Isolated Singularities for Fully Nonlinear Elliptic Equations

Abstract

We obtain Serrin type characterization of isolated singularities for solutions of fully nonlinear uniformly elliptic equations F(D 2 u)=0. The main result states that any solution to the equation in the punctured ball bounded from one side is either extendable to the solution in the entire ball or can be controlled near the centre of the ball by means of special fundamental solutions. In comparison with semi-and quasilinear results the proofs use the viscosity notion of generalised solution rather than distributional or Sobolev weak solutions. We also discuss one way of defining

Fundamental solutions of homogeneous fully nonlinear elliptic equations

“…If u is bounded above or below in a neighborhood of the origin, then precisely one of the following five alternatives holds: (i) the singularity is removable; that is, u can be defined at the origin so that This theorem generalizes a result of Labutin [17], who proved (i)-(iii) above under the supplementary assumptions that F is rotationally invariant and there exist (fundamental) solutions u and v of F .D 2 u/ D 0 in R n n f0g such that u.x/ ! 1 and v.x/ !…”

“…For fully nonlinear operators this is not true, as noticed by Labutin [17], who showed that if ¤ ƒ, then P C ;ƒ . ƒ.n 1/= 1 / vanishes near the origin in a reasonable weak sense.…”

“…The work most closely related to ours is the one by Labutin [17], who gave, among other things, a partial extension of Bocher's theorem to solutions of Pucci extremal equations. Below we discuss in more detail that paper and the additional hypotheses it involved.…”

“…Though it is more complicated, our proof of Theorem 1.9 borrows some ideas from the arguments of Labutin [17], who proved Theorem 1.9 for F D P ;ƒ in the case that˛ .P ;ƒ / 0. The idea is to show that either the singularity at the origin of a solution u of…”

Fundamental solutions and two properties of elliptic maximal and minimal operators

“…See [18], [19] and [9] for further discussion on the fundamental solutions for the Pucci operators M In this section we provide a proof of Theorem 1.3, which is a consequence of a more general theorem that involves more general nonlinearities. But before going to that, we state a Comparison Principle due to Ishi and Lions [16], which will be used repeatedely in the proof of our results.…”