The ground state and maximum principle for second order elliptic operators in general domains. (État fondamental et principe du maximum pour les opérateurs elliptiques du second ordre dans des domaines généraux.) (French. Abridged English version) Zbl 0798.35038

Berestycki, Henri; Nirenberg, Louis; Varadhan, Srinivasa

C. R. Acad. Sci., Paris, Sér. I 317, No. 1, 51-56 (1993).

Summary: For an elliptic operator \(L\) in a general bounded domain \(\Omega\subset\mathbb{R}^ N\) (no assumption of smoothness is made here), we define the principal eigenvalue by \[ \lambda_ 1= - \inf_{\{\phi>0\}}\sup_{x\in \Omega}\bigl\{L\phi(x)/\phi(x)\bigr\}= \sup\bigl\{\lambda;\;\exists\varphi>0\text{ such that } L\varphi+ \lambda\varphi\leq 0 \text{ in }\Omega\bigr\}.\tag{1} \] We show that the Krein-Rutman theory extends to this general setting. Indeed, we show that there exists a function \(\phi_ 1\in L^ \infty(\Omega)\) such that \((L+ \lambda_ 1)\phi_ 1= 0\) in \(\Omega\), \(\phi_ 1\) vanishes on \(\partial\Omega\) in a sense which is made precise. This function is unique up to a multiplicative constant. Furthermore, the Maximum Principle (in a conveniently refined formulation) holds for \(L\) in \(\Omega\) if and only if \(\lambda_ 1> 0\). We establish several properties of \(\lambda_ 1\) about the dependence on the coefficients, the domain, etc. and several estimates which are new – even in the case of a regular domain \(\Omega\). In deriving these estimates we emphasize the structural aspect of the various constants – independently of the particular operator under consideration. In particular we show that the maximum principle holds for domains which are sufficiently “narrow” or have small measure.

MSC:

35J15	Second-order elliptic equations
35R05	PDEs with low regular coefficients and/or low regular data
35B50	Maximum principles in context of PDEs
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B45	A priori estimates in context of PDEs

Keywords:

principal eigenvalue; Krein-Rutman theory