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A minimax formula for the principal eigenvalues of Dirichlet problems and its applications. (English) Zbl 1153.35057
Consider the (generally nonselfadjoint) Dirichlet problem \(L(u) = \lambda m(x) u\) in \(\Omega\), \(u = 0 \) on \(\partial \Omega\), with \(\Omega\) bounded in \(\mathbb R^N\), \(L\) being a second order elliptic operator of the form \(Lu := -\operatorname{div} (A(x) \nabla u) + <a(x)\), \(\nabla u> + a_0 (x) u\) and \(m(x)\) is a possibly indefinite weight. Under the condition that \(m(x)\) may change sign, the authors first clarify the status of the principal eigenvalues.
The main result is a minimax formula for the principal eigenvalue \(\lambda^*\) for \(m^+ \not \equiv 0\). As an application of the minimax formula, the antimaximum principle (in short AMP) for \(Lu = \lambda m(x) u + h(x)\) in \(\Omega\), \(u = 0\) on \(\partial \Omega\) is further considered. The AMP principle means that for any \(h \geq 0, h \not \equiv 0\), there exists \(\delta > 0\) such that for \(\lambda \in (\lambda^*, \lambda^* + \delta)\), the solution \(u\) is negative. The authors prove that the \(AMP\) holds nonuniformly for \(m \in L^\infty (\Omega)\) with \(m \not \equiv 0\).

MSC:
35P15 Estimates of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
49J35 Existence of solutions for minimax problems
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