On shifting the principal eigenvalue of Dirichlet problem to infinity with non-transversal incompressible drift
DOI:
https://doi.org/10.7146/math.scand.a-139656Abstract
We prove that it is always possible to add some divergence free drift vector field to some two dimensional spherical Dirichlet problem, such that the resulting principal eigenvalue lies above a prescribed bound. By construction those drift vector fields vanish on the boundary and their flow lines individually stay away from the boundary. The capacity of those drift vector fields to accelerate diffusivity originates from high frequency oscillation of the associated flow lines. The lower bounds for the spectrum are obtained through isoperimetric inequalities for flow invariant functions.
References
Adams, R. A., Sobolev spaces, Academic Press New York-London 1975.
Bérard, P. H., Spectral geometry: direct and inverse problems, Springer Lecture Notes in Mathematics 1207, Springer-Verlag, Berlin, 1986. https://doi.org/10.1007/BFb0076330
Berestycki, H., Hamel, F., and Nadirashvili, N., Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys. 253 (2005), no. 2, 451–480. https://doi.org/10.1007/s00220-004-1201-9
Chavel, I., Eigenvalues in Riemannian geometry, Academic Press, Inc., Orlando, FL, 1984.
Damak, M., Franke, B., and Yaakoubi, N., Accelerating planar Ornstein-Uhlenbeck diffusion with suitable drift, Discrete Contin. Dyn. Syst. 40 (2020), no. 7, 4093–4112. https://doi.org/10.3934/dcds.2020173
Feng, Y., and Iyer, G., Dissipation enhancement by mixing, Nonlinearity 32 (2019), no. 5, 1810–1851. https://doi.org/10.1088/1361-6544/ab0e56
Franke, B., Integral inequalities for the fundamental solutions of diffusions on manifolds with divergence-free drift, Math. Z. 246 (2004), no. 1–2, 373–403. https://doi.org/10.1007/s00209-003-0604-1
Franke, B., and Yaakoubi, N., On how to push the spectral gap of a diffusion on $S^2$ to infinity, Quart. Appl. Math. 74 (2016), no. 2, 321–335. https://doi.org/10.1090/qam/1426
Franke, B., and Yaakoubi, N., Accelerating diffusion on compact surfaces by drift, Anal. Appl. (Singap.) 15 (2017), no. 5, 653–666. https://doi.org/10.1142/S0219530516500184
Franke, B., Hwang, C.-R., Pai, H.-M., and Sheu S.-J., The behavior of the spectral gap under growing drift, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1325–1350. https://doi.org/10.1090/S0002-9947-09-04939-3
Gilbarg, D., and Trudinger, N. S., Elliptic partial differential equations of second order, Grundlehren der mathematischen Wissenschaften, 224. Springer-Verlag, Berlin-New York 1977.
Pai, H.-M., and Hwang, C.-R., Accelerating Brownian motion on $N$-torus, Statist. Probab. Lett. 83 (2013), no. 5, 1443–1447. https://doi.org/10.1016/j.spl.2013.02.009
Pommerenke, C., Boundary behavior of conformal maps, Grundlehren der mathematischen Wissenschaften, 299. Springer-Verlag, Berlin, 1992. https://doi.org/10.1007/978-3-662-02770-7
