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Multiple solutions for some strongly degenerate second order elliptic equations

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 Added by Gaetano Siciliano
 Publication date 2020
  fields
and research's language is English




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We consider a boundary value problem in a bounded domain involving a degenerate operator of the form $$L(u)=-textrm{div} (a(x) abla u)$$ and a suitable nonlinearity $f$. The function $a$ vanishes on smooth 1-codimensional submanifolds of $Omega$ where it is not allowed to be $C^{2}$. By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where $a$ vanishes.



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We obtain asymptotic mean value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. We study both when the set of coefficients is bounded and unbounded (each case requires different techniques). The families of equations that we consider include well-known operators such as Pucci, Issacs, and $k-$Hessian operators.
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