit, while the Poincaré inequality supplies the coercivity that keeps the minimizing sequence bounded.

Historical & philosophical context Master

Henri Poincaré introduced the inequality bearing his name in his 1890 American Journal of Mathematics memoir on the partial differential equations of mathematical physics ^{[Poincaré 1890]}, where he needed to bound a function by its gradient to control the eigenvalue problems arising in heat conduction and potential theory. The one-dimensional sharp form, bounding a periodic mean-zero function by its derivative with the optimal constant attained by the first trigonometric mode, is attributed to Wilhelm Wirtinger and was popularized through Wilhelm Blaschke's 1916 Kreis und Kugel ^{[Wirtinger 1916]}; the combined name Poincaré-Wirtinger reflects this dual ancestry in potential theory and in the isoperimetric problem.

The compactness theorem originates with Franz Rellich's 1930 Göttingen note Ein Satz über mittlere Konvergenz ^{[Rellich 1930]}, which proved that an $H^1$-bounded sequence on a bounded domain has an $L^2$-convergent subsequence — the case $p=q=2$. Vladimir Kondrachov extended the result across the full subcritical range of exponents in his 1945 Doklady note ^{[Kondrachov 1945]}, establishing the compact embedding $W^{1,p}\hookrightarrow L^q$ for $q<p^{\ast }$. The compactness criterion underlying the modern proof is the Fréchet-Kolmogorov / Kolmogorov-Riesz theorem, whose threads run from Maurice Fréchet's 1907 work on compact sets of functions ^{[Fréchet 1907]} through Andrei Kolmogorov's 1931 Göttingen note characterizing precompact families in $L^p$ by mean continuity ^{[Kolmogorov 1931]} and Marcel Riesz's 1933 Acta Szeged completion to general exponents ^{[Riesz 1933]}; the consolidated statement and its history are surveyed by Harald Hanche-Olsen and Helge Holden ^{[Hanche-Olsen-Holden 2010]}. Vladimir Maz'ya's capacitary theory later identified the exact geometric conditions on domains and measures for both boundedness and compactness, completing the picture begun by Rellich and Kondrachov.

Bibliography Master

@article{Rellich1930,
  author  = {Rellich, Franz},
  title   = {Ein Satz \"uber mittlere Konvergenz},
  journal = {Nachrichten von der Gesellschaft der Wissenschaften zu G\"ottingen, Mathematisch-Physikalische Klasse},
  year    = {1930},
  pages   = {30--35}
}

@article{Kondrachov1945,
  author  = {Kondrachov, Vladimir I.},
  title   = {Sur certaines propri\'et\'es des fonctions dans l'espace $L^p$},
  journal = {Doklady Akademii Nauk SSSR},
  volume  = {48},
  year    = {1945},
  pages   = {535--538}
}

@article{Poincare1890,
  author  = {Poincar\'e, Henri},
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  year    = {1890},
  pages   = {211--294}
}

@incollection{Wirtinger1916,
  author    = {Blaschke, Wilhelm},
  title     = {Kreis und Kugel},
  publisher = {Veit, Leipzig},
  year      = {1916},
  note      = {\S1.6; the one-dimensional Wirtinger inequality}
}

@article{Kolmogorov1931,
  author  = {Kolmogorov, Andrei N.},
  title   = {\"Uber Kompaktheit der Funktionenmengen bei der Konvergenz im Mittel},
  journal = {Nachrichten von der Gesellschaft der Wissenschaften zu G\"ottingen},
  year    = {1931},
  pages   = {60--63}
}

@article{Riesz1933,
  author  = {Riesz, Marcel},
  title   = {Sur les ensembles compacts de fonctions sommables},
  journal = {Acta Scientiarum Mathematicarum (Szeged)},
  volume  = {6},
  year    = {1933},
  pages   = {136--142}
}

@article{HancheOlsenHolden2010,
  author  = {Hanche-Olsen, Harald and Holden, Helge},
  title   = {The Kolmogorov-Riesz compactness theorem},
  journal = {Expositiones Mathematicae},
  volume  = {28},
  year    = {2010},
  pages   = {385--394}
}

@book{Mazya2011,
  author    = {Maz'ya, Vladimir},
  title     = {Sobolev Spaces, with Applications to Elliptic Partial Differential Equations},
  edition   = {2},
  publisher = {Springer Grundlehren 342},
  year      = {2011}
}