A family of degenerate elliptic operators: Maximum principle and its consequences

Abstract

In this paper we investigate the validity and the consequences of the maximum principle for degenerate elliptic operators whose higher order term is the sum of k eigenvalues of the Hessian. In particular we shed some light on some very unusual phenomena due to the degeneracy of the operator. We prove moreover Lipschitz regularity results and boundary estimates under convexity assumptions on the domain. As a consequence we obtain the existence of solutions of the Dirichlet problem and of principal eigenfunctions.2010 Mathematics Subject Classification. 35J60, 35J70, 49L25 .

Evolution equations involving nonlinear truncated Laplacian operators

“…As a by product, we can construct a very surprising example of an initial data driving the solution to zero everywhere in finite time, see Example 5.5, which is referred as a quenching phenomena. On the other hand, and as already suspected since Section 3, the Heat equation (4) in dimension N involving P + k behaves like the Heat equation in lower dimension k < N . Finally, Section 6 is devoted to the analysis of the Cauchy problems (5) and (6).…”

“…In Section 3 we inquire on the existence of self similar solutions to (3) and (4). A key observation is that when a solution u is "one dimensional", its Hessian has an eigenvalue of multiplicity (at least) N − 1 and therefore, computing P ± k u reduces to localize the last eigenvalue, see assumption (8).…”

Towards a Liouville Theorem for Continuous Viscosity Solutions to Fully Nonlinear Elliptic Equations in Conformal Geometry

“…We establish the strong comparison principle and Hopf Lemma when one competitor is C 1,1 , and obtain as a consequence a Liouville theorem in this regularity. (6). Assume that (i) u 1 ∈ USC(Ω; [0, ∞)) and u 2 ∈ LSC(Ω; (0, ∞]) are a sub-solution and a supersolution to f (λ(A u )) = 1 in Ω in the viscosity sense, respectively, (ii) and that u 1 ≤ u 2 in Ω.…”

“…Let Ω be an open subset of R n , n ≥ 3, such that ∂Ω is C 2 near some pointx ∈ ∂Ω, Γ be a non-empty open subset of R n and f ∈ C 0 (Γ) satisfying (2)- (6). Assume that (i) u 1 ∈ USC(Ω ∪ {x}; [0, ∞)) and u 2 ∈ LSC(Ω ∪ {x}; (0, ∞]) are a sub-solution and a super-solution to f (λ(A u )) = 1 in Ω in the viscosity sense, respectively, (ii) and that u 1 < u 2 in Ω, and u 1 (x) = u 2 (x).…”

The Dirichlet problem for fully nonlinear degenerate elliptic equations with a singular nonlinearity

“…As usual B R (y) stands for the ball centered at y ∈ R N with radius R. When Ω is C 2 this is equivalent to require that all principal curvatures of the boundary are uniformly bounded from below by a positive constant, see [1,Proposition 2.7].…”