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The “hot spots” conjecture on the Vicsek set. (English) Zbl 1410.28009
Summary: We prove the “hot spots” conjecture on the Vicsek set. Specifically, we will show that every eigenfunction of the second smallest eigenvalue of the Neumann Laplacian on the Vicsek set attains its maximum and minimum on the boundary.
MSC:
28A80 Fractals
47A75 Eigenvalue problems for linear operators
Software:
Maxima
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References:
[1] Atar R., Burdzy K., On Neumann eigenfunctions in lip domains, J. Amer. Math. Soc., 2004, 17(2), 243-265 · Zbl 1151.35322
[2] Bañuelos R., Burdzy K., On the “hot spots” conjecture of J. Rauch, J. Funct. Anal., 1999, 164(1), 1-33
[3] Jerison D., Nadirashvili N., The “hot spots” conjecture for domains with two axes of symmetry, J. Amer. Math. Soc., 2000, 13(4), 741-772 · Zbl 0948.35029
[4] Miyamoto Y., The “hot spots” conjecture for a certain class of planar convex domains, J. Math. Phys., 2009, 50(10), 103530 · Zbl 1283.35016
[5] Krejcirik D., Tušek M., Location of hot spots in thin curved strips, 2017, arXiv e-prints arXiv:1709.01279
[6] Burdzy K., The hot spots problem in planar domains with one hole, Duke Math. J., 2005, 129(3), 481-502 · Zbl 1154.35330
[7] Burdzy K., Werner W., A counterexample to the “hot spots” conjecture, Ann. of Math. (2), 1999, 149(1), 309-317 · Zbl 0919.35094
[8] Kigami J., Analysis on fractals, volume 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001 · Zbl 0998.28004
[9] Strichartz R. S., Differential Equations on Fractals: A Tutorial, Princeton University Press, Princeton, NJ, 2006 · Zbl 1190.35001
[10] Fukushima M., Shima T., On a spectral analysis for the Sierpinski gasket, Potential Anal., 1992, 1(1), 1-35 · Zbl 1081.31501
[11] Rammal R., Toulouse G., Random walks on fractal structures and percolation clusters, J. Phys. Lett., 1983, 44(10), L13-L22
[12] Shima T., On eigenvalue problems for the random walks on the Sierpinski pre-gaskets, Japan J. Indust. Appl. Math., 1991, 8(1), 127-141 · Zbl 0715.60088
[13] Shima T., On eigenvalue problems for Laplacians on p.c.f. self-similar sets, Japan J. Indust. Appl. Math., 1996, 13(1), 1-23 · Zbl 0861.58047
[14] Li X.-H., Ruan H.-J., The “hot spots” conjecture on higher dimensional Sierpinski gaskets, Commun. Pure Appl. Anal., 2016, 15(1), 287-297 · Zbl 1331.28018
[15] Ruan H.-J., The “hot spots” conjecture for the Sierpinski gasket, Nonlinear Anal., 2012, 75(2), 469-476 · Zbl 1229.28020
[16] Ruan H.-J., Zheng Y.-W., The “hot spots” conjecture on the level-3 Sierpinski gasket, Nonlinear Anal., 2013, 81, 101-109 · Zbl 1347.28011
[17] Lau K.-S., Li X.-H., Ruan H.-J., A counterexample to the “hot spots” conjecture on nested fractals, J. Fourier Anal. Appl., 2018, 24(1), 210-225 · Zbl 1388.28007
[18] Barlow M. T., Diffusions on fractals, In: Lectures on Probability Theory and Statistics (Saint-Flour, 1995), Lecture Notes in Math., Springer Berlin Heidelberg, 1998, 1690, 1-121
[19] Malozemov L., Teplyaev A., Self-similarity, operators and dynamics, Math. Phys. Anal. Geom., 2003, 6(3), 201-218 · Zbl 1021.05069
[20] Metz V., How many diffusions exist on the Vicsek snowflake?, Acta Appl. Math., 1993, 32(3), 227-241 · Zbl 0795.31011
[21] Zhou D., Spectral analysis of Laplacians on the Vicsek set, Pacific J. Math., 2009, 241(2), 369-398 · Zbl 1177.28029
[22] Constantin S., Strichartz R. S., Wheeler M., Analysis of the Laplacian and spectral operators on the Vicsek set, Commun. Pure Appl. Anal., 2011, 10(1), 1-44 · Zbl 1242.28009
[23] Barnsley M. F., Rising H., Fractals everywhere, Academic Press Professional, Boston, MA, second edition, 1993
[24] Hutchinson J. E., Fractals and self-similarity, Indiana Univ. Math. J., 1981, 30(5), 713-747 · Zbl 0598.28011
[25] Maxima, Maxima, a computer algebra system, version 5.41.0, 2017, http://maxima.sourceforge.net/
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