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Can anyone give me the name of this equation and what references i can find such equations :

$\left\{\begin{array}{ll} -\mu \Delta u+(\lambda+\mu)\nabla(\mbox{div }u)=Au,\ \ \ \ \mu>0,\ \ \lambda>0\\ u=0 \ \ \ \ \partial \Omega \end{array} \right.$

Student
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1 Answers1

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If it were $$\mu\Delta u + (\lambda + \mu) \nabla(\textrm{div} u)$$ (maybe your Laplacian has a different sign convention than mine?) it would be the "Static Lam\'e System" (see also "isotropic elasticity", "elastic wave equation" (a related, time-dependent PDE))

BaronVT
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  • Ok, i find it and it's the same as you metioned in the comment, but do you have any references or books i can find or read so i can know the existence of the solution and the regularity of it? – Student Feb 17 '14 at 20:15
  • Unfortunately, I don't know of a good "handbook", and I know of more sources for the wave version than this static version. That said, you might look at Stolk's "On the modeling and inversion of seismic data" (he deals with the general anisotropic elasticity, of which this is a spec. case) and McLaughlin/Yoon's "Unique identifiability of elastic parameters from time-dependent interior displacement measurement" (not the whole thing, just the general facts for the elastic wave such as finite speed of propagation). The basic idea is to treat this as a system of scalar PDE's and go from there. – BaronVT Feb 17 '14 at 20:28
  • do you have a name of a book that work on the well-posedness of this system? – Student Feb 18 '14 at 15:39
  • The Stolk book covers the well-posedness of the wave equation. You might also try Gould's "Intro. to Linear Elasticity" (I don't think it refers to "well-posedness", but I believe he does existence/uniquness) – BaronVT Feb 18 '14 at 18:03