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Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations

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  • In this work, a biharmonic equation with an impedance (non standard) boundary condition and more general equations are considered. The study is performed in the space $L^{p}(-1,0$ $;$ $X)$, $1 < p < \infty $, where $X$ is a UMD Banach space. The problem is obtained through a formal limiting process on a family of boundary and transmission problems $(P^{\delta})_{\delta > 0}$ set in a domain having a thin layer. The limiting problem models, for instance, the bending of a thin plate with a stiffness on a part of its boundary (see Favini et al. [13]).
        We build an explicit representation of the solution, then we study its regularity and give a meaning to the non standard boundary condition.
    Mathematics Subject Classification: 31A30, 35J40, 31A10, 35Q74, 34K30.

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