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Vector space of functions in mathematics
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
Sobolev_space
Theorem about inclusions between Sobolev spaces
analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving
Sobolev_inequality
Type of vector space in math
Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized
Hilbert_space
Generalized function whose value is zero everywhere except at zero
delta function defines a bounded linear functional. The Sobolev embedding theorem for Sobolev spaces on the real line R implies that any square-integrable
Dirac_delta_function
Area of mathematical analysis
point of view is especially useful for fractional Sobolev spaces, Besov spaces, Triebel–Lizorkin spaces, and nonlinear problems where estimates must be
Harmonic_analysis
Auxiliary functions used to probe equations, distributions, and weak formulations
can be approximated in the Sobolev norm by compactly supported smooth functions. For this reason, functions in Sobolev spaces such as H 0 1 ( U ) {\displaystyle
Test_function
In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving
Sobolev spaces for planar domains
Sobolev_spaces_for_planar_domains
Type of function space
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
Orlicz_space
Generalization of Sobolev spaces
spaces, serve to generalize more elementary function spaces such as Sobolev spaces and are effective at measuring regularity properties of functions. Several
Besov_space
Russian mathematician (1908-1989)
differential equations. Sobolev introduced notions that are now fundamental for several areas of mathematics. Sobolev spaces can be defined by some growth
Sergei_Sobolev
Class of inequalities
In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f {\displaystyle f} , its logarithm, and
Logarithmic Sobolev inequalities
Logarithmic_Sobolev_inequalities
Vector space in mathematics
interpolation space is a space which lies "in between" two other Banach spaces. The main applications are in Sobolev spaces, where spaces of functions
Interpolation_space
Boundary condition for generalized functions
function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential
Trace_operator
Theorem limiting types of conformal mappings in Euclidean space of dimension > 2
domain in Euclidean space that is also conformal is a Möbius transformation. This equivalent statement justifies using the Sobolev space W1,n, since f ∈ W1
Liouville's theorem (conformal mappings)
Liouville's_theorem_(conformal_mappings)
Mathematical set with some added structure
Quadratic space Quotient space (disambiguation) Riemann's Moduli space Sample space Sequence space Sierpiński space Sobolev space Standard space State space Stone
Space_(mathematics)
Mathematical inequality in Sobolev space theory
In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality
Poincaré_inequality
Theorem in mathematical analysis
Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the L p {\displaystyle L^{p}} -norms of different weak
Gagliardo–Nirenberg interpolation inequality
Gagliardo–Nirenberg_interpolation_inequality
Method for constructing existence proofs and calculating solutions in variational calculus
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})}
Direct method in the calculus of variations
Direct_method_in_the_calculus_of_variations
In mathematics, Souček spaces are generalizations of Sobolev spaces, named after the Czech mathematician Jiří Souček. One of their main advantages is that
Souček_space
Representation theory of the symplectic group
functions, for example using Fourier series. The Sobolev spaces Hs, sometimes called Hermite-Sobolev spaces, are defined to be the completions of S {\displaystyle
Oscillator_representation
Type of topological space
Here H 0 1 ( Ω ) {\displaystyle H_{0}^{1}(\Omega )} denotes the Sobolev Hilbert space of once-weakly differentiable functions with first weak derivative
Bochner_space
Mathematical space with a notion of distance
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
Metric_space
and let W n , 2 {\displaystyle W^{n,2}} be the corresponding Sobolev space. The Sobolev orthogonal polynomials { p n } n ≥ 0 {\displaystyle \{p_{n}\}_{n\geq
Sobolev orthogonal polynomials
Sobolev_orthogonal_polynomials
analysis, a Triebel–Lizorkin space is a generalization of many standard function spaces such as Lp spaces and Sobolev spaces. It is named after Hans Triebel [de;
Triebel–Lizorkin_space
Compact embedding theorem concerning Sobolev spaces
Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician Franz Rellich and
Rellich–Kondrachov_theorem
Concept in mathematical analysis
Pólya–Szegő inequality (or Szegő inequality) states that the Sobolev energy of a function in a Sobolev space does not increase under symmetric decreasing rearrangement
Pólya–Szegő_inequality
In mathematics, a Sobolev mapping is a mapping between manifolds which has smoothness in some sense. Sobolev mappings appear naturally in manifold-constrained
Sobolev_mapping
Set of functions between two fixed sets
)} , the space of all Lipschitz functions on Ω {\displaystyle \Omega } that vanish at zero. W k , p {\displaystyle W^{k,p}} Sobolev space of functions
Function_space
Construction for adding objects to a Hilbert space
{S}}'(\mathbb {R} ).} Another example is given by Sobolev spaces: Here (in the simplest case of Sobolev spaces on R n {\displaystyle \mathbb {R} ^{n}} ) H =
Rigged_Hilbert_space
Mathematical theorem
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
Rademacher's_theorem
concerning Banach spaces. It is often used in functional analysis to demonstrate the equivalence of certain norms on Sobolev spaces. It was named after
Ehrling's_lemma
Elliptic partial differential operator
weak solutions. For example, we say that a function u belonging to the Sobolev space W 1 , p ( Ω ) {\displaystyle W^{1,p}(\Omega )} is a weak solution of
P-Laplacian
Belgian mathematician (1954–2018)
Brezis, Haim; Mironescu, Petru (2001). "Another look at Sobolev spaces". pp. 439–455. (See Sobolev space.) Bourgain, J. (2002). "Nonlinear partial differential
Jean_Bourgain
analysis on Sobolev spaces. It is named after Neil Trudinger (and Jürgen Moser). It provides an inequality between a certain Sobolev space norm and an
Trudinger's_theorem
Normed vector space that is complete
Function spaces generalizing finite-dimensional p norm spaces Sobolev space – Vector space of functions in mathematics Banach lattice – Banach space with
Banach_space
Algebraic structure in linear algebra
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
Vector_space
The Sobolev conjugate of p for 1 ≤ p < n {\displaystyle 1\leq p<n} , where n is space dimensionality, is p ∗ = p n n − p > p {\displaystyle p^{*}={\frac
Sobolev_conjugate
{G}}} are some unspecified function spaces, such as Hardy space, Lp space, Sobolev space, or, more vaguely, the space of holomorphic functions. List of
List_of_mathematic_operators
Mathematical measure of a function's variability
a function is. More abstractly, it is a quadratic functional on the Sobolev space H1. The Dirichlet energy is intimately connected to Laplace's equation
Dirichlet_energy
the sense of trace (that is, u {\displaystyle u} is a limit in the Sobolev space H 1 ( Ω ) {\displaystyle H^{1}(\Omega )} of a sequence of smooth functions
Ladyzhenskaya's_inequality
weak sense" (see the article on Sobolev spaces for details.) The space H 1 {\displaystyle H^{1}} is a Hilbert space, which accounts for much of the ease
Elliptic boundary value problem
Elliptic_boundary_value_problem
Lions–Magenes lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a criterion for moving a time
Lions–Magenes_lemma
Mathematical theorem
vector space of possible solutions, e.g. a Sobolev space W k,p. Abstractly, consider two real normed spaces U and V with their continuous dual spaces U∗ and
Babuška–Lax–Milgram_theorem
Vector space with a notion of nearness
examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces. Many topological vector spaces are spaces of functions, or linear operators
Topological_vector_space
Topics referred to by the same term
an old photometric unit of luminous intensity Hexokinase, an enzyme Sobolev space Hk, in mathematics Kyūkyoku!! Hentai Kamen, simplified as Hentai Kamen
HK_(disambiguation)
Russian-French mathematician
Korevaar and Schoen, establishing extensions of most of the standard Sobolev space theory. A sample application of Gromov and Schoen's methods is the fact
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Conformal structure admits a Hodge dual of 1-forms without even specifying a metric
precursor of the modern theory of elliptic differential operators and Sobolev spaces. These techniques were originally applied to prove the uniformization
Differential forms on a Riemann surface
Differential_forms_on_a_Riemann_surface
Certain vector fields are the sum of an irrotational and a solenoidal vector field
{u} =\nabla \varphi +\nabla \times \mathbf {A} } where φ is in the Sobolev space H1(Ω) of square-integrable functions on Ω whose partial derivatives
Helmholtz_decomposition
Inequality in mathematical analysis
involving Sobolev spaces, the Gagliardo–Nirenberg interpolation inequality. The following example, this one allowing interpolation of non-integer Sobolev spaces
Interpolation_inequality
American mathematician
the global well-posedness of the Korteweg–De Vries equation in the Sobolev space H−1. Killip was an undergraduate at the University of Auckland. He completed
Rowan_Killip
Property of functions which is weaker than continuity
function over a bounded domain in Euclidean space. The integrand is convex in an appropriate Sobolev space, so the limit of a minimizing sequence is a
Semi-continuity
Type of continuity of a complex-valued function
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
Hölder_condition
Index of lists with the same name
Morrey–Campanato space Orlicz space Riesz space Schwartz space Sobolev space Tsirelson space This set index article is a list of articles associated with
List of vector spaces in mathematics
List_of_vector_spaces_in_mathematics
Concept in fluid dynamics
n 2 {\displaystyle |\mathbf {u} |_{H^{1}(\Omega )^{n}}^{2}} in the Sobolev space H 1 ( Ω ) n {\displaystyle H^{1}(\Omega )^{n}} . In the case that the
Enstrophy
Mathematical inequality relating the derivative of a function to its covariant derivative
the usual Hilbert space of square-integrable functions, and H 1 ( R n ) {\displaystyle H^{1}(\mathbb {R} ^{n})} the Sobolev space of square-integrable
Diamagnetic_inequality
Inequality on Lp norm with weak derivatives
that u : Ω → R {\displaystyle u:\Omega \to \mathbb {R} } lies in the Sobolev space W 0 k , p ( Ω ) {\displaystyle W_{0}^{k,p}(\Omega )} , i.e., u ∈ W k
Friedrichs'_inequality
Type of differential operator
states the sufficient condition for a weak solution u to exist in the Sobolev space Hk. For example, for a second-order elliptic operator as in Example
Elliptic_operator
Type of continuous linear operator
linear operators, and in applications to differential equations and Sobolev spaces. For example, compactness often implies that the nonzero spectrum of
Compact_operator
Method of smoothing using a spline function
function f {\displaystyle f} is defined to be the unique minimizer, in the Sobolev space W 2 2 {\displaystyle W_{2}^{2}} on a compact interval, of ∑ i = 1 n
Smoothing_spline
Form of continuity for functions
absolutely continuous function; this provides a characterization of Sobolev spaces on intervals of the real line. If f: [a,b] → R is absolutely continuous
Absolute_continuity
Function spaces generalizing finite-dimensional p norm spaces
{\displaystyle \blacksquare } Adams, Robert A.; Fournier, John F. (2003), Sobolev Spaces (Second ed.), Academic Press, ISBN 978-0-12-044143-3. Bahouri, Hajer;
Lp_space
Matrix used in finite element analysis
operator L of order 2k, there is associated a bilinear form B on the Sobolev space Hk, so that the weak formulation of the equation Lu = f is B [ u , v
Stiffness_matrix
Degree of differentiability of a function or map
using Sobolev spaces: Fourier-transform decay gives Sobolev regularity, and the Sobolev embedding theorem gives conditions under which Sobolev regularity
Smoothness
Real function with finite total variation
Sobolev Spaces, Berlin–Heidelberg–New York: Springer-Verlag, ISBN 0-387-13589-8, Zbl 0692.46023; particularly chapter 6, "On functions in the space BV(Ω)"
Bounded_variation
Function which is integrable on its domain
Maz'ya and Shaposhnikova define only the "localized" version of the Sobolev space W k , p ( Ω ) {\textstyle W^{k,p}(\Omega )} , nevertheless explicitly
Locally_integrable_function
Norman George Meyers, states that smooth functions are dense in the Sobolev space W k , p ( Ω ) {\displaystyle W^{k,p}(\Omega )} for arbitrary domains
Meyers–Serrin_theorem
functions of bounded variation Sobolev spaces The Birnbaum–Orlicz spaces L A ( μ ) . {\displaystyle L^{A}(\mu ).} Hölder spaces C k ( Ω ) . {\displaystyle
List_of_Banach_spaces
Australian and American mathematician (born 1975)
Keel, G. Staffilani, and H. Takaoka, on global regularity in optimal Sobolev spaces for KdV and other equations, as well as his many deep contributions
Terence_Tao
Swedish Mathematician
early achievements include: his work on Sobolev spaces, in particular the discovery of the equivalence between Sobolev and isoperimetric/isocapacitary inequalities
Vladimir_Mazya
One of Fredholm's theorems in mathematics
and dom ( L ) {\displaystyle \operatorname {dom} (L)} is then the Sobolev space W 2 , 2 ( Ω ) ∩ W 0 1 , 2 ( Ω ) {\displaystyle W^{2,2}(\Omega )\cap
Fredholm_alternative
Class of partial differential equations
partial differential equation Maximum principle (property of solutions) Sobolev space Evans 2010, Chapter 6. Zauderer 2006, chpt. 3.3 Classification of equations
Elliptic partial differential equation
Elliptic_partial_differential_equation
Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , n ≥ 2 {\displaystyle n\geq 2} . Let H 1 ( Ω ) {\displaystyle H^{1}(\Omega )} be the Sobolev space of all
Korn's_inequality
On weak solutions of differential equations
x ∈ U {\displaystyle x\in U} denotes the space variable, t denotes the time variable, Hs is a Sobolev space of functions with square-integrable weak derivatives
Regularity_theory
Mathematical description of quantum state
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
Wave_function
Feature of certain mathematical spaces
analysis, compact embedding is usually about Banach spaces of functions. Several of the Sobolev embedding theorems are compact embedding theorems. When
Compact_embedding
R m ) {\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})} denote the Sobolev space of mappings from Ω {\displaystyle \Omega } to R m {\displaystyle \mathbb
Polyconvex_function
Homeomorphism between plane domains
differentiability of f can be replaced by the weaker condition that f be in the Sobolev space W1,2(D) of functions whose first-order distributional derivatives are
Quasiconformal_mapping
Mathematical result in the theory of Sobolev spaces
Aubin–Lions lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that
Aubin–Lions_lemma
Value approached by a mathematical object
examples of function spaces with some notion of convergence are Lp spaces and Sobolev space. Suppose f is a real-valued function and c is a real number. Intuitively
Limit_(mathematics)
Differential operator in mathematics
locally square-integrable, then u {\displaystyle u} is locally in the Sobolev space H 2 {\displaystyle H^{2}} . In particular, harmonic functions are smooth
Laplace_operator
Class of partial differential equations
enlarged to a Sobolev space class H s {\displaystyle H^{s}} , for s > dim M / 2 + 1 {\displaystyle s>\dim M/2+1} . The corresponding tangent space consists
Euler–Arnold_equation
Topics referred to by the same term
certain set of operators in a Hilbert space Trace operator, a restriction-to-boundary operator in a Sobolev space TRACE (Transition Region and Coronal
Trace
Mathematical theory
Green functions which, up to logarithmic singularities, belong to the Sobolev space L 1 2 {\displaystyle L_{1}^{2}} . In this context, Bost obtains an arithmetic
Arakelov_theory
Theorem of analytic continuations
locally to the same function in a higher Sobolev space. For k large enough, this convergence is uniform by the Sobolev embedding theorem. By the argument for
Edge-of-the-wedge_theorem
Russian-born Swedish mathematician
operators acting between pairs of Sobolev spaces in 1995. In 1989 she showed that multipliers in Bessel potential spaces are traces of multipliers belonging
Tatyana_Shaposhnikova
Number, approximately 3.14
value of the derivative operator on the space of functions on [0, 1] vanishing at both endpoints (the Sobolev space H 0 1 [ 0 , 1 ] {\displaystyle H_{0}^{1}[0
Pi
Generalisation of convexity
the unit ball and W 0 1 , ∞ {\displaystyle W_{0}^{1,\infty }} is the Sobolev space of essentially bounded functions with essentially bounded derivative
Quasiconvexity (calculus of variations)
Quasiconvexity_(calculus_of_variations)
Type of mathematical function
Spectral Geometry Phenomenon Riesz rearrangement inequality Sobolev space – Vector space of functions in mathematics Szegő inequality – Concept in mathematical
Symmetric decreasing rearrangement
Symmetric_decreasing_rearrangement
Mathematical concept of energy in physics
{\displaystyle C>0.} The energetic space in respect to the operator B {\displaystyle B} is then the Sobolev space H 0 1 ( a , b ) . {\displaystyle H_{0}^{1}(a
Energetic_space
Chinese mathematician and Ph.D. at Peking University 2018
Weiren. "Linear inviscid damping for a class of monotone shear flow in Sobolev spaces". Communications on Pure and Applied Mathematics. 71 (2018), no. 4,
Wei_Dongyi
On when a family of real, continuous functions has a uniformly convergent subsequence
be seen as a simple instance of the fact that the injection from the Sobolev space H 0 1 ( Ω ) {\displaystyle H_{0}^{1}(\Omega )} into L2(Ω), for Ω a bounded
Arzelà–Ascoli_theorem
Domain in a Euclidean space whose boundary is sufficiently regular
strongly Lipschitz domain is given by the two-bricks domain Many of the Sobolev embedding theorems require that the domain of study be a Lipschitz domain
Lipschitz_domain
Kind of linear transformation
domain is a Sobolev space and it takes values in a space of square-integrable functions) is bounded. The unilateral shift operator on the Lp space ℓ 2 {\displaystyle
Bounded_operator
on the same space. Several of the Sobolev embedding theorems are continuous embedding theorems. Let X and Y be two normed vector spaces, with norms ||·||X
Continuous_embedding
Numerical method for solving physical or engineering problems
assumed that v ∈ H 0 1 ( Ω ) {\displaystyle v\in H_{0}^{1}(\Omega )} (see Sobolev spaces). The existence and uniqueness of the solution can also be shown. We
Finite_element_method
Harmonic functions as solutions to Laplace's equation
spaces. In this fashion, one obtains such spaces as the Hardy space, Bloch space, Bergman space and Sobolev space. Subharmonic function – Class of mathematical
Potential_theory
{\displaystyle n} -dimensional Euclidean space and let H k ( Ω ) {\displaystyle H^{k}(\Omega )} denote the Sobolev space of k {\displaystyle k} -times weakly
Gårding's_inequality
Relates three different kinds of weak compactness in a Banach space
in the theory of PDEs, and particularly in Sobolev spaces. Many Sobolev spaces are reflexive Banach spaces and therefore bounded subsets are weakly precompact
Eberlein–Šmulian_theorem
considered in the context of Sobolev spaces. The term cocompact embedding is inspired by the notion of cocompact topological space. Let G {\displaystyle G}
Cocompact_embedding
PDE to describe nonlinear wave motion
Benjamin–Bona–Mahony equation in the L 2 {\displaystyle L^{2}} -based Sobolev space H x k ( R ) {\displaystyle H_{x}^{k}(\mathbb {R} )} for all k ≥ 1 {\displaystyle
Kadomtsev–Petviashvili equation
Kadomtsev–Petviashvili_equation
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