Applications of the implicit function theorem to quasilinear elliptic boundary value problems with non-smooth data. (English) Zbl 0838.35044

Sufficient conditions are given in order to apply the implicit function theorem to quasilinear elliptic systems with mixed boundary conditions, where the coefficients depend on the space variable \(x\) and the functions \(u= (u_1,\dots, u_n)\) but not on their gradients. Non-smooth data are allowed in the sense that the boundary of the spatial domain is only Lipschitz, coefficients are only bounded and “genuine” mixed boundary conditions are included. The main tools for the proof are some theorems proving that superposition (or Nemitskii) operators between some Sobolev spaces of type \(W^{1,p}\) and their duals are continuously differentiable. This implies local existence and uniqueness of solutions. That the implicit function theorem cannot be applied in general to get such results was shown by J. M. Ball, R. J. Knops and J. E. Marsden [Two examples in nonlinear elasticity, Springer Lect. Notes Math. 665, 41-48 (1978; Zbl 0386.73012)].


35J65 Nonlinear boundary value problems for linear elliptic equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)


Zbl 0386.73012
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