# The strong Legendre condition and the well-posedness of mixed Robin problems on manifolds with bounded geometry

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Bernd Ammann, Nadine Große, Victor Nistor

**The strong Legendre condition and the well-posedness of mixed Robin problems on manifolds with bounded geometry**
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Let M be a smooth manifold with (smooth) boundary ∂M and bounded geometry and ∂_{D}M ⊂ ∂M be an open and closed subset. We prove the well-posedness of the mixed Robin boundary value problem Pu = f in M, u = 0 on ∂_{D}M, ∂^{P}_{ν} u + bu = 0 on ∂M \ ∂_{D}M under the following assumptions. First, we assume that P satisfies the strong Legendre condition (which reduces to the uniformly strong ellipticity condition in the scalar case) and that it has totally bounded coefficients (that is, that the coefficients of P and all their derivatives are bounded). Let ∂_{R}M ⊂ ∂M \ ∂_{D}M be the set where b≠ 0.
**An extended abstract is in the file.**

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The Paper was written on 8.8.2018

Last update 11.9.2018