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A plane wave in a homogeneous linear medium satisfies

A group is nilpotent when it has a finite central series, equivalently when its lower central series reaches the identity.

A weak solution satisfies an integrated identity in which derivatives are transferred to test functions. It can therefore exist with fewer classical derivatives.

A classical solution has enough ordinary derivatives for the differential equation and its data to hold pointwise. For the heat equation, $u_t$ and $u_{xx}$ should be continuous in the interior, with the specified initial and boundary traces. A weak solution instead imposes an integrated identity, allowing less regular data; uniqueness must always be stated for the solution class used.

This is the two-dimensional form of Ladyzhenskaya's inequality. For $u\in C_c^{\infty }({\mathbb{R}}^2)$,Apply the fundamental theorem of calculus to $|u{|}^2$ in each coordinate, multiply the resulting one-dimensional bounds, integrate, and use Cauchy--Schwarz.

If $s>d/2$, then $H^s({\mathbb{R}}^d)$ embeds continuously into bounded continuous functions.

If $({\Delta }^m+1)u=f$ distributionally on ${\mathbb{R}}^n$, $m$ is even, and $f\in H^r$, thenand $u\in H^{r+2m}({\mathbb{R}}^n)$.

This theorem describes the continuous extension of the Trace operator. For $s>1/2$, restriction to the coordinate hyperplane extends uniquely to a bounded linear mapIn Fourier variables, Cauchy--Schwarz bounds the integral over the normal frequency because $(1+t^2{)}^{-s}$ is integrable exactly when $s>1/2$.

A sequence bounded in $H^2({\mathbb{R}}^n)$ whose members are supported in one bounded set has a subsequence converging strongly in $H^1({\mathbb{R}}^n)$. Apply Rellich compactness simultaneously to the functions and their first derivatives.

For every real $s$,is an isomorphism. Its inverse is the Fourier multiplier $(1+|\xi {|}^2{)}^{-1}$.

For every multi-index $\alpha$ and real $s$,This follows from the Fourier multiplier $(i\xi {)}^{\alpha }$.

A Poincare inequality controls a function by its derivatives after the constant ambiguity has been removed. For example, on a bounded connected Lipschitz domain,

The Campanato seminorm measures mean oscillation byOn regular bounded domains, ${\mathcal{L}}^{p,n+p\alpha }=C^{0,\alpha }$ for $0<\alpha \le 1$, with equivalent norms.

For prescribed boundary data $\phi$, the affine Sobolev spaceconsists of the $W^{1,p}$ functions whose trace equals the trace of $\phi$. It is the natural admissible set for a variational Dirichlet problem.

For a boundary portion ${\Gamma }_D\subseteq \partial U$, the spaceis the kernel of the restricted trace map, hence a closed vector subspace of $H^1(U)$.

For an open set $U\subseteq {\mathbb{R}}^n$ and $1\le p\le \infty$, the first-order Sobolev space iswhere each $D_{i}u$ is a weak derivative. Its standard norm is $\Vert u{\Vert }_{W^{1,p}}=\Vert u{\Vert }_p+{\sum }_i\Vert D_{i}u{\Vert }_p$.

A smooth manifold is orientable when it admits an atlas whose transition maps have positive Jacobian determinant wherever they are defined.

This atlas criterion expresses orientability for a differentiable manifold.

Every open cover of a smooth paracompact manifold admits a smooth partition of unity subordinate to it: nonnegative smooth functions with locally finite supports in the cover whose sum is one.

A map between smooth manifolds is smooth when its expression in every pair of coordinate charts is a smooth function.

A map between smooth manifolds is smooth when its coordinate expressions in compatible charts have derivatives of every order. The condition is invariant under smooth chart changes and expresses smoothness in charts.

An abstract smooth surface is a two-dimensional smooth manifold. Thus it is a Hausdorff second-countable space with an atlas of plane-valued coordinate charts whose transition maps are smooth.