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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.

A hypersurface in an $n$-dimensional smooth manifold is a smooth submanifold of dimension $n-1$. Locally, a regular level set $\{\varphi =0\}$ with $\nabla \varphi \ne 0$ is a hypersurface whose normal vector is $\nabla \varphi$.

An embedded submanifold $M\subseteq N$ is a subset whose inclusion is an embedding. Around every point there are coordinates on $N$ in which $M$ is a coordinate plane.

An embedded submanifold is a submanifold for which the inclusion gives the subspace topology, unlike a merely immersed submanifold.

A smooth parametrized curve is regular when its velocity never vanishes.

Regularity is an additional differential condition on a smooth curve, not a synonym for an arbitrary curve.

Two nonzero functions with disjoint supports have identically zero pointwise product. Separated continuous tent functions or smooth bumps provide examples.

Normal frequencies satisfy $\det (\mathsf{K}-{\Omega }^2\mathsf{M})=0$, and each null vector gives the fixed amplitude ratio of a mode.

The Hessians of kinetic and potential energy at equilibrium give the symmetric mass matrix $\mathsf{M}$ and stiffness matrix $\mathsf{K}$.