Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 68, pp. 1-23.

Title: Maximum principles, sliding techniques and applications to
       nonlocal equations

Author: Jerome Coville (Univ. Paris Dauphine, France)

Abstract:
 This paper is devoted to the study of maximum principles holding
 for some nonlocal diffusion operators defined in (half-) bounded
 domains and its applications to obtain  qualitative behaviors of
 solutions of some nonlinear problems.
 It is shown that, as in the classical case, the nonlocal diffusion
 considered satisfies a  weak and  a strong maximum principle.
 Uniqueness and monotonicity of solutions of nonlinear equations are
 therefore expected as in the classical case.  It is first presented a
 simple proof of this qualitative behavior and  the weak/strong
 maximum principle. An optimal condition to have a strong maximum
 for operator $\mathcal{M}[u] :=J\star u -u$ is also obtained.
 The proofs of the uniqueness and monotonicity essentially rely on
 the sliding method and the strong maximum principle.
  
Submitted August 7, 2006. Published May 10, 2007.
Math Subject Classifications: 35B50, 47G20, 35J60.
Key Words: Nonlocal diffusion operators; maximum principles; 
           sliding methods.