A relation between pointwise convergence of functions and convergence of functionals

1983/07/01 by Haïm Brézis, Elliott Lieb, Élliott H. Lieb · 81 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.1090/s0002-9939-1983-0699419-3

Abstract

We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace f Subscript n Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:mo></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex"> \ f_n \</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a sequence of uniformly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bounded functions on a measure space, and if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript n Baseline right-arrow f"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f_n → f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> pointwise a.e., then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts left-brace double-vertical-bar f Subscript n Baseline double-vertical-bar Subscript p Superscript p Baseline minus double-vertical-bar f Subscript n Baseline minus f double-vertical-bar Subscript p Superscript p Baseline right-brace equals double-vertical-bar f double-vertical-bar Subscript p Superscript p"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:munder> </mml:mrow> <mml:mrow> <mml:mo></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msubsup> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> <mml:mo> − </mml:mo> <mml:msubsup> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo> − </mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:mrow> <mml:mo></mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mi>f</mml:mi> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">lim _{n → ∞} \ ‖ f_n ‖_p^p - ‖ f_n - f ‖_p^p \ = ‖ f ‖_p^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">0 &gt; p &gt; ∞</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This result is also generalized in Theorem 2 to some functionals other than the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm, namely <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral StartAbsoluteValue j left-parenthesis f Subscript n Baseline right-parenthesis minus j left-parenthesis f Subscript n Baseline minus f right-parenthesis minus j left-parenthesis f right-parenthesis EndAbsoluteValue right-arrow 0"> <mml:semantics> <mml:mrow> <mml:mo> ∫

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