Eigenvalues of the Laplace operator and evolution equations in presence of an obstacle.(Italian)Zbl 0613.35024

Differential problems and theory of critical points, Meet. Bari/Italy 1984, 137-155 (1984).
[For the entire collection see Zbl 0594.00007.]
The paper studies the eigenvalue problem: $$u\in K\cap S_{\rho}$$, $$\lambda\in R$$ satisfying: $\int_{\Omega}Du.D(v-u)+g(u).(v-u)\geq \lambda \int_{\Omega}u(v-u)$ for all $$v\in K=\{u\in H^ 1_ 0(\Omega):\phi_ 1\leq u\leq \phi_ 2$$ in $$\Omega\}$$ where $$\Omega$$ is open and bounded in $$R^ n$$, $$\phi_ 1,\phi_ 2\in H^ 2(\Omega)\cap C^ 0(\Omega)$$, $$S_{\rho}=\{u\in L^ 2(\Omega):\int_{\Omega}u^ 2=\rho^ 2\}$$ and g is lipschitzian. The problem is in fact equivalent to finding $$u\in H^ 2(\Omega)$$ with $$\lambda u+\Delta u-g(x,u)\leq,=,\geq 0$$ respectively for x so that $$\phi_ 1=u<\phi_ 2$$, $$\phi_ 1<u<\phi_ 2$$, $$\phi_ 1<u=\phi_ 2$$. The problem is studied by means of an evolution problem. Methods similar to those of Lusternik- Schnirelman are employed. Unfortunately no proofs are available in the paper, so that the reader has to look for them in other of the authors’ papers.
Reviewer: B.Vernescu

MSC:

 35J20 Variational methods for second-order elliptic equations 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000) 49R50 Variational methods for eigenvalues of operators (MSC2000) 35G30 Boundary value problems for nonlinear higher-order PDEs

Zbl 0594.00007