Tonelli's theorem (functional analysis)

In mathematics, Tonelli's theorem in functional analysis is a fundamental result on the weaklower semicontinuity of nonlinearfunctionals 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 ItalianmathematicianLeonida Tonelli.

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

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

• Discontinuous linear functional

• 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)