Boundary Yamabe Problem with Minimal Boundary Case: A Complete Solution.

2021 
This article provides a complete solution of the boundary Yamabe problem with minimal boundary scenario, or equivalently, the existence of a real, positive, smooth solution of $ -\frac{4(n -1)}{n - 2} \Delta_{g} u + S_{g} u = \lambda u^{\frac{n+2}{n - 2}} $ in $ M $, $ \frac{\partial u}{\partial \nu} + \frac{n-2}{2} h_{g} u = 0 $ on $ \partial M $. In contrast to the classical method of calculus of variation with assumptions on Weyl tensors and classification of types of points on $ \partial M $, boundary Yamabe problem is fully solved here in cases classified by the sign of the first eigenvalue $ \eta_{1} $ of the conformal Laplacian with Robin condition. The signs of scalar curvature $ S_{g} $ and mean curvature $ h_{g} $ play an important role in this existence result.
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