Tonelli's theorem (functional analysis)

In mathematics, Tonelli's theorem in functional analysis is a fundamental result on the weak lower semicontinuity of nonlinear functionals on L^p spaces. As such, it has major implications for functional analysis and the calculus of variations. Roughly, it shows that weak lower semicontinuity for integral functionals is equivalent to convexity of the integral kernel. The result is attributed to the Italian mathematician Leonida Tonelli.

Statement of the theorem

Let $\Omega$ be a bounded domain in $n$ -dimensional Euclidean space $\mathbb {R} ^{n}$ and let $f:\mathbb {R} ^{m}\to \mathbb {R} \cup \{\pm \infty \}$ be a continuous extended real-valued function. Define a nonlinear functional $F$ on functions $u:\Omega \to \mathbb {R} ^{m}$ by $F[u]=\int _{\Omega }f(u(x))\,\mathrm {d} x.$

Then $F$ is sequentially weakly lower semicontinuous on the $L^{p}$ space $L^{p}(\Omega )$ for $1<p<+\infty$ and weakly-∗ lower semicontinuous on $L^{\infty }(\Omega )$ if and only if $f$ is convex.

References

Renardy, Michael & Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 347. ISBN 0-387-00444-0. (Theorem 10.16)

v t e Convex analysis and variational analysis
Basic concepts	Convex combination Convex function Convex set
Topics (list)	Choquet theory Convex geometry Convex metric space Convex optimization Duality Lagrange multiplier Legendre transformation Locally convex topological vector space Simplex
Maps	Convex conjugate Concave (Closed K- Logarithmically Proper Pseudo- Quasi-) Convex function Invex function Legendre transformation Semi-continuity Subderivative
Main results (list)	Carathéodory's theorem Ekeland's variational principle Fenchel–Moreau theorem Fenchel-Young inequality Jensen's inequality Hermite–Hadamard inequality Krein–Milman theorem Mazur's lemma Shapley–Folkman lemma Robinson–Ursescu Simons Ursescu
Sets	Convex hull (Orthogonally, Pseudo-) Convex set Effective domain Epigraph Hypograph John ellipsoid Lens Radial set/Algebraic interior Zonotope
Series	Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))
Duality	Dual system Duality gap Strong duality Weak duality
Applications and related	Convexity in economics

v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point Sobczyk's theorem
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

Statement of the theorem

See also

References