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

Marino, Antonio; Scolozzi, Donato

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

Keywords:

unilateral problems; eigenvalue problem; evolution problem; Lusternik- Schnirelman

Citations:

Zbl 0594.00007