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Differentiability of Weak Solutions of Nonlinear Second Order Parabolic Systems with Quadratic Growth and Non Linearity q ≥ 2

  • L. Fattorusso
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)

Abstract

Let Ω be a bounded open subset of ℝn, let X = (x,t) be a point of ℝn × ℝN. In the cylinder Q = Ω × (−T,0), T>0, we deduce the local differentiability result
$$ u \in L^2 \left( { - a,0,H^2 \left( {B\left( \sigma \right),\mathbb{R}^N } \right)} \right) \cap H^l \left( { - a,0,L^2 \left( {B\left( \sigma \right),\mathbb{R}^N } \right)} \right) $$
for the solutions u of the class \( L^q \left( { - T,0,H^{l,q} \left( {\Omega ,\mathbb{R}^N } \right)} \right) \cap C^{0,\lambda } \left( {\bar Q,\mathbb{R}^N } \right) \) (0 < λ < 1, N integer ≥ 1) of the non linear parabolic system
$$ - \sum\limits_{i = 1}^n {D_i a^i \left( {X,u,Du} \right) + \partial u\partial t = B^0 \left( {X,u,Du} \right)} $$
with quadratic growth and non linearity q ≥ 2. This result had been obtained making use of the interpolation theory and an imbedding theorem of Gagliardo-Nirenberg type for functions u belonging to W l,q C 0,λ .

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References

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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • L. Fattorusso
    • 1
  1. 1.D.I.M.E.T. Faculty of EngineeringUniversity of Reggio CalabriaReggio CalabriaItaly

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