Variational problems with concentration
This book is mainly concerned with the following class of problems. Given a (possibly unbounded) domain \(\Omega\) in \(\mathbb{R}^n\), find a solution \(u\) in a suitable space solving \[ -\mu\text{ div}(|\nabla u|^{p- 2}\nabla u)= f(u)\quad\text{in }\Omega,\tag{1} \] \[ u= 0\quad\text{on }\partial\Omega,\tag{2} \] and study its behaviour as \(\mu\to 0\). Here \(1< p\leq n\). The variational version consists in looking for \[ \sup\Biggl\{ \int_\Omega F(u): \int_\Omega|\nabla u|^p\leq \varepsilon^p, u= 0\text{ on }\partial\Omega\Biggr\}\tag{3} \] with \(F'= f\), \(F(0)= 0\) and consider the low energy limit \(\varepsilon\to 0+\). Appropriately, although a bit abruptly, the author begins by sketching two important examples: 1) the problem of designing an annular condenser in which the boundary of the internal conductor \(A\) satisfies the requirement that the normal derivative of the electric potential \(u\) on it is a prescribed constant (Bernoulli's free boundary problem). In variational form one wants to maximize the measure of the set \(\{u\geq 1\}\) in a space of functions with zero boundary data and having a fixed value of the Dirichlet integral \(\int_\Omega|\nabla u|^2\) (i.e. \(p= n= 2\), \(F(t)= 0\) for \(t< 1\), \(F(t)= 1\) for \(t\geq 1\)). The special aim is to analyze those situations in which the measure of \(A\) becomes small. 2) The problem of determining the shape of the plasma ring in a tokomak machine (with special attention to the case of thin rings). In this case \(f(u)\) is proportional to \((u- 1)^{q-1}_+\) and \(F(t)= (t-1)\) for \(t\geq 1\), \(F(t)= 0\) for \(t< 1\) (\(p=2\), \(n=3\)). To the Bernoulli problem the whole chapter 14 is devoted, including the question of optimizing the shape of a small conductor. Other applications are described in Chap. 13 (the rest point of a ball on an elastic membrane, etc.) and in Chap. 14 (vortex motion). The core of the book is the illustration of the properties of the solutions \(u_\varepsilon\) of problem (3) with emphasis on concentration and limit shape of low energy limits. The main general results are effectively summarized in Chap. 7, analyzing the conditions for which \(u_\varepsilon\) tends to a Dirac mass and the behaviour of the rescaled function \(w_\varepsilon(y)= u_\varepsilon(x_\varepsilon+ \varepsilon^{{p\over n-p}}y)\) along sequences \(x_\varepsilon\to x_0\), \(x_0\) being the concentration point. This material is very rich, because various alternatives are possible. The book provides a good systematic overview of this difficult topic.
variational methods for elliptic equations
scientific article; zbMATH DE number 1329686