In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its...

analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving...

Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized...

In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality...

Orlicz spaces are many of the most important Sobolev spaces. In addition, the Orlicz sequence spaces are examples of Orlicz spaces. These spaces are called...

delta function defines a bounded linear functional. The Sobolev embedding theorem for Sobolev spaces on the real line R implies that any square-integrable...

function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential...

differential equations. Sobolev introduced notions that are now fundamental for several areas of mathematics. Sobolev spaces can be defined by some growth...

For example, the space L 2 {\displaystyle L^{2}} is a Hilbert space. The Hardy spaces, the Sobolev spaces are examples of Banach spaces that are related...

interpolation space is a space which lies "in between" two other Banach spaces. The main applications are in Sobolev spaces, where spaces of functions...

)} compact support in limit topology W k , p {\displaystyle W^{k,p}} Sobolev space of functions whose weak derivatives up to order k are in L p {\displaystyle...

spaces, serve to generalize more elementary function spaces such as Sobolev spaces and are effective at measuring regularity properties of functions. Several...

then u is Hölder continuous. Functions in Sobolev space can be embedded into the appropriate Hölder space via Morrey's inequality if the spatial dimension...

functions, for example using Fourier series. The Sobolev spaces Hs, sometimes called Hermite-Sobolev spaces, are defined to be the completions of S {\displaystyle...

completion (a Sobolev space) rather than the original space of nice functions for which the differential equation actually makes sense. A metric space M is bounded...

In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving...

derivatives leads to Sobolev spaces. Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert. The Hilbert space L 2 ( Ω ) , {\displaystyle...

isomorphism in the category of Hilbert spaces. One such relaxation is that the wave function must belong to the Sobolev space W1,2. It means that it is differentiable...

Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician Franz Rellich and...

Hölder space LF-space Lp space Minkowski space Montel space Morrey–Campanato space Orlicz space Riesz space Schwartz space Sobolev space Tsirelson space...

concerning Banach spaces. It is often used in functional analysis to demonstrate the equivalence of certain norms on Sobolev spaces. It was named after...

{G}}} are some unspecified function spaces, such as Hardy space, Lp space, Sobolev space, or, more vaguely, the space of holomorphic functions. List of...

examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces. Many topological vector spaces are spaces of functions, or linear operators...

Rademacher's theorem can be used to prove that, for any p ≥ 1, the Sobolev space W1,p(Ω) is preserved under a bi-Lipschitz transformation of the domain...

applying the direct method, the functional is usually defined on a Sobolev space W 1 , p ( Ω , R m ) {\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}...

Quadratic space Quotient space (disambiguation) Riemann's Moduli space Sample space Sequence space Sierpiński space Sobolev space Standard space State space Stone...

^{2})^{s}S_{ff}(\ell )<\infty ,} then f is in the Sobolev space Hs(S2). In particular, the Sobolev embedding theorem implies that f is infinitely differentiable...

In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient ∇ f {\displaystyle...

Here H 0 1 ( Ω ) {\displaystyle H_{0}^{1}(\Omega )} denotes the Sobolev Hilbert space of once-weakly differentiable functions with first weak derivative...

A weak solution of this system is defined to be an element f of the Sobolev space W1,n loc(Ω, Rn) with non-negative Jacobian determinant almost everywhere...