Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Exact Fields, Direct Methods, Sobolev Spaces

In course

MATH-437: Calculus of variations

Introduction to classical Calculus of Variations and a selection of modern techniques. The Calculus of Variations aims at showing the existence of minimisers (or critical points) of functionals that naturally appear in mathematics and physics (Dirichlet en ...

Description

This lecture covers Hilbert's theorem, direct methods, and Sobolev spaces, including classical and modern methods. The instructor discusses the properties of exact fields and their applications, such as minimizing functionals and weak derivatives.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Ontological neighbourhood

Mathematics

Analysis: Functional analysis

Related concepts (32)

Sobolev space

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 derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

Sobolev inequality

In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev. Let W k,p(Rn) denote the Sobolev space consisting of all real-valued functions on Rn whose first k weak derivatives are functions in Lp.

Direct method in the calculus of variations

In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy. The calculus of variations deals with functionals , where is some function space and .

Windows 10

Windows 10 is a major release of Microsoft's Windows NT operating system. It is the direct successor to Windows 8.1, which was released nearly two years earlier. It was released to manufacturing on July 15, 2015, and later to retail on July 29, 2015. Windows 10 was made available for download via MSDN and TechNet, as a free upgrade for retail copies of Windows 8 and Windows 8.1 users via the Windows Store, and to Windows 7 users via Windows Update.

Windows 10 Mobile

Windows 10 Mobile is a discontinued mobile operating system developed by Microsoft. First released in 2015, it is a successor to Windows Phone 8.1, but was marketed by Microsoft as being an edition of its PC operating system Windows 10. Windows 10 Mobile aimed to provide greater consistency with its counterpart for PCs, including more extensive synchronization of content, Universal Windows Platform apps, as well as the capability, on supported hardware, to connect devices to an external display and use a desktop interface with mouse and keyboard input support (reminiscent of Windows on PCs).

Related lectures (53)

Determinantal Point Processes and Extrapolation

Covers determinantal point processes, sine-process, and their extrapolation in different spaces.

Hilbert Spaces: Definition and Properties

Covers the definition and properties of Hilbert spaces, including the Cauchy-Schwarz inequality and norm definition.

Seifert van Kampen: proof and identification

Covers the proof and identification of isomorphisms in the Seifert van Kampen theorem.

Recollements: Building New Spaces by Gluing Together

Introduces recollements, showing how to build new spaces by assembling and gluing them together.

Topology: Disk Deprivation