The hot spots conjecture

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The hotspots conjecture can be expressed in simple English as:

Suppose a flat piece of metal, represented by a two-dimensional bounded connected domain, is given an initial heat distribution which then flows throughout the metal. Assuming the metal is insulated (i.e. no heat escapes from the piece of metal), then given enough time, the hottest point on the metal will lie on its boundary.

In mathematical terms, we consider a two-dimensional bounded connected domain D and let u(x,t) (the heat at point x at time t) satisfy the heat equation with Neumann boundary conditions. We then conjecture that

For sufficiently large t > 0, u(x,t) achieves its maximum on the boundary of D

This conjecture has been proven for some domains and proven to be false for others. In particular it has been proven to be true for obtuse and right triangles, but the case of an acute triangle remains open. The proposal is that we prove the Hot Spots conjecture for acute triangles!

Note: strictly speaking, the conjecture is only believed to hold for generic solutions to the heat equation. As such, the conjecture is then equivalent to the assertion that the generic eigenvectors of the second eigenvalue of the Laplacian attain their maximum on the boundary.

Threads



Possible approaches

Combinatorial approach

Special cases

Isosceles triangles

Equilateral triangles

  • Quoting from [McC2011]: "in 1833, Gabriel Lam´e discovered analytical formulae for the complete eigenstructure of the Laplacian on the equilateral triangle under either Dirichlet or Neumann boundary conditions and a portion of the corresponding eigenstructure under a Robin boundary condition. Surprisingly, the associated eigenfunctions are also trigonometric. The physical context for his pioneering investigation was the propagation of heat throughout polyhedral bodies."
  • [McC2002] gives a complete description of Lame’s eigenfunctions for an

equilateral triangle with a Neumann boundary coundition (del U)/(del n) = 0 , n the normal to the triangular boundary, U an eigenfuntion. In section 8 on Modal Properties, he gives beautiful expressions for the eigenfunctions in terms of pairs of integers (m, n) in Equations 8.1 and 8.2, where (8.1) covers what McCartin calls the symmetric, and (8.2) the antisymmetric modes (respectively).

I’m intrigued by whether the modes in equations (8.1) and (8.2) always attain their max and min on the boundary of the triangular region (equilateral triangle); the 3D plots of several modes by McCartin seems consistent with max/min always being attained on the boundary; Lame solved the equilateral triangle case, in the sense that he gave a complete set of eigenfunctions for the Laplacian eigenvalue problem with the (del U)/(del n) = 0 boundary value condition.

Obtuse triangles

Right-angled triangles

General domains

Bibliography

  • [BB1999] Rodrigo Bañuelos and Krzysztof Burdzy. On the “hot spots” conjecture of J. Rauch. J.

Funct. Anal., 164(1):1–33, 1999.

  • [Ch1997] Fan R. K. Chung. Spectral graph theory, volume 92 of CBMS Regional Conference Series

in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997.

  • [Fi1973] Miroslav Fiedler. Algebraic connectivity of graphs. Czechoslovak Math. J., 23(98):298–

305, 1973.

  • [Fi1975] Miroslav Fiedler. A property of eigenvectors of nonnegative symmetric matrices and its

application to graph theory. Czechoslovak Math. J., 25(100)(4):619–633, 1975.