A minimax formula for principal eigenvalues and application to an antimaximum principle (Q1884786)

The authors deal with the problem \[ Lu=\lambda m(x)\cdot u+h (x),\text{ in } \Omega\qquad Bu=0,\text{ on }\partial\Omega,\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) and \(L\) is a second-order elliptic operator of the form \[ Lu:=-\text{div}\bigl(A(x)\nabla u \bigr)+\bigl\langle a(x),\nabla u\bigr\rangle +a_0(x)u,\tag{2} \] with \(\langle,\rangle\) the scalar product in \(\mathbb{R}^N\), \(m(x)\) is a possibly indefinite weight, and \(B\) is a first-order boundary operator of the form: \[ Bu:=\bigl\langle b(x),\nabla u\bigr\rangle+b_0(x)\cdot u.\tag{3} \] Here the authors extend a minimax formula for the principle eigenvalue of a nonselfadjoint elliptic problem to the case, where an indefinite weight \(m(x)\) is present.