We prove a C1-elliptic estimate of the form (Equation Presented) valid on any complete Riemannian manifold M and for any smooth solution of the Poisson equation Δψ = f which is defined in a neighbourhood of the geodesic ball B(x, r). Above, C is a constant which only depends on dim(M) and ∈ > 0 is arbitrary. In case of global solutions, the estimate is sensitive of the curvature growth on large balls and can be applied to deduce global results such as the zero-mean value property of f as in the compact setting.
Guneysu, B., Pigola, S. (2018). Quantitative C1-estimates on Manifolds. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2018(13), 4103-4119 [10.1093/imrn/rnx016].
Quantitative C1-estimates on Manifolds
Pigola S.
2018
Abstract
We prove a C1-elliptic estimate of the form (Equation Presented) valid on any complete Riemannian manifold M and for any smooth solution of the Poisson equation Δψ = f which is defined in a neighbourhood of the geodesic ball B(x, r). Above, C is a constant which only depends on dim(M) and ∈ > 0 is arbitrary. In case of global solutions, the estimate is sensitive of the curvature growth on large balls and can be applied to deduce global results such as the zero-mean value property of f as in the compact setting.
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