Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential

Moschini, Luisa ; Tesei, Alberto

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 16 (2005), p. 171-180 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

In this preliminary Note we outline some results of the forthcoming paper [11], concerning positive solutions of the equation $\partial_{t} u = \triangle u + \frac{c}{|x^{2}|} u \big( 0 < c < \frac{(n-2)^{2}}{4}; \, n \ge 3 \big)$. A parabolic Harnack inequality is proved, which in particular implies a sharp two-sided estimate for the associated heat kernel. Our approach relies on the unitary equivalence of the Schrödinger operator $Hu = - \triangle u - \frac{c}{|x|^{2}} u$ with the opposite of the weighted Laplacian $\triangle_{\lambda} v = \frac{1}{|x|^{\lambda}} \text{div} (|x|^{\lambda} \nabla v)$ when $\lambda = 2 - n + 2 \sqrt{c_{0} - c}$.

Résumé

In questa Nota preliminare si presentano alcuni risultati del successivo lavoro [11], riguardanti soluzioni positive dell'equazione $\partial_{t} u = \triangle u + \frac{c}{|x^{2}|} u \big( 0 < c < \frac{(n-2)^{2}}{4}; \, n \ge 3 \big)$. Si dimostra una disuguaglianza di Harnack parabolica, che in particolare implica una stima bilatera sul nucleo del calore associato. Il nostro approccio si basa sull'equivalenza unitaria dell'operatore di Schrödinger $Hu = - \triangle u - \frac{c}{|x|^{2}} u$ con l'opposto dell'operatore di Laplace pesato $\triangle_{\lambda} v = \frac{1}{|x|^{\lambda}} \text{div} (|x|^{\lambda} \nabla v)$ quando $\lambda = 2 - n + 2 \sqrt{c_{0} - c}$.

Détail

Publié le : 2005-09-01

MR 2227741 | Zbl 1225.35112

Bibliographie

[1] Barbatis, G. - Filippas, S. - Tertikas, A., Critical heat kernel estimates for Schrödinger operators via Hardy-Sobolev inequalities. J. Funct. Anal., 208, 2004, 1-30. | MR 2034290 | Zbl 1058.35049

[2] Brezis, H. - Dupaigne, L. - Tesei, A., On a semilinear elliptic equation with inverse-square potential. Selecta Math., 11, 2005, 1-7. | MR 2179651 | Zbl 1161.35383

[3] Chiarenza, F.M. - Serapioni, R.P., A remark on a Harnack inequality for degenerate parabolic equations. Rend. Sem. Mat. Univ. Padova, 73, 1985, 179-190. | MR 799906 | Zbl 0588.35013

[4] Davies, E.B., Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics, 92, Cambridge University Press, 1989. | MR 990239 | Zbl 0699.35006

[5] Fabes, E.D. - Stroock, D.W., A new proof of Moser's parabolic Harnack inequality via the old ideas of Nash. Arch. Rat. Mech. Anal., 96, 1986, 327-338. | MR 855753 | Zbl 0652.35052

[6] Grigoryan, A., The heat equation on non-compact Riemannian manifolds. Mat. Sb., 182, 1991, 55-87 (in Russian); Engl. transl.: Math. USSR Sb., 72, 1992, 47-77. | MR 1098839

[7] Grigoryan, A., Heat kernels on weighted manifolds and applications. Cont. Math., to appear; http:// www.ma.ic.ac.uk/~grigor/wma.pdf | MR 2218016 | Zbl 1106.58016

[8] Grigoryan, A. - Saloff-Coste, L., Stability results for Harnack inequalities. Ann. Inst. Fourier (Grenoble), 55, 2005, to appear; http://www.ma.ic.ac.uk/~grigor/vc1eps.pdf | MR 2149405 | Zbl 1115.58024

[9] Milman, P.D. - Semenov, Y.A., Heat kernel bounds and desingularizing weights. J. Funct. Anal., 202, 2003, 1-24. | MR 1994762 | Zbl 1036.35044

[10] Milman, P.D. - Semenov, Y.A., Global heat kernel bounds via desingularizing weights. J. Funct. Anal., 212, 2004, 373-398. | MR 2064932 | Zbl 1057.47043

[11] Moschini, L. - Tesei, A., Parabolic Harnack Inequality for the Heat Equation with Inverse-Square Potential. Forum Math., to appear. | MR 2328115 | Zbl 1145.35051

[12] Saloff-Coste, L., A note on Poincaré, Sobolev and Harnack inequalities. Int. Math. Res. Notes, 2, 1992, 27-38. | MR 1150597 | Zbl 0769.58054

[13] Saloff-Coste, L., Parabolic Harnack inequality for divergence form second order differential operators. Potential Anal., 4, 1995, 429-467. | MR 1354894 | Zbl 0840.31006

[14] Saloff-Coste, L., Aspects of Sobolev-Type Inequalities. London Math. Soc. Lecture Notes, 289, Cambridge University Press, 2002. | MR 1872526 | Zbl 0991.35002