Abstract
In the last decades, comparison results of Talenti type for Elliptic Problems with Dirichlet boundary conditions have been widely investigated. In this paper, we generalize the results obtained in Alvino et al. (Commun Pure Appl Math, to appear) to the case of p-Laplace operator with Robin boundary conditions. The point-wise comparison, obtained in Alvino et al. (to appear) only in the planar case, holds true in any dimension if p is sufficiently small.
Highlights
Let be a positive parameter and let Ω be a bounded open set of Rn, n ≥ 2, with Lipschitz boundary.Let f ∈ Lp (Ω) be a non-negative function
Different kinds of boundary conditions are considered by Alvino, Nitsch and Trombetti in [3], where they establish a comparison between a suitable norm of u and v, respectively, solution to
If we show that the functional admits a minimum, our problem will always have a solution
Summary
Let be a positive parameter and let Ω be a bounded open set of Rn , n ≥ 2 , with Lipschitz boundary. The first step is contained in [10], where Talenti proved a pointwise comparison result between u♯ and v in the case of Dirichlet Laplacian. Different kinds of boundary conditions are considered by Alvino, Nitsch and Trombetti in [3], where they establish a comparison between a suitable norm of u and v, respectively, solution to. Another work that is worth to be mentioned is [1], where the authors obtained similar results to [3] in the case of mixed Dirichlet and Robin boundary conditions. 3, we prove the main results about comparison of the two solutions in terms of the Lorentz norm.
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