Abstract
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.
Highlights
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
One of the most powerful tools in the study of partial di erential equations and nonlinear analysis is without any doubts the Maximum Principle (MP in the sequel)
It turns out to be fundamental in obtaining existence, uniqueness and regularity results in the theory of linear elliptic equations, as well as to establish qualitative properties of solutions to nonlinear equations
Summary
One of the most powerful tools in the study of partial di erential equations and nonlinear analysis is without any doubts the Maximum Principle (MP in the sequel). Let us merely mention that the roots of MP date back two centuries in the work of Gauss on harmonic functions, up to the ultimate version of Hopf [16], and further extended in the seminal work of Nirenberg [20], Alexandrov [2] and Serrin [24], within the foundations of modern theory of PDEs. The underlying idea is simple: positivity of a suitable set of derivatives of a function induces positivity of the function itself. The underlying idea is simple: positivity of a suitable set of derivatives of a function induces positivity of the function itself This is elementary true for real functions of one variable which vanish at the endpoints of an interval where −u′′(x) ≥ and the validity can be extended to second order uniformly elliptic operators for which a prototype is the Laplace operator:.
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