A Gagliardo–nirenberg Inequality, with Application to Duality-based a Posteriori Estimation in the L Norm
نویسنده
چکیده
We prove the Gagliardo–Nirenberg-type multiplicative interpolation inequality ‖v‖L1(Rn) ≤ C‖v‖ 1/2 Lip′(Rn)‖v‖ 1/2 BV(Rn) ∀v ∈ Lip ′(Rn) ∩ BV(R), where C is a positive constant, independent of v. Here ‖·‖Lip′(Rn) is the norm of the dual to the Lipschitz space Lip 0(R) := C 0,1 0 (Rn) = C0,1(Rn) ∩C0(R) and ‖ · ‖BV(Rn) signifies the norm in the space BV(Rn) consisting of functions of bounded variation on Rn. We then use a local version of this inequality to derive an a posteriori error bound in the L1(Ω′) norm, with Ω̄′ ⊂ Ω = (0, 1)n, for a finite element approximation to a boundary-value problem for a first-order linear hyperbolic equation, under the limited regularity requirement that the solution to the problem belongs to BV(Ω).
منابع مشابه
A Gagliardo – Nirenberg inequality , with application to duality - based a posteriori error estimation in the L 1 norm
A Gagliardo–Nirenberg inequality, with application to duality-based a posteriori error estimation in the L 1 norm Dedicated to Professor Boško S. Jovanovi´c on the occasion of his sixtieth birthday Endre Süli We establish the Gagliardo–Nirenberg-type multiplicative interpolation inequality v L 1 (R n) ≤ Cv 1/2 Lip (R n) v 1/2 BV(R n) ∀v ∈ BV(R n), where C is a positive constant, independent of ...
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