## 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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### References:

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