The Alexandrov-Bakelman-Pucci Weak Maximum Principle for Fully Nonlinear Equations in Unbounded Domains

Alexandroff–Bakelman–Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations

“…As explained in [6], a fondamental tool in this approach is the Alexandroff-Bakelman-Pucci estimate. Many authors extended these results since then; see for instance [14,20,7,26] and references therein.…”

“…In order to get such an estimate, we use the techniques from [5]. As we already mentioned it in [17], the ABP estimate that we are able to obtain differs slightly from classical ones in the sense that we can prove it under a weaker condition than (7); moreover, the supersolution is only lower semi-continuous. We recall that this is an a priori estimate: structure conditions ensuring the uniqueness of the solution are not required.…”

“…We will explain how to deal with non-linearities, after redefining them if necessary, growing quadratically with respect to the gradient. Precisely, (7) and (8) are replaced with…”

Eigenfunctions for singular fully nonlinear equations in unbounded domains

“…Let us recall that in general, even for the Laplacian operator, the maximum principle does not hold in unbounded domain, hence we cannot define the "principal" eigenvalue in the same way as in the case of bounded domains. In [10] and [11] Capuzzo Dolcetta, Leoni and Vitolo study the conditions on the domain Ω in order for the Maximum principle to hold for fullynonlinear operators, extending the result of Cabré [9].…”

Surface Evolution Equations

“…The reader is referred to a book of L. A. Caffarelli and X. Cabré (1995) for a basic Alexandrov-Bakelman-Pucci inequality as well as a recent book of S. Koike (2004). See also I. Capuzzo-Dolcetta, F. Leoni and A. Vitolo (2004) for further references and extension to unbounded domains. We shall not touch this type of results in this book.…”