Fundamental solutions and two properties of elliptic maximal and minimal operators

Abstract

Abstract. For a large class of nonlinear second order elliptic differential operators, we define a concept of dimension, upon which we construct a fundamental solution. This allows us to prove two properties associated to these operators, which are generalizations of properties for the Laplacian and Pucci's operators. If M denotes such an operator, the first property deals with the possibility of removing singularities of solutions to the equationwhere B is a ball in R N . The second property has to do with existence or nonexistence of solutions in R N to the inequalityIn both cases a common critical exponent defined upon the dimension number is obtained, which plays the role of N/(N − 2) for the Laplacian.

Existence and non-existence results for fully nonlinear elliptic systems

“…We start by proving a version, for the extremal operators of the previous section, of the Hadamard Three Spheres Theorem which can be found in [21]. For completeness we give the proof here.…”

“…So we use a different argument to get (21). In case h (r) +â ann h (r)/r ≥ 0 we have trivially (21).…”

“…Nonexistence in a half-space for this equation is proved in [30]. Recently, Liouville results for scalar equations with general operators of type (8) were obtained in [20] and [21]. To our knowledge, Theorem 1.3 is the first result on nonexistence for systems involving fully nonlinear operators.…”

“…In our situation we are led to proving Liouville results for systems involving a general class of extremal operators, related to ones recently defined in [20] and [21]. Specifically, we are going to consider the HJB operators…”

“…or the last two inequalities with A and a interchanged (here a is a positive lower bound for d 21 or d 12 from (4)) 3 .…”