Lecture

Polyconvexity

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

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 the concept of polyconvexity in the context of vectorial calculus, focusing on open bounded sets with Lipschitz boundaries. The instructor explains the conditions for polyconvexity and provides examples and exercises to illustrate the concept. The lecture also delves into weak continuity theorems related to determinants and Morrey-Reshetnyak results. The content progresses to discuss the minimization of sequences in function spaces and the weak lower semicontinuity of certain functions, emphasizing the application of these concepts in the context of convexity assumptions.

Official source

Ontological neighbourhood

Mathematics

Analysis: Functional analysis

Related concepts (31)

Function space

In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space. Vector space#Function spaces Let be a vector space over a field and let be any set.

Schwartz space

In mathematics, Schwartz space is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space of , that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function. Schwartz space is named after French mathematician Laurent Schwartz.

Square-integrable function

In mathematics, a square-integrable function, also called a quadratically integrable function or function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line is defined as follows. One may also speak of quadratic integrability over bounded intervals such as for . An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable.

Vector calculus

Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations.

Hardy space

In complex analysis, the Hardy spaces (or Hardy classes) Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real analysis Hardy spaces are certain spaces of distributions on the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the Lp spaces of functional analysis.

Related lectures (64)

Distribution & Interpolation Spaces

Explores distribution and interpolation spaces, showcasing their importance in mathematical analysis and the computations involved.

Distributions and Derivatives

Covers distributions, derivatives, convergence, and continuity criteria in function spaces.

Embedding in Function Spaces

Explores embedding in function spaces, including elliptic equations, Hölder inequality, Lorentz space, and Hardy-Young inequality.

Weak Derivatives: Definition and Properties

Covers weak derivatives, their properties, and applications in functional analysis.

Calculus of Variations: Ground States in Quantum Mechanics