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Every superposition operator mapping one Sobolev space into another is continuous. (English) Zbl 0418.46024


MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47H99 Nonlinear operators and their properties
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References:

[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101
[2] Coffman, C. V.; Duffin, R.; Mizel, V. J., Positivity of weak solutions of non-uniformly elliptic equations, Ann. Mat. Pura Appl., 104, 209-238 (1975) · Zbl 0302.35040
[3] Deuel, J.; Hess, P., A criterion for the existence of solutions of nonlinear elliptic boundary value problems, (Proc. Roy. Soc. Edinburgh Sect. A, 74 (1974-1975)), 49-54 · Zbl 0331.35028
[4] Marcus, M.; Mizel, V. J., Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal., 45, 294-320 (1972) · Zbl 0236.46033
[5] Marcus, M.; Mizel, V. J., Nemitsky operators on Sobolev spaces, Arch. Rational Mech. Anal., 51, 347-370 (1973) · Zbl 0266.46029
[6] Marcus, M.; Mizel, V. J., Complete characterization of functions which act, via superposition, on Sobolev spaces, Trans. Amer. Math. Soc., 251, 187-218 (1979) · Zbl 0417.46035
[7] Milman, V. D., Geometric theory of Banach spaces, part II, Russian Math. Surveys, 26, No. 6, 79-163 (1971)
[8] Shmulyan, V. L., Some geometrical properties of the sphere in spaces of type (B), Dokl. Akad. Nauk SSSR, 24, 648-652 (1939) · JFM 65.1312.03
[9] Stampacchia, G., Équations elliptiques du second ordre à coefficients discontinue (1966), Les Presses de l’Université de Montréal · Zbl 0151.15501
[10] Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton Univ. Press: Princeton Univ. Press Princeton, N. J · Zbl 0207.13501
[11] de la Valée Poussin, Ch. J., Sur l’intégrale de Lebesgue, Trans. Amer. Math. Soc., 16, 435-501 (1915) · JFM 45.0441.06
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