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On the continuity of the minima of variational integrals in orlicz-sobolev spaces

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1165)

Abstract

In this paper I investigate the minima of the variational integral ∫Ωf(x,u,▽u)dx, where f(x,u,p) : ω × R1 × Rn → R1 satisfies the following condition: M(|p|) − K ≤ f(x,u,p) ≤ aM(|p|) + K with K ≥ 0, a ≥ 1, and M(t) − an N-function. I prove the continuity of such minima when M(t) satisfies the Δ2 condition and the following further condition: there exists an mε with \(\mathop {\lim }\limits_{\varepsilon \to 0^ + } \varepsilon m_\varepsilon = 0\)for which M(βt) ≤ ≤ βn-ε M(t) if 0 < β < 1 and t > 0.

Keywords

  • Banach Space
  • Variational Integral
  • Orlicz Space
  • Linear Hull
  • Distributional Derivative

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The author is a member of the GNAFA of the CNR.

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References

  1. ADAMS, R.A., Sobolev spaces, Academic Press, New York 1975.

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  2. GIAQUINTA, M. and E. GIUSTI, On the regularity of the minima of variational integrals, in print.

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  3. KRASNOSEL'SKII, M.A. and Y.B. RUTICKII, Convex functions and Orlicz spaces, P. Noordhoff LTD, Groningen 1961.

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  4. LADYZHENSKAJA, O.A. and N.N. URAL'TSEVA, Linear and quasilinear elliptic equations, Academic Press, New York 1968.

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  5. MORREY, C.B., Multiple integrals in the calculus of variations, Springer-Verlag, Berlin 1968.

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© 1985 Springer-Verlag

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Porru, G. (1985). On the continuity of the minima of variational integrals in orlicz-sobolev spaces. In: Ławrynowicz, J. (eds) Seminar on Deformations. Lecture Notes in Mathematics, vol 1165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076158

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  • DOI: https://doi.org/10.1007/BFb0076158

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16050-2

  • Online ISBN: 978-3-540-39734-2

  • eBook Packages: Springer Book Archive