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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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