The obstacle problem. (English) Zbl 1084.49001
Lezioni Fermiane. Rome: Accademia Nazionale dei Lincei; Pisa: Scuola Normale Superiore. ii, 54 p. (1998).
These lecture notes are devoted to the obstacle problem
\[
\begin{gathered} \int_\Omega|\nabla u(x)|^2 dx\to\min,\quad u\in K,\\ K= \{u\in H^1(\Omega)| u= \varphi\text{ on }\partial\Omega,\;u\geq \psi\text{ in }\Omega\}.\end{gathered}\tag{1}
\]
The notes are self-contained and give the basic results on regularity of solutions of (1), especially on the behavior of solutions near the boundary of the coincidence set.
Reviewer: Uldis Raitums (Riga)
MSC:
49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |
35J20 | Variational methods for second-order elliptic equations |
35J85 | Unilateral problems; variational inequalities (elliptic type) (MSC2000) |
35R35 | Free boundary problems for PDEs |
49J40 | Variational inequalities |
49J45 | Methods involving semicontinuity and convergence; relaxation |
49K20 | Optimality conditions for problems involving partial differential equations |
35B65 | Smoothness and regularity of solutions to PDEs |
49N60 | Regularity of solutions in optimal control |