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Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators on Bounded Domains

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Abstract

On a smooth bounded domain \(\Omega \subset {\bf {\rm R}}^N\) we consider the Schrödinger operators − Δ − V, with V being either the critical borderline potential V(x) =  (N − 2)2/4 |x|−2 or V(x) =  (1/4) dist(x, ∂Ω)−2, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Schrödinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a series of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincaré. As a byproduct of our technique we are able to answer positively to a conjecture of E. B. Davies.

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References

  1. Aronson D.G. (1967). Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73: 890–896

    MATH  MathSciNet  Google Scholar 

  2. Baras P. and Goldstein J. (1984). The heat equation with a singular potential. Trans. Amer. Math. Soc. 284: 121–139

    Article  MATH  MathSciNet  Google Scholar 

  3. Barbatis G., Filippas S. and Tertikas A. (2003). A unified approach to improved L p Hardy inequalities with best constants. Trans. Amer. Math. Soc. 356(6): 2169–2196

    Article  MathSciNet  Google Scholar 

  4. Barbatis G., Filippas S. and Tertikas A. (2004). Critical heat kernel estimates for Schrödinger operators via Hardy-Sobolev inequalities. J. Funct. Anal. 208: 1–30

    Article  MATH  MathSciNet  Google Scholar 

  5. Brezis H. and Marcus M. (1997). Hardy’s inequalities revised. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1–2): 217–237

    MATH  MathSciNet  Google Scholar 

  6. Brezis H. and Vazquez J.L. (1997). Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madrid 10(2): 443–469

    MATH  MathSciNet  Google Scholar 

  7. Cabré X. and Martel Y. (1999). Existence versus explosion instantanée pour des équationes de la chaleur linéaires avec potentiel singulier. C.R. Acad. Sci. Paris Ser. I Math. 329(11): 973–978

    MATH  MathSciNet  ADS  Google Scholar 

  8. Chiarenza F.M. and Serapioni R.P. (1985). A remark on a Harnack inequality for degenerate parabolic equations. Rend. Sem. Mat. Univ. Padova 73: 179–190

    MathSciNet  MATH  Google Scholar 

  9. Davies E.B. (1986). Perturbations of ultracontractive semigroups. Quart. J. Math. Oxford 2(37): 167–176

    Article  Google Scholar 

  10. Davies E.B. (1987). The equivalence of certain heat kernel and Green function bounds. J. Funct. Anal. 71: 88–103

    Article  MATH  MathSciNet  Google Scholar 

  11. Davies E.B. (1987). Explicit constants for Gaussian upper bounds on heat kernels. Amer. J. of Math. 109: 319–334

    Article  MATH  Google Scholar 

  12. Davies E.B. (1989). Heat kernels and spectral theory. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  13. Davies E.B. and Simon B. (1984). Ultracontractivity and the heat kernels for Schrodinger operators and Dirichlet Laplacians. J. Funct. Anal. 59: 335–395

    Article  MATH  MathSciNet  Google Scholar 

  14. Davies E.B. and Simon B. (1991). L p norms of non-critical Schrödinger semigroups. J. Funct. Anal. 102: 95–115

    Article  MATH  MathSciNet  Google Scholar 

  15. Dávila J. and Dupaigne L. (2004). Hardy type inequalities. J. Eur. Math. Soc. 6(3): 335–365

    Article  MATH  MathSciNet  Google Scholar 

  16. De Giorgi, E.: Sulla differenziabilitá e l’analiticitá delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (3), n. 3, 25–43 (1957)

  17. Fabes E.B., Garofalo N. and Salsa S. (1986). A backward Harnack inequality and Fatou Theorem for nonnegative solutions of parabolic equations. Ill. J. Math. 30(4): 536–565

    MATH  MathSciNet  Google Scholar 

  18. Fabes, E.B., Kenig, C.E., Jerison, D.: Boundary behavior of solutions to degenerate elliptic equations. Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Belmont, CA: Wadsworth, 1983, pp. 577–589

  19. Fabes E.B., Kenig C.E. and Serapioni R.P. (1982). The local regularity of solutions of degenerate elliptic equations. Comm. Part. Diff. Eq. 7: 77–116

    MATH  MathSciNet  Google Scholar 

  20. Fabes E.B. and Stroock D.W. (1986). A new proof of Moser’s parabolic Harnack inequality via the old ideas of Nash. Arch. Rat. Mech. Anal. 96: 327–338

    Article  MATH  MathSciNet  Google Scholar 

  21. Filippas S., Maz’ya V.G. and Tertikas A. (2004). Sharp Hardy-Sobolev inequalities. C.R. Math. Acad. Sci. Paris 339(7): 483–486

    MATH  MathSciNet  Google Scholar 

  22. Filippas S., Maz’ya V.G. and Tertikas A. (2007). Critical Hardy-Sobolev inequalities. J. Math. Pures Appl. 87: 37–56

    MATH  MathSciNet  Google Scholar 

  23. Filippas S. and Tertikas A. (2002). Optimizing Improved Hardy inequalities. J. Funct. Anal. 192: 186–233

    Article  MATH  MathSciNet  Google Scholar 

  24. Gao P. (1993). The boundary Harnack principle for some degenerate elliptic operators. Comm. Partial Differ. Eq. 18(12): 2001–2022

    MATH  Google Scholar 

  25. Grigoryan, A.: The heat equation on non-compact Riemannian manifolds (in Russian). Matem. Sbornik 182(1), 55–87 (1991). Engl. Transl.: Math. USSR Sb. 72(1), 47–77 (1992)

    Google Scholar 

  26. Grigoryan A. (2006). Heat kernels on weighted manifolds and applications. Cont. Math. 398: 93–191

    MathSciNet  Google Scholar 

  27. Grigoryan A. and Saloff-Coste L. (2005). Stability results for Harnack inequalities. Ann. Inst. Fourier, Grenoble 55(3): 825–890

    MathSciNet  Google Scholar 

  28. Kufner, A.: Weighted Sobolev spaces. Teubner-Texte zur Mathematik, 31, Stüttgart, Teubner, 1981

  29. Kufner, A., Opic, B.: Hardy type inequalities. Pitman Research Notes in Math Series 219, London: Pitman, 1990

  30. Li P. and Yau S.-T. (1986). On the parabolic kernel of the Schrödinger operator. Acta Math. 156(3–4): 153–201

    Article  MathSciNet  Google Scholar 

  31. Mazya V.G. (1985). Sobolev spaces. Springer-Verlag, Berlin-Heidelberg, New York

    Google Scholar 

  32. Milman P.D. and Semenov Yu.A. (2004). Global heat kernel bounds via desingularizing weights. J. Funct. Anal. 212: 373–398

    Article  MathSciNet  Google Scholar 

  33. Moschini L. and Tesei A. (2005). Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential. Rend. Mat. Acc. Lincei 16: 171–180

    Article  MATH  MathSciNet  Google Scholar 

  34. Moschini, L., Tesei, A.: Parabolic Harnack inequality for the heat equation with inverse-square potential. To appear in Forum Mathematicum

  35. Moser J. (1961). On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14: 577–591

    Article  MATH  MathSciNet  Google Scholar 

  36. Moser, J.: A Harnack inequality for parabolic differential equations. Comm. Pure. Appl. Math. 17, 101–134 (1964); Correction: 20, 231–236 (1967)

    Google Scholar 

  37. Nash J. (1958). Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80: 931–954

    Article  MATH  MathSciNet  Google Scholar 

  38. Saloff-Coste L. (1992). A note on Poincaré, Sobolev and Harnack inequalities. Internat. Math. Res. Notices 2: 27–38

    Article  MathSciNet  Google Scholar 

  39. Saloff-Coste, L.: Aspects of Sobolev-type inequalities. London Math. Soc. Lecture Notes Series 289, Cambridge: Cambridge University Press, 2002

  40. Vázquez J.L. and Zuazua E. (2000). The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. J. Funct. Anal. 173: 103–153

    Article  MATH  MathSciNet  Google Scholar 

  41. Zhang Qi. S. (2002). The boundary behaviour of heat kernels of Dirichlet Laplacians. J. Diff. Eq. 182: 416–430

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Crete, 71409, Heraklion, Greece

    Stathis Filippas

  2. Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, University of Rome “La Sapienza”, 00161, Rome, Italy

    Luisa Moschini

  3. Department of Mathematics, University of Crete, 71409, Heraklion, Greece

    Achilles Tertikas

  4. Institute of Applied and Computational Mathematics, FORTH, 71110, Heraklion, Greece

    Stathis Filippas & Achilles Tertikas

Authors
  1. Stathis Filippas
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  2. Luisa Moschini
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  3. Achilles Tertikas
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Correspondence to Achilles Tertikas.

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Communicated by B. Simon

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Filippas, S., Moschini, L. & Tertikas, A. Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators on Bounded Domains. Commun. Math. Phys. 273, 237–281 (2007). https://doi.org/10.1007/s00220-007-0253-z

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  • DOI: https://doi.org/10.1007/s00220-007-0253-z

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