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Take $u\in W_0^{1,p}(U)$ and extend it by zero outside $U$. The zero extension of W01 belongs to $W^{1,p}({\mathbb{R}}^n)$ and the Sobolev inequality givesBecause $U$ has finite measure, the Holder inequality givesThusThe reverse estimate follows directly from $\Vert Du{\Vert }_{L^p}\le \Vert u{\Vert }_{W^{1,p}}$. After adjusting constants,This is the Poincaré inequality on $W_0^{1,p}(U)$.

For $1\le p<n$, define the Sobolev conjugate exponentThe Sobolev inequality, also called the Gagliardo--Nirenberg--Sobolev inequality, states that there is a constant $C=C(n,p)$ such thatfor every $u\in C_c^{\infty }({\mathbb{R}}^n)$, and hence by completion for every $u\in W^{1,p}({\mathbb{R}}^n)$ for which the right formulation applies.

Interpolation between $L^2$ and $L^6$, followed by the three-dimensional Sobolev inequality, givesApply this with $v=P_{N}w$ and use part i:Orthogonal projection is contractive in $H$, soThe Sobolev constant is dimensionless after using the periodic-domain normalization, so the estimate is scale invariant.

Let a solenoidal flow be confined to a sphere $V$, with the insulating boundary condition for the radial magnetic scalar. Define $N={\int }_V{P}^2$, $D={\int }_{{\mathbb{R}}^3}|\nabla P{|}^2$, $M={\int }_{{\mathbb{R}}^3}|\mathbf{B}{|}^2$ and $q={\max }_V|\mathbf{u}\cdot \mathbf{x}|$. Integration by parts inside and outside the sphere givesThe Cauchy-Schwarz inequality therefore makes $q^2\ge {\eta }^{2}D/M$ necessary whenever $N^{\prime }\ge 0$ and $D>0$. A uniform strict violation with a positive gap forces decay, using a Sobolev inequality to bound $N$ by a constant times $D$. A bounded statistically steady field instead requires $q_{\ast }^2\ge {\eta }^2⟨D⟩/⟨M⟩$, with $q_{\ast }={\sup }_{t}q(t)$. This is a necessary condition for dynamo action, not a sufficient criterion or a pointwise assertion at every instant of an arbitrary time-dependent solution.

// Target: analysis.bigb

For real $f\in L^2({\mathbb{R}}^3)$, this energy functional on $H^1({\mathbb{R}}^3)$ is coercive and attains its minimum by the direct method in the calculus of variations. The nonnegative potential term is weakly lower semicontinuous by local Sobolev compactness gives lower semicontinuity of a nonnegative integral. Its Euler-Lagrange equation is $-\Delta u+u+\sin u=f$ in the weak solution sense. The extra linear restoring term screens the Sine-Gordon equation nonlinearity. The scalar potential $V(s)=s^2/2+1-\cos s$ is convex because $V^{\prime \prime }(s)=1+\cos s\ge 0$; hence any weak critical point is a minimizer. The elliptic regularity estimate for $-\Delta +1$, together with $|\sin u|\le |u|$, gives $u\in H^2$. The Sobolev inequality then puts $u$ in $W^{1,6}$, so Morrey's inequality and uniformly continuous integrable functions vanish at infinity give a continuous representative tending to zero.

For $1\le p<n$, the Sobolev conjugate exponent $p^{\ast }$ is defined byThe first-order Sobolev inequality maps one derivative in $L^p({\mathbb{R}}^n)$ to the function in $L^{p^{\ast }}({\mathbb{R}}^n)$.