Maximum principle for higher order operators in general domains

Abstract

We first prove De Giorgi type level estimates for functions in W 1,t (Ω), Ω ⊂ R N $ \Omega\subset{\mathbb R}^N $ , with t > N ≥ 2 $ t \gt N\geq 2 $ . This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi’s classes as obtained in Di Benedetto–Trudinger [10] for functions in W 1,2(Ω). As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.

Uniqueness results for higher order Lane-Emden systems

“…If m = 1 then (6) reduces to (5) with α = α 1 , and ( 7) coincides with the Serrin exponent N N −2α . In the case m = 2, α 1 = α, α 2 = β, p 1 = q, p 2 = p, (7) reduce to the Serrin curves:…”

“…Lemma 2 (Theorem 7.1 in [13]). Let u be a nontrivial solution to (5). Then it is radially symmetric and strictly decreasing in the radial variable.…”

“…We next prove a key ingredient for what follows: Lemma 3. Let u be a nontrivial solution to (5). Then, ∆ s u(0) < 0 if 1 ≤ s < α is odd, and in this case it is increasing until the first zero, ∆ s u(0) > 0 if 1 ≤ s < α is even, and in this case it is decreasing up to the first zero.…”

“…Clearly, this definition does not take into account boundary conditions according to which the iterated operator can be a power or not of the Laplacian. Higher order problems are more sensitive to boundary conditions with respect to the second order case, and the richness of plenty of physically significant boundary values is the challenge which prevents to using standard tools from elliptic theory for second order operators, such as the maximum principle which fails in general for domains which are not slight perturbations of the ball; sufficient conditions for the validity of a general maximum principle have been addressed in [5].…”

“…Notice that in the case α = 1, (5) reduces to the Lane-Emden equation (1) and for α = 2 to the biharmonic equation (2). The boundary conditions in (5) are the higher order Dirichlet boundary conditions for which the polyharmonic operator fails to be the power of the Laplace operator. Those conditions are particularly relevant form the point of view of applications, see [13], as well as from the theoretical point of view, as they prevent to use reduction methods, such as decomposing the equation into a system of lower order equations.…”