## Summer 2024

I am teaching the course

- V5B6 - Selected Topics in Analysis and Calculus of Variations:

**Γ-convergence of Integral Functionals**

It takes place at room **SR 1.008** on **Mondays from 14 (c.t.) to 16**.

If you have any question, don't hesitate to contact me!

**Info:** the exam window for the first appointment is **from 29.07.2024 to 02.08.2024**, for the second **from 16.09.2024 to 20.09.2024**.

Here a quick report of past lectures:

**Lecture 14**(on 15.07.2024): gradient theory of phase transition, physical motivation, Γ-convergence of Modica-Mortola energy. [notes]**Lecture 13**(on 08.07.2024): cell formula for convex homogenization, homogenization of nonlinear PDEs^{*}, a one-dimensional example, homogenization of quadratic forms. [notes]

*there was a mistake (f_{hom}is not isotropic), this is fixed in the notes.**Lecture 12**(on 01.07.2024): setting of periodic homogenization, cluster points (wrt Γ-convergence) are homogeneous integral functionals, asymptotic homogenization formula, homogenization formula on periodic functions. [notes]**Lecture 11**(on 24.06.2024): stability of Γ-convergence under compatible boundary conditions and equi-coerciveness, definition of quasiconvexification (and some properties), Γ-convergence and relaxation of homogeneous functionals, example of 1-dimensional problem. [notes]**Lecture 10**(on 17.06.2024): inner regularity and subadditivity of (localized) Γ-liminf and Γ-limsup, compactness of (localized) Γ-limit and its integral representation. [notes]**Lecture 9**(on 10.06.2024): localization method of Γ-convergence, integral representation of functionals in Sobolev spaces, the case of homogeneous functionals, fundamental estimate. [notes]**Lecture 8**(on 03.06.2024): quasiconvexity (and growth conditions) implies rank1-convexity, characterization of (seq.) weak lower-semicontinuity in Sobolev spaces, equivalence between (seq.) weak lower-semicontinuity and L^{p}-strong lower-semicontinuity under growth conditions. [notes]**Lecture 7**(on 27.05.2024): integral functionals (on the gradient) in Sobolev spaces, weak convergence of oscillating piecewise affine functions, necessary conditions for (seq.) weak lower-semicontinuity, notions of polyconvexity, quasiconvexity and rank-1 convexity. [notes]**Lecture 6**(on 13.05.2024): weak coerciveness, equivalence between (seq.) weak lower-semicontinuity and weak lower-semicontinuity under growth conditions, Γ-convergence of integral functionals in Lebesgue spaces. [notes]**Lecture 5**(on 10.05.2024): sufficient conditions for (seq.) weak lower-semicontinuity, strong lower-semicontinuity, (seq.) weak(*) lower-semicontinuous envelope. [notes]**Lecture 4**(on 06.05.2024): Direct Method with relaxation, compactness of Γ-convergence, integral functionals in Lebesgue spaces, necessary conditions for (seq.) weak* lower-semicontinuity. [notes]**Lecture 3**(on 29.04.2024): lower semicontinuity of Γ-limits, stability under continuous perturbations, Γ-limit of constant and monotone sequences, coerciveness notions, the fundamental Theorem of Γ-convergence. [notes]**Lecture 2**(on 22.04.2024): upper and lower limits, lower semicontinuity (definitions and properties), definition of Γ-convergence, example on the real line, upper and lower Γ-limits. [notes]**Lecture 1**(on 15.04.2024): quick overview of the course, CalcVar and Direct Method, the meaning (and need) of Γ-convergence, example of a homogenization problem. [notes]