Singular solutions of fully nonlinear elliptic equations and applications

Scott Armstrong (1) , Boyan Sirakov (2) , Charles Smart (3)

1 Department of Mathematics [Chicago]
2 MODAL'X - Modélisation aléatoire de Paris X
3 CIMS - Courant Institute of Mathematical Sciences [New York]

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\rn$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragmén-Lindelöf result as well as a principle of positive singularities in certain Lipschitz domains.

Domaines: Equations aux dérivées partielles [math.AP]

Scott Armstrong, Boyan Sirakov, Charles Smart. Singular solutions of fully nonlinear elliptic equations and applications. Archive for Rational Mechanics and Analysis, 2012, 205 (2), pp.345-394. ⟨10.1007/s00205-012-0505-8⟩. ⟨hal-00687702⟩