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Interior integral estimates on weak solutions of certain degenerate elliptic systems

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We prove the existence of second order derivatives of any weak solution of the system

$$\frac{\partial }{{\partial x_\alpha }}A_i^\alpha (\nabla u) = 0(i = 1,...,N)$$

under very mild conditions on the functions A i α . These conditions include the special case: A i α (ξ)=0 if ξ=0, A i α (ξ)=|ξ|p−2ξ i α if ξ≠0 (ξ∈ℝnN;α=1,...,n,i=1,...,N;1<p<2). Under a stronger condition on A i α we establish an appropriate Caccioppoli inequality which enables us to prove the integrability of (1+|∇u|2)(p−1)/42u to a certain power t > 2.


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Naumann, J. Interior integral estimates on weak solutions of certain degenerate elliptic systems. Annali di Matematica pura ed applicata 156, 113–125 (1990).

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  • Weak Solution
  • Mild Condition
  • Order Derivative
  • Elliptic System
  • Strong Condition