We first prove De Giorgi type level estimates for functions in W 1, t (Ω), Ω⊂RN $ \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.

中文翻译：

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一般域中高阶算子的最大原理

我们首先证明了 W 1, t (Ω), Ω⊂RN $ \Omega\subset{\mathbb R}^N $ 中函数的 De Giorgi 类型估计，其中 t>N≥2 $ t \gt N\geq 2 $ . 这种增强的可积性使我们能够为不一定属于 De Giorgi 类的函数建立新的 Harnack 类型不等式，如 Di Benedetto-Trudinger [10] 中针对 W 1,2 (Ω) 中的函数获得的。因此，如果考虑二阶导数，我们证明了强最大值原理对于任何偶数阶一致椭圆算子的有效性，在相当一般的二维和三维域中。