A symmetry problem

By A. Ramm. 2004

663 citations recorded by the source corpus.

The following result is proved: {\bf Theorem.} Let $D\subset \R^3$ be a bounded domain homeomorphic to a ball, $|D|$ be its volume, $|S|$ be the surface area of its smooth boundary $S$, $D\subset B_R:=\{x:|x|\leq R\}$, and $H_R$ is the set of all harmonic in $B_R$ functions. If $$\frac 1 {|D|}\int_Dhdx=\frac 1 {|S|}\int_Shds\quad \forall h\in H_R,$$ then $D$ is a ball.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4bfe0c8a32080658cfe25c69dbedb69f59e39167

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In Geometry and Topology

Citations in the corpus

Inversion of terrestrial ecosystem model parameter values against eddy covariance measurements by Monte Carlo sampling (2005, methodology): “Developed for applications in nuclear physics (Metropolis et al., 1953), and later geophysics (Mosegaard & Tarantola, 1995; Mosegaard, 1998; Mosegaard & Rygaard-Hjalsted, 1999), it is now widely used in other fields of environmental modeling.”
Overdetermined problems with fractional Laplacian (2013, neutral): “Since a complete list of references cannot be given, we only quote: [1, 4, 7, 8, 11, 10, 12, 19, 20, 22, 23, 25, 28, 31, 32, 34, 33, 39, 38].”
Inverse problems for elliptic equations with power type nonlinearities (2019, methodology): “First and second order linearizations were also used in [KN02] for a nonlinear conductivity equation (see also [CNV19]).”
Inverse Problems in Heat Transfer (2009, background): “Finally, in section 2.6, directions for further investigation in the use of spectral methods for stochastic inverse problems are suggested.”
Spatial functional prediction from spatial autoregressive Hilbertian processes (2012, background): “The ill-posed nature of the moment-based estimation problem is then removed in terms of these bases (see, for example, Ramm, 2005).”
Bilinear equations, Bell polynomials and linear superposition principle (2013, background): “However, Hirota bilinear equations are special and there are many other bilinear differential equations which are not written in the Hirota bilinear form.”
Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves (2015, methodology): “As well, unique continuation results for waves have been useful tools to solve inverse problems, for instance using the boundary control method [Bel87] (see also the review article [Bel07] and the book [KKL01]).”
RAFT: Recurrent All-Pairs Field Transforms for Optical Flow (2020, methodology): “Another approach is to learn iterative updates directly from data [Adler and Öktem, 2017].”