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Interaction between nonlinear diffusion and mean curvature of a surface with initial discontinuities
Abstract
We consider the Cauchy problems for a class of nonlinear diffusion equations where each initial data is given by a characteristic function of an open set Ω in the whole Euclidean space. If the boundary ∂Ω is of class of C2 in a neighborhood of a point x on it, then we extract the mean curvature of ∂Ω at x from the initial diffusion at x. This corresponds to the author's previous result for the two-phase linear diffusion equation. Applications to some overdetermined problems for nonlinear diffusion equations are given.
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BibTeX
@misc{sakaguchi2026interaction,
title = {{Interaction between nonlinear diffusion and mean curvature of a surface with initial discontinuities}},
author = {Shigeru Sakaguchi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14358},
eprint = {2609.14358},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14358},
}
RIS
TY - UNPB
TI - Interaction between nonlinear diffusion and mean curvature of a surface with initial discontinuities
AU - Shigeru Sakaguchi
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14358
UR - https://arxiv.org/abs/2609.14358
AB - We consider the Cauchy problems for a class of nonlinear diffusion equations where each initial data is given by a characteristic function of an open set Ω in the whole Euclidean space. If the boundary ∂Ω is of class of C^2 in a neighborhood of a point x on it, then we extract the mean curvature of ∂Ω at x from the initial diffusion at x. This corresponds to the author's previous result for the two-phase linear diffusion equation. Applications to some overdetermined problems for nonlinear diffusion equations are given.
ER -