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A directed spanning tree rooted towards a vertex $r$ is a spanning tree whose edges are oriented so that every other vertex has a unique directed path to $r$.

A uniform spanning tree is sampled uniformly from all spanning trees of a finite connected graph.

This is the sequence space of sequences with a finite limit.

The space $c$ consists of convergent scalar sequences with the supremum norm.

If $\epsilon :\mathbb{R}\to [0,\infty )$ satisfies $\epsilon (t)\le M|t|$, thenThus these square-root perturbations converge uniformly to $|x|$.

With the supremum norm, $C_0({\mathbb{R}}^d)$ is complete. A uniform Cauchy sequence has a uniform continuous limit, and one member controls the limit uniformly outside a large ball. Every member is also uniformly continuous: use uniform continuity on a large compact ball and smallness of the function beyond it.

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 $1\le p<\infty$, an open set $\Omega$ and $\phi \in W^{1,p}(\Omega )$, the Affine Sobolev spaceis a translate of a closed linear subspace of the first-order Sobolev space. This definition makes sense without assuming an ordinary boundary trace.

On a bounded Lipschitz domain, the Sobolev trace theorem identifies it exactly with the Sobolev space elements having the same boundary trace as $\phi$, since the trace kernel is $W_0^{1,p}(\Omega )$. It is the natural admissible set for a variational Dirichlet problem. Specifying a trace datum alone also requires an extension $\phi$ in the stated Sobolev space; arbitrary boundary data need not admit one.

Let $U$ be a bounded Lipschitz domain and ${\Gamma }_D$ a measurable portion of its boundary. The spaceis a closed vector subspace of the H1 space. The Sobolev trace operator maps $H^1(U)$ continuously into $L^2(\partial U)$, and restriction to ${\Gamma }_D$ is bounded, so $V$ is the kernel of a continuous linear map. Boundary regularity is what makes this ordinary trace formulation available.

No positive-measure assumption is needed for closedness. If $U$ is also connected and ${\Gamma }_D$ has positive surface measure, the Poincare inequality with a partial Dirichlet boundary controls the full Sobolev norm by the gradient norm on $V$.