Chapter 14 Regularity of the Solutions 14.1 Some Inequalities We recall for beginners some very helpful inequalities in the following. Theorem 14.1.1 Let  an open set from the space IR n . We have the following clas- sical inequalities: 1 o . The inequality of Cauchy–Buniakovski–Schwartz. If the functions f and g are from L 2 (), then the product f g is a function from L 1 () and we have       f (x )g(x )dx     ≤   ( f (x )) 2 dx  1/2   (g(x )) 2 dx  1/2 . 2 o . The inequality of H˝ older. If the function f is from the space L P () where 1 < p < ∞, and the function g is from the space L q () where q is so that 1 = 1 p + 1 q , then the product f g is a function from L 1 () and we have       f (x )g(x )dx     ≤   ( f (x )) p dx  1/ p   (g(x )) q dx  1/q . 3 o . The inequality of Young. If the function f ∈ L p () ∩ L q (), where p and q are so that 1 ≤ p < q ≤∞, then f ∈ L r (), for any r ∈[ p, q ] and we have ‖ f ‖ L r () ≤‖ f ‖ α L p () ‖ f ‖ 1−α L q () , where α is chosen so that α p + 1 − α q = 1 r . The proofs of these inequalities can be found in many books, especially those dedi- cated to functional analysis. © Springer International Publishing AG, part of Springer Nature 2019 M. Marin and A. Öchsner, Essentials of Partial Differential Equations, https://doi.org/10.1007/978-3-319-90647-8_14 327