An optimal pointwise Morrey-Sobolev inequality.
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2020
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Let Ω be a bounded, smooth domain of RN , N ≥ 1. For each p > N we study the optimal function s = sp in the pointwise inequality |v(x)| ≤ s(x) ∇vLp(Ω) , ∀ (x, v) ∈ Ω × W1,p 0 (Ω). We show that sp ∈ C0,1−(N/p) 0 (Ω) and that sp converges pointwise to the distance function to the boundary, as p → ∞. Moreover, we prove that if Ω is convex, then sp is concave and has a unique maximum point.
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Dirac delta distribution, Infinity Laplacian
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ERCOLE, G.; PEREIRA, G. de A. An optimal pointwise Morrey-Sobolev inequality. Journal of Mathematical Analysis and Applications, v. 489, n. 1, artigo 124143, 2020. Disponível em: <https://www.sciencedirect.com/science/article/pii/S0022247X2030305X>. Acesso em: 06 jul. 2022.