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In mathematics, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that estimates the weak derivatives of a function. The estimates are in terms of ''L''''p'' norms of the function and its derivatives, and 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== The inequality concerns functions ''u'': R''n'' → R. Fix 1 ≤''q'', ''r'' ≤ ∞ and a natural number ''m''. Suppose also that a real number ''α'' and a natural number ''j'' are such that : and : Then # every function ''u'': R''n'' → R that lies in ''L''''q''(R''n'') with ''m''th derivative in ''L''''r''(R''n'') also has ''j''th derivative in ''L''''p''(R''n''); # and, furthermore, there exists a constant ''C'' depending only on ''m'', ''n'', ''j'', ''q'', ''r'' and ''α'' such that :: The result has two exceptional cases: # If ''j'' = 0, ''mr'' < ''n'' and ''q'' = ∞, then it is necessary to make the additional assumption that either ''u'' tends to zero at infinity or that ''u'' lies in ''L''''s'' for some finite ''s'' > 0. # If 1 < ''r'' < ∞ and ''m'' − ''j'' − ''n'' ⁄ ''r'' is a non-negative integer, then it is necessary to assume also that ''α'' ≠ 1. For functions ''u'': Ω → R defined on a bounded Lipschitz domain Ω ⊆ R''n'', the interpolation inequality has the same hypotheses as above and reads : where ''s'' > 0 is arbitrary; naturally, the constants ''C''1 and ''C''2 depend upon the domain Ω as well as ''m'', ''n'' etc. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gagliardo–Nirenberg interpolation inequality」の詳細全文を読む スポンサード リンク
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