Minimizing movements and evolution problems in Euclidean spaces

AbstractWe construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions {S^{J}\times\mathbb{R}^{n-J}}, {1\leq J\leq n-1}. These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.

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Equiangular lines in Euclidean spaces

Journal of Combinatorial Theory Series A ◽

10.1016/j.jcta.2015.09.008 ◽

2016 ◽

Vol 138 ◽

pp. 208-235 ◽

Cited By ~ 25

Author(s):

Gary Greaves ◽

Jacobus H. Koolen ◽

Akihiro Munemasa ◽

Ferenc Szöllősi

Keyword(s):

Euclidean Spaces ◽

Equiangular Lines

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Volumetric bounds for intersections of congruent balls in Euclidean spaces

Aequationes Mathematicae ◽

10.1007/s00010-021-00814-w ◽

2021 ◽

Author(s):

Károly Bezdek

Keyword(s):

Euclidean Spaces

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Regularity and time behavior of the solutions to weak monotone parabolic equations

Journal of Evolution Equations ◽

10.1007/s00028-021-00709-y ◽

2021 ◽

Author(s):

Maria Michaela Porzio

Keyword(s):

Heat Equation ◽

Parabolic Equations ◽

Parabolic Problems ◽

Summable Function ◽

Time Behavior ◽

Decay Estimates ◽

Laplacian Equation ◽

Evolution Problems ◽

Degenerate Equations ◽

Uniqueness Results

AbstractIn this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum $$u_0$$ u 0 is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of $$u_0$$ u 0 , immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.

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