Fundamental solutions of homogeneous fully nonlinear elliptic equations

Abstract

We prove the existence of two fundamental solutionsˆand ẑ of the PDEfor any positively homogeneous, uniformly elliptic operator F . Corresponding to F are two unique scaling exponents˛ ; z > 1 that describe the homogeneity ofˆand ẑ . We give a sharp characterization of the isolated singularities and the behavior at infinity of a solution of the equation F .D 2 u/ D 0, which is bounded on one side. A Liouville-type result demonstrates that the two fundamental solutions are the unique nontrivial solutions of F .D 2 u/ D 0 in R n n f0g that are bounded on one side in both a neighborhood of the origin as well as at infinity. Finally, we show that the sign of each scaling exponent is related to the recurrence or transience of a stochastic process for a two-player differential game.

Stationary States of Reaction-Diffusion and Schrödinger Systems with Inhomogeneous or Controlled Diffusion

“…The proof is well known and is the same as for the Laplacian. The result is also included in Theorem 1.7 in [5].…”

“…These operators are usually referred to as Hamilton-Jacobi-Bellman (HJB) operators, and are in turn a subclass of the so-called Isaacs min-max operators, basic in game theory. In the following we will consider Isaacs operators as in (5) below, that is, sup-inf over arbitrary index sets of linear operators as in (4)…”

“…Similarly, we denote with α + (F, C ω ) the supremum of all positive α such that −F (D 2 u) ≥ 0 has a (−α)-homogeneous solution in C ω \ {0} (such α always exist) See [5] and [4] for more details on these scaling exponents and the way they describe properties of the operators F . Obviously α * (F Q,y ) ≤ α + (F Q,y , C).…”

Harnack Inequalities and Bôcher‐Type Theorems for Conformally Invariant, Fully Nonlinear Degenerate Elliptic Equations

“…For fully nonlinear uniformly elliptic equations, extensions of Bôcher's theorem have been established in the literature. See Labutin [17], Felmer and Quass [10] and Armstrong, Sirakov and Smart [1].…”

Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

“…If F is nonlinear but positively homogeneous, the fully nonlinear equation (1.4) is a Bellman-Isaacs equation which arises in the theory of stochastic optimal control and two-player stochastic differential games, and the homogenization result yields similar information about these more general diffusion processes. Although we do not explore this point here, we remark that the recurrence verses transience of such controlled diffusion processes in an isotropic environment was characterized in [3], and this result applied to the effective operator F , together with its proof, gives information about the corresponding questions for controlled diffusions in random environments.…”