Optimal trace Nash inequality
Copyright policy )Optimal trace Nash inequality
The author gives lower and upper bounds for the optimal constant~\(C_n\) in the trace Nash inequality \[ \left(\int_{\partial\mathbb R^n_+}u^2 ds\right)^{n\over n-1} \leq C_n\int_{\mathbb R^n_+}\left|\nabla u\right|^2 dx \left(\int_{\partial\mathbb R^n_+}|u|ds\right)^{2\over n-1} . \] A version for compact Riemannian manifolds with boundary is stated. The author gives detailed proofs in [Geom. Funct. Anal. 11, 759-772 (2001; Zbl 1005.58014)].
Partial differential inequalities and systems of partial differential inequalities, trace Nash inequality, Elliptic equations on manifolds, general theory, Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.), Methods of global Riemannian geometry, including PDE methods; curvature restrictions, optimal constants, manifolds with boundary, Sobolev spaces, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, Geometric theory, characteristics, transformations in context of PDEs
Partial differential inequalities and systems of partial differential inequalities, trace Nash inequality, Elliptic equations on manifolds, general theory, Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.), Methods of global Riemannian geometry, including PDE methods; curvature restrictions, optimal constants, manifolds with boundary, Sobolev spaces, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, Geometric theory, characteristics, transformations in context of PDEs
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