Asymptotic dirichlet problem for schrodinger operator and rough isometry

Abstract

The asymptotic Dirichlet problem for harmonic functions on a noncompact complete Riemannian manifold has a long history. It is to find the harmonic function satisfying the given Dirichlet boundary condition at infinity. By now, it is well understood [A, AS, Ch, S], when M is a Cartan-Hadamard manifold with sectional curvature $-b^2 \leq K_M \leq -a^2 < 0$. (By a Cartan-Hadamard manifold, we mean a complete simply connected manifold of non-positive sectional curvature.)

Keywords

• Rough isometry;
• asymptotic boundary;
• boundary at infinity;
• Dirichlet problem;
• Schrodinger operator;
• net;
• Poincare inequality;
• Sobolev inequality

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References

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