Gagliardo–Nirenberg interpolation inequality
In mathematics, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the formula_1 norms of the weak derivatives of a function. The inequality “interpolates” among various values of "p" and orders of differentiation, hence the name. The result is of particular importance in the theory of elliptic partial differential equations. It was proposed by Louis Nirenberg and Emilio Gagliardo. Statement of the inequality. Suppose formula_2 are non-negative integers and that formula_3 and formula_4 are real numbers such that and Suppose furthermore that formula_7 is a function in formula_8  with "m" weak derivative in formula_9. Then the "j" weak derivative of formula_10 lies in formula_11 and there exists a constant formula_12 depending on formula_13 and formula_14 but independent of formula_10 such thatformula_16 The result has two exceptional cases: For functions formula_25 defined on a bounded Lipschitz domain formula_26, the interpolation inequality has the same hypotheses as above and reads for arbitrary formula_28, where the constants formula_29 depend on the domain formula_30 and on formula_31 in addition to the other parameters.