A relation between pointwise convergence of functions and convergence of functionals
Authors:
Haïm Brézis and Elliott Lieb
Journal:
Proc. Amer. Math. Soc. 88 (1983), 486-490
MSC:
Primary 28A20; Secondary 46E30, 49A99
DOI:
https://doi.org/10.1090/S0002-9939-1983-0699419-3
MathSciNet review:
699419
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Abstract: We show that if $\left \{ {{f_n}} \right \}$ is a sequence of uniformly ${L^p}$-bounded functions on a measure space, and if ${f_n} \to f$ pointwise a.e., then ${\lim _{n \to \infty }}\left \{ {\left \| {{f_n}} \right \|_p^p - \left \| {{f_n} - f} \right \|_p^p} \right \} = \left \| f \right \|_p^p$ for all $0 < p < \infty$. This result is also generalized in Theorem 2 to some functionals other than the ${L^p}$ norm, namely $\int \left | {j({f_n}) - j({f_n} - f) - j(f)} \right | \to 0$ for suitable $j:{\mathbf {C}} \to {\mathbf {C}}$ and a suitable sequence $\left \{ {{f_n}} \right \}$. A brief discussion is given of the usefulness of this result in variational problems.
- Elliott H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. MR 717827, DOI https://doi.org/10.2307/2007032
- Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. MR 709644, DOI https://doi.org/10.1002/cpa.3160360405
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Additional Information
Keywords:
Convergence of functionals,
pointwise convergence,
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Article copyright:
© Copyright 1983
American Mathematical Society