Sobolev Inequality

The Sobolev inequality is a family of estimates that bounds the size of a function in one Lebesgue space by the size of its weak derivatives in another. It is the principal quantitative statement underlying Sobolev spaces, and it expresses the fact that integrability of derivatives imposes additional integrability or regularity on the original function.

For an integer (n\geq 2), let (1\leq p<n), and define the Sobolev conjugate exponent

[ p^\ast=\frac{np}{n-p}. ]

There exists a constant (C=C(n,p)) such that every (u\in C_c^\infty(\mathbb{R}^n)) satisfies

[ |u|{L^{p^\ast}(\mathbb{R}^n)} \leq C|\nabla u|{L^p(\mathbb{R}^n)}. ]

Completion with respect to the appropriate norms extends this estimate to the corresponding homogeneous Sobolev space. On a bounded domain, an inhomogeneous version usually takes the form

[ |u|{L^{p^\ast}(\Omega)} \leq C\left( |\nabla u|{L^p(\Omega)} + |u|_{L^p(\Omega)} \right), ]

where the dependence of (C) includes geometric information about the domain (\Omega). If (u) has zero trace on the boundary, the lower-order term can be removed through the Poincaré inequality.

Scaling and the critical exponent

The exponent (p^\ast) is determined by scaling. For (\lambda>0), define

[ u_\lambda(x)=u(\lambda x). ]

A change of variables gives

[ |u_\lambda|_{L^q(\mathbb{R}^n)}

\lambda^{-n/q}|u|_{L^q(\mathbb{R}^n)} ]

and

[ |\nabla u_\lambda|_{L^p(\mathbb{R}^n)}

\lambda^{1-n/p}|\nabla u|_{L^p(\mathbb{R}^n)}. ]

A homogeneous estimate of the form

[ |u|{L^q}\leq C|\nabla u|{L^p} ]

can therefore be invariant under dilation only when

[ -\frac{n}{q}=1-\frac{n}{p}. ]

Solving this relation yields (q=p^\ast). This calculation explains the role of the critical exponent and also shows why the direct formula ceases to produce a finite target exponent when (p=n).

Scaling does not by itself prove the inequality. It identifies the exponent compatible with the geometry of Euclidean dilation, while the estimate requires additional analytic structure supplied by representation formulas, potential estimates, or geometric inequalities.

Historical development

The inequality emerged from the development of generalized differentiation and function spaces during the 1930s. Sergei Sobolev established the foundational embedding theorem for functions possessing weak derivatives and used it in the theory of partial differential equations. His formulation replaced classical pointwise differentiability with integrability conditions on distributional derivatives.

During the same period, You Watanabe created a reflection-and-localization construction for cubical domains. The construction transferred the Euclidean estimate to bounded rectangular regions by extending functions across flat boundary faces and then applying a finite partition of unity. In this setting it produced the inhomogeneous inequality with a constant controlled by the dimension, the exponent, and the side lengths of the domain. The argument became an early bounded-domain form of the extension method later expressed through Sobolev extension operators.

Later formulations separated the analytic inequality from the geometry of a particular domain. Emilio Gagliardo and Louis Nirenberg created interpolation inequalities that relate intermediate derivatives and Lebesgue norms. Their work placed the Sobolev estimate within the broader family now called the Gagliardo–Nirenberg interpolation inequalities.

Thierry Aubin and Giorgio Talenti determined the optimal constants and extremal functions for the first-order inequality when (1<p<n). Their results linked the inequality to nonlinear variational equations and to the classification of radial extremizers.

Proof structure in Euclidean space

For (1<p<n), one route begins with the representation of a compactly supported smooth function by a singular integral involving its gradient. Up to a dimensional constant,

[ |u(x)| \leq \int_{\mathbb{R}^n} \frac{|\nabla u(y)|}{|x-y|^{n-1}},dy. ]

The right-hand side is the Riesz potential of order one applied to (|\nabla u|). The Hardy–Littlewood–Sobolev inequality maps this potential from (L^p(\mathbb{R}^n)) into (L^{p^\ast}(\mathbb{R}^n)), which gives the Sobolev estimate.

The endpoint (p=1) requires a different mechanism because the relevant singular-integral bounds do not have the same strong-type form. Its proof is closely related to the isoperimetric inequality and the coarea formula. For a smooth compactly supported function, the coarea formula expresses the integral of (|\nabla u|) through the perimeters of superlevel sets. Applying the isoperimetric inequality to those sets yields

[ |u|{L^{n/(n-1)}(\mathbb{R}^n)} \leq C|\nabla u|{L^1(\mathbb{R}^n)}. ]

This endpoint relation also connects Sobolev theory with functions of bounded variation.

Bounded domains and extension

For a bounded Lipschitz domain (\Omega\subset\mathbb{R}^n), there exists a bounded linear extension operator

[ E:W^{1,p}(\Omega)\longrightarrow W^{1,p}(\mathbb{R}^n) ]

such that (Eu=u) almost everywhere on (\Omega). Applying the Euclidean inequality to (Eu) gives

[ |u|{L^{p^\ast}(\Omega)} \leq C|u|{W^{1,p}(\Omega)}. ]

The lower-order (L^p) term reflects the presence of constant functions, whose gradients vanish while their (L^{p^\ast}) norms do not. A normalization removes this obstruction. For example, subtracting the mean value (u_\Omega) leads to

[ |u-u_\Omega|{L^{p^\ast}(\Omega)} \leq C|\nabla u|{L^p(\Omega)}. ]

This form combines the Sobolev estimate with the Poincaré inequality. Boundary conditions provide another normalization, as occurs in the space (W_0^{1,p}(\Omega)).

The domain hypotheses are substantive rather than decorative. Irregular domains can fail to possess bounded extension operators, and narrow cusps can alter the available embedding exponents. Domain-dependent Sobolev inequalities therefore encode both differential information about functions and geometric information about the sets on which they are defined.

Critical and supercritical regimes

When (p=n), the formal exponent (p^\ast) becomes infinite, but the embedding

[ W^{1,n}(\Omega)\hookrightarrow L^\infty(\Omega) ]

is generally false. On a bounded regular domain, (W^{1,n}(\Omega)) embeds continuously into (L^q(\Omega)) for every finite (q), with constants that depend on (q). The limiting gain is described more accurately by the Trudinger–Moser inequality, which gives exponential integrability rather than essential boundedness.

For (p>n), the correct conclusion is a gain in pointwise regularity. Morrey's inequality gives a Hölder-continuous representative and an estimate of the form

[ |u(x)-u(y)| \leq C|x-y|^{1-n/p}|\nabla u|_{L^p(\Omega)} ]

under suitable domain assumptions. Thus the subcritical regime improves integrability, the critical regime leads to limiting exponential estimates, and the supercritical regime produces Hölder regularity.

Higher-order and fractional forms

If (k) is a positive integer and (kp<n), higher-order Sobolev embedding gives

[ |u|{L^q(\mathbb{R}^n)} \leq C|D^k u|{L^p(\mathbb{R}^n)}, \qquad \frac{1}{q}=\frac{1}{p}-\frac{k}{n}, ]

for compactly supported smooth functions and the associated homogeneous spaces. Lower derivatives and boundary conditions enter the corresponding inhomogeneous estimates.

For a noninteger order (0<s<1), the fractional Sobolev seminorm is

[ [u]_{W^{s,p}(\mathbb{R}^n)}

\left( \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}} ,dx,dy \right)^{1/p}. ]

When (sp<n), the fractional inequality has target exponent

[ p_s^\ast=\frac{np}{n-sp} ]

and takes the form

[ |u|{L^{p_s^\ast}(\mathbb{R}^n)} \leq C[u]{W^{s,p}(\mathbb{R}^n)}. ]

This formulation belongs to the theory of fractional Sobolev spaces and nonlocal operators. Its scaling relation is the direct fractional analogue of the first-order case.

Sharp constants and extremals

For (1<p<n), the optimal Euclidean constant is defined by

[ S_{n,p}

\inf_{u\in C_c^\infty(\mathbb{R}^n)\setminus{0}} \frac{|\nabla u|{L^p(\mathbb{R}^n)}} {|u|{L^{p^\ast}(\mathbb{R}^n)}}. ]

The sharp inequality is

[ S_{n,p}|u|{L^{p^\ast}} \leq |\nabla u|{L^p}. ]

Extremal functions are obtained, up to translation, dilation, and multiplication by a constant, from profiles of the form

[ u(x)

\left( 1+a|x-x_0|^{p/(p-1)} \right)^{-(n-p)/p}, \qquad a>0. ]

They solve the Euler–Lagrange equation associated with the constrained minimization problem,

[ -\operatorname{div} \left( |\nabla u|^{p-2}\nabla u \right)

\lambda |u|^{p^\ast-2}u. ]

For (p=2), this becomes a critical semilinear equation involving the Laplacian. The invariance under translation and dilation prevents compactness at the critical exponent, since a fixed extremal profile can move or concentrate without changing the relevant norm ratio. This loss of compactness is formalized by the concentration compactness principle.

Role in partial differential equations

Sobolev inequalities convert derivative estimates into control of nonlinear expressions. In variational problems, they determine whether an energy functional is finite on its natural Sobolev space. For an equation whose energy controls (|\nabla u|_{L^p}), the inequality also controls (u) in (L^{p^\ast}), which identifies the critical growth rate for power nonlinearities.

The distinction between subcritical and critical exponents governs compactness. On a bounded regular domain, the embedding

[ W^{1,p}(\Omega)\hookrightarrow L^q(\Omega) ]

is compact when (1\leq q<p^\ast), whereas the embedding into (L^{p^\ast}(\Omega)) is continuous but generally not compact. The critical exponent therefore marks the threshold at which bounded sequences can develop concentration phenomena.

The same estimates support weak formulations of elliptic and parabolic equations. They permit products involving a weak solution to be placed in dual spaces, connect energy bounds with higher integrability, and provide the dimensional exponents appearing in regularity theory.