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Multiplicity of eigenvalues for the Laplace operator with respect to an obstacle, and nontangency conditions. (English) Zbl 0716.49009

The authors study existence and multiplicity of the eigenvalues of the Laplace operator “with respect to an obstacle”. More precisely, given a bounded open subset \(\Omega\) of \({\mathbb{R}}^ N\) and given: \(\phi_ 1,\phi_ 2: \Omega \to {\mathbb{R}}\), g: \(\Omega\times {\mathbb{R}}\to {\mathbb{R}}\), \(\rho >0\) they study existence of solutions (u,\(\lambda\)) of the following variational inequality \[ \int_{\Omega}\nabla u\nabla (v- u)dx+\int_{\Omega}g(x,u)(v-u)dx\geq \lambda \int_{\Omega}u(v-u)dx\quad \forall v\in {\mathbb{K}},\quad u\in {\mathbb{K}}\cap S_{\rho},\quad \lambda \in {\mathbb{R}}, \] where \({\mathbb{K}}=\{u\in W_ 0^{1,2}(\Omega)|\phi_ 1\leq u\leq \phi_ 2\}\), \(S_{\rho}=\{u\in L^ 2(\Omega)|\int_{\Omega}u^ 2dx=\rho^ 2\}\). They prove the existence of infinitely many solutions under symmetry assumptions. Such solutions are regarded as “lower critical points” of a suitable (classical) functional f restricted on the constraint \({\mathbb{K}}\cap S_{\rho}\), which is neither convex nor regular.
The main tools used are the flow of the solutions of the corresponding parabolic problem on \({\mathbb{K}}\cap S_{\rho}\) (namely the “curves of maximal slope for f on \({\mathbb{K}}\cap S_{\rho}''\), studied in a different paper) and some results of subdifferential analysis, which where developed in several previous papers, in collaboration with other authors.
In other following articles the corresponding bifurcation problem has been studied, as well as the problem of well clamped elastic plates in presence of an obstacle; see e.g. M. Degiovanni and the second author [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 81, No.3, 259-270 (1987; Zbl 0671.58029)] and M. Degiovanni [in: Nonlinear variational problems, Vol. II, Proc. Int. Conf., Elba/Italy 1986, Pitman Research Notes Math. Ser. 193, 161-180 (1989; Zbl 0684.73022)].
Reviewer: A.Marino

MSC:

49J40 Variational inequalities
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
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