Elliptic systems of phase transition type (Q1991032)

The monograph discusses the structure of the set of bounded solutions to the semilinear elliptic system $\Delta u-f(u)=0$ on a domain $\Omega\subset\mathbb{R}^n$ emphasizing the case where the system is of gradient type, that is $f(u)=W_u(u)$ for a potential $W:\mathbb{R}^m\to\mathbb{R}$. Except of Chapter 3, the study is performed for a phase transition potential, which means that the set $\{W=0\}$ has finitely many points (called phases), and for minimal solutions in the sense that they are minimizers of the energy subject to their Dirichlet values. It is worth mentioning that Chapters 2, 3, 4, 5, and 9 do not require symmetry hypotheses on $W$ and that they are independent from the rest of the book. Chapters 6, 7, and 8 require symmetry hypotheses and depend on Chapter 5. Every chapter ends with comments and a list of references. Chapter 1 provides an overview of the book offering a comprehensive motivation. Chapter 2 considers the case $n=1$ studying the so-called connections, which are solutions $u:\mathbb{R}\to\mathbb{R}^m$ to the ordinary differential system $u''-W_u(u)=0$ with fixed limits $u(\pm \infty)$. Chapter 3 presents properties of solutions to the elliptic system $\Delta u-W_u(u)=0$ with $W\geq 0$ mainly exploiting the associated stress-energy tensor. Chapter 4 centers around a special maximum principle allowing control on minimal solutions. Chapter 5 produces basic estimates. Chapter 6 constructs entire solutions connecting as $|x|\to \infty$ the minima $\{W=0\}$. Chapter 7 continues the study of entire symmetric solutions under the assumption that $W$ is invariant with respect to a finite reflection group. Chapter 8 revisits the density estimate in a symmetry setting targeting the stationary layered solutions. Chapter 9 describes minimal solutions $u:\mathbb{R}^2\to\mathbb{R}^m$ without symmetry hypotheses. An appendix is devoted to radial solutions. The monograph sets forth rich material with novel insight for the nonlinear elliptic theory and applications to phase transitions.