Maximum principle and existence of Lp-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms

Abstract

We study L p -viscosity solutions of fully nonlinear, second-order, uniformly elliptic partial differential equations (PDE) with measurable terms and quadratic nonlinearity. We present a sufficient condition under which the maximum principle holds for L p -viscosity solution. We also prove stability and existence results for the equations under consideration.2000 Mathematics Subject Classification: 49L25, 35J60, 35D10.

Maximum principle for fully nonlinear equations via the iterated comparison function method

“…When m > 1, we will show that the maximum principle holds provided that µ or f is small enough in a certain norm. This will generalize a result of [15].…”

“…It is known that the maximum principle fails in general for such equations in the elliptic case. In [15] an example was given for the equation − u − µ|Du| 2 = C 0 in B 1 with certain constants µ, C 0 > 0, where B r = {y ∈ R n : |y| < r } for r > 0. Below we present a counter-example for which the maximum principle fails even when the PDE has any super-linear nonlinearity with respect to Du.…”

“…The basis for our analysis is the iterated comparison function method which we introduced in [15]. We will show several results.…”

Entire solutions of fully nonlinear elliptic equations with a superlinear gradient term

“…Such result has been extended in the case m > 1 of superlinear growth in the gradient by Koike-Świȩch [11]. In order to get the following ABP-estimates, deduced by [11, Theorems 3.1-3.2], we also need the restriction m ≤ 2.…”

“…As far as the growth is concerned, we have uniqueness, for instance if m = 2 and s > 2 provided (1.15) holds for some ρ < 2(s−2) s . The continuity of f ensures that the two notions of L n -viscosity solutions and classical viscosity solutions are equivalent, see [13]. For further comments about the assumptions, see Section 5.…”