## 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:

  On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, pp. 1952 in: Methods of Functional Analysis and Theory of Elliptic Equations, D. Greco, ed., Liguori Ed. Napoli, 1983.  Convex Functions and Elliptic Equations, manuscript of book.  Barta, C. R. Acad. Sci. Paris 204 pp 472– (1937)  Berestycki, J. Geom. Phys. 5 pp 237– (1988)  Berestycki, Boll. Soc. Brasil Mat. Nova Ser. 22 pp 1– (1991)  Second-Order Elliptic Equations in General Domains: Hopf’s Lemma and Anti-Maximum Principle, Doctoral Dissertation, New York University, 1992.  Bony, C. R. Acad. Sci. Paris Série A 265 pp 333– (1967)  Donsker, Proc. Nat. Acad. Sci. USA 72 pp 780– (1975)  Donsker, Comm. Pure Appl. Math. 29 pp 595– (1976)  and , Elliptic Partial Differential Equations of Second Order; 2nd ed., Grundlehren der mathematischen Wissenschaften, No. 224, Springer-Verlag, Berlin-New York, 1983.  Nonlinear elliptic and parabolic equations of the second order, Math. and its Applications, Reidel, Norwell, Massachusetts, 1987.  Lieb, Invent. Math. 74 pp 441– (1983)  Lions, Proc. AMS 88 pp 503– (1983)  Miller, Ann. Mat. Pura Appl. 76 pp 93– (1967)  Nussbaum, J. Analyse Math. 59 pp 161– (1992)  Protter, Bull. AMS 72 pp 251– (1966)  and , Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1967.  Pucci, Proc. AMS 17 pp 788– (1966)  Théorie du degré topologique et applications à des problèmes aux limites non linéaires, Lecture Notes Lab. Analyse Numerique, Université Paris VI, 1975.  Serrin, Arch. Rat. Mech. Anal. 44 pp 183– (1972)  Stroock, Comm. Pure Appl. Math. 25 pp 651– (1972)  Venturino, Boll. Un. Math. It. 5 pp 576– (1978)  Walter, Math. Z. 200 pp 293– (1989)  Zhao, Bull. AMS 23 pp 513– (1990)  Brezis, Boll. Un. Math. Ital. A 17 pp 503– (1980)
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