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Abstract Let Ω be a bounded, smooth domain of R N , N ≥ 1 . For each p > N we study the optimal function s = s p… Click to show full abstract
Abstract Let Ω be a bounded, smooth domain of R N , N ≥ 1 . For each p > N we study the optimal function s = s p in the pointwise inequality | v ( x ) | ≤ s ( x ) ‖ ∇ v ‖ L p ( Ω ) , ∀ ( x , v ) ∈ Ω ‾ × W 0 1 , p ( Ω ) . We show that s p ∈ C 0 0 , 1 − ( N / p ) ( Ω ‾ ) and that s p converges pointwise to the distance function to the boundary, as p → ∞ . Moreover, we prove that if Ω is convex, then s p is concave and has a unique maximum point.
Keywords:
morrey sobolev;
pointwise morrey;
sobolev inequality;
inequality;
optimal pointwise;
pointwise
Journal Title: Journal of Mathematical Analysis and Applications
Year Published: 2020
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Journal Title: Journal of Mathematical Analysis and Applications
Year Published: 2020
Link to full text (if available)
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recommendations!