[For the entire collection see Zbl 0671.00007.]
The paper gives a survey on techniques of calculus of variations for lower semi continuous functionals, which have been developed in collaboration with other authors. The applications concern the multiplicity of solutions to elliptic variational inequalities of the form \[ \begin{aligned} &u\in \mathbb{K} := \{w\in W_0^{1,2}(\Omega) : \phi_1\leq w\leq \phi_2\} \\ &\int_\Omega \{DuD(v-u)+g(x,u)(v-u)\}dx\geq 0\;\forall v\in\mathbb{K} \end{aligned} \tag{1} \] and the existence of solutions to parabolic variational inequalities with non-c onvex constraints of the form \[ \begin{aligned} &\int_\Omega \mathcal U'(t)(v-\mathcal U(t))dx + \int_\Omega \{D\mathcal U(t)D(v-\mathcal U)+g(x,\mathcal U(t))(v-\mathcal U(t))\}dx\geq \\ &\qquad \geq \Lambda(t) \int_\Omega \mathcal U(t)(v-\mathcal U(t))dx \qquad \forall v\in\mathbb{K} \\ &\Lambda(t)\in\mathbb{R}; \\ &\mathcal U(t)\in\mathbb{K} \cap \left\{w: \int_\Omega w^2 dx = \rho^2\right\}; \\ &\mathcal U(0) = u_0 \end{aligned} \tag{2} \] Chapter 1 is devoted to preliminary relations among the notion of “slope”, some known regularity results and the use of super-solutions as “fictitious obstacles”.
In chapter II some multiplicity results are given for the solutions of (1), when \(\phi_2 \equiv +\infty\) and \(g(x,u) = p(x,u)-h(x)\). In particular, in sections 1 and 2, it is shown that the question of existence of solutions to (1) in dependence of h resembles to the so-called “folding” in the unconstrained case \((\phi_1 \equiv -\infty)\) with “jumping nonlinearity. Further developments in this direction are contained in [the author and D. Passaseo, “A jumping behaviour induced by an obstacle.” Variational methods in Hamiltonian systems and elliptic equations, Proc. Conf., L’Aquila/Italy 1990 (to appear)].
In chapter III some abstract tools concerning \(\phi\)-convex functions are given. A first application is an existence theorem for problem (2). Moreover,also the results of the following chapter depend on such tools.
Chapter IV deals with existence and multiplicity of eigenvectors of the Laplace operator with obstacle, namely the solutions \((\lambda,u)\) of (1), when \(g(x,u) = p(x,u) - \lambda u\) and the further constraint \(\int_\Omega u^2 = \rho^2\) is imposed. Moreover, the corresponding bifurcation problem is also studied in section 2. It is worthwhile to point out that the “linearized” problem is individuated by means of \(\Gamma\)-convergence.