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The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. (English) Zbl 0806.35129

Let \(L\) be a uniformly elliptic operator in a general bounded domain (i.e., open connected set) \(\Omega \subset \mathbb{R}^ n\), of the form \(L=M + c(x) = a_{ij} (x) \partial_{ij} + b_ i(x) \partial_ i + c(x)\), where for some positive constants \(c_ 0\), \(C_ 0\), \(c_ 0 | \xi |^ 2 \leq a_{ij} (x) \xi_ i \xi_ j \leq C_ 0 | \xi |^ 2\) for all \(\xi \in \mathbb{R}^ n\). It is assumed that \(a_{ij} \in C (\Omega)\), \(b_ i\), \(c \in L^ \infty\), \((\sum b_ i^ 2)^{1/2}\), \(| c | \leq b\) for some constant \(b \geq 0\). The authors find a principal eigenvalue \(\lambda_ 1\) and eigenfunction \(\varphi_ 1\) for the Dirichlet problem for \(-L\) and study their relationship with a refined maximum principle.
A brief outline of their work is the following: The principal eigenvalue is defined by \(\lambda_ 1 = \sup \{\lambda \mid \exists \varphi > 0\) in \(\Omega\) satisfying \((L + \lambda) \varphi \leq 0\}\). Various bounds on \(\lambda_ 1\) are established, the dependence of \(\lambda_ 1\) on \(\Omega\) and on the coefficients \(b_ i\) and \(c\) is studied and a principal eigenfunction \(\varphi_ 1\) is constructed. \(L\) is said to satisfy the refined maximum principle in \(\Omega\) if for any function \(w(x)\) on \(\Omega\), \(w \leq 0\) in \(\Omega\) is implied by the conditions \(Lw \geq 0\) in \(\Omega\), \(w\) bounded above, and \(\lim \sup w(x_ j) \leq 0\) for every sequence \(x_ j \to \partial \Omega\) for which \(u_ 0 (x_ j) \to 0\). Here, \(u_ 0\) is a special function which is constructed in the paper and is a positive function in \(\Omega\) for which \(Mu_ 0 = - 1\) and \(u_ 0\) vanishes, in a suitable sense, on \(\partial \Omega\). It is proved that the refined maximum principle holds for \(L\) if and only if \(\lambda_ 1>0\).

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35B50 Maximum principles in context of PDEs
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