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Schröndiger Type and Relaxed Dirichlet Problems for the Subelliptic p-Laplacian

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Abstract

We study at first the solutions of a Schrödinger type problem relative to the subelliptic p-Laplacian: we prove, for potentials that are in the Kato space, an Harnack inequality on enough small intrinsic balls; the continuity of the solutions to the homogeneous Dirichlet problem follows from some estimates in the proof of the Harnack inequality. In the second part of the paper we study a relaxed Dirichlet problem for the subelliptic p-Laplacian and we prove a Wiener type criterion for the regularity of a point (with respect to our problem).

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References

  1. Adams, D.R. and Hedberg, L.I.: Function Spaces and Potential Theory, Grundlehren fur Mathematik 314, Springer-Verlag, Berlin, Heidelberg, New York, 1996.

    Google Scholar 

  2. Attouch, H.: Variational Convergence for Functions and Operators, Pitman, Applicable Mathematics Series, London, Marshfield, 1984.

    Google Scholar 

  3. Aizeman, M. and Simon, B.: 'Brownian motion and Harnack's inequality for Schródinger operators', Comm. Pure Appl. Math. 35(1982), 209-271.

    Google Scholar 

  4. Biroli, M.: 'The Wiener test for Poincaré-Dirichlet forms', in G. Sankaran et al. (eds), Classical and Modern Potential Theory and Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 1994, pp. 93-103.

    Google Scholar 

  5. Biroli, M.: 'An embedding property of Schechter type for Dirichlet forms on an homogeneous space', Rend. Ist. Lombardo A 128(1994), 247-255.

    Google Scholar 

  6. Biroli, M.: 'Nonlinear Kato measures and nonlinear subelliptic Schrödinger problems', Rend. Acc. Naz. Sc. detta dei XL, Memorie di Matematica e Appl. 21(1) (1997), 235-252.

    Google Scholar 

  7. Biroli, M. and Mosco, U.: 'A Saint Venant type Principle for Dirichlet forms on discontinuous media', Annali Mat. Pura Appl. 169(4) (1995), 125-181.

    Google Scholar 

  8. Biroli, M. and Mosco, U.: 'Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces', Rend. Acc. Naz. Lincei, Mat. e Appl. 6(9) (1995), 33-44.

    Google Scholar 

  9. Biroli, M. and Mosco, U.: 'Sobolev inequalities on homogeneous spaces', Potential Analysis 4(4) (1995), 311-324.

    Google Scholar 

  10. Biroli, M. and Mosco, U.: 'Kato spaces for Dirichlet forms', Potential Analysis, to appear.

  11. Biroli, M. and Tchou, N.: 'Subelliptic nonlinear equation with measure data', Rend. Acc. Naz. Sc. detta dei XL, Memorie di Matematica e Appl., to appear.

  12. Biroli, M. and Tchou, N.: 'Relaxed Dirichlet problem for the subelliptic p-Laplacian', Ann. Mat. Pura e Appl., to appear.

  13. Biroli, M., Picard, C. and Tchou, N.: (in preparation).

  14. Capogna, L., Danielli, D. and Garofalo, N.: 'An embedding theorem and the Harnack inequality for nonlinear subelliptic equations', Commun. in P.D.E. 18(1993), 1765-1794.

    Google Scholar 

  15. Capogna, L., Danielli, D. and Garofalo, N.: 'Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations', Am. J. ofMathematics 118(1997), 1153-1196.

    Google Scholar 

  16. Chiarenza, F., Fabes, E. and Garofalo, N.: 'Harnack's inequality for Scrödinger operators and the continuity of solutions', Proc. Amer. Math. Soc. 98(1986), 415-425.

    Google Scholar 

  17. Citti, G., Garofalo, N. and Lanconelli, E.: 'Harnack's inequality for sum of squares of vector fields plus a potential', Am. J. of Math. 115(3) (1993), 699-734.

    Google Scholar 

  18. Coifman, R.R. and Weiss, G.: Analyse harmonique noncommutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer-Verlag, Berlin, Heidelberg, New York, 1971.

    Google Scholar 

  19. Dal Maso, G.: 'Comportamento asintotico delle soluzioni di problemi di Dirichlet', Boll. U.M.I., VII Ser. A 11(2) (1997), 253-277.

    Google Scholar 

  20. DalMaso, G. and Mosco, U.: 'Wiener criteria and energy decay for relaxed Dirichlet problems', tiArch. Rat. Mech. Analysis} 95(1986), 345-387.

  21. Danielli, D.: 'Regularity at the boundary for solutions of nonlinear subelliptic equations', Indiana Un. Math. J. 44(1) (1995), 269-286.

    Google Scholar 

  22. Fefferman, C.L. and Phong, D.: 'Subelliptic eigenvalue problems', Harmonic Analysis, Wadsworth, Chicago, 1981, pp. 590-606.

    Google Scholar 

  23. Fefferman, C.L. and Sanchez Calle, A.: 'Fundamental solution for second order subelliptic operators', Ann. of Math. 124(1986), 247-272.

    Google Scholar 

  24. Franchi, B., Lu, G. and Wheeden, R.L.: 'Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations', Potential Analysis 4(4) (1995), 361-376.

    Google Scholar 

  25. Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Markov Processes, W. de Gruyter and Co., Berlin, Heidelberg, New York, 1994.

    Google Scholar 

  26. Gariepy, R. and Ziemer, P.W.: 'A regularity condition at the boundary for solutions of quasilinear elliptic equations', Arch. Rat. Mech. Analysis 67(1977), 25-39.

    Google Scholar 

  27. Jerison, D.: 'The Poincaré inequality for vector fields satisfying an Hörmander's condition', Duke Math. J. 53(1986), 502-523.

    Google Scholar 

  28. Jerison, D. and Sanchez Calle, A.: 'Subelliptic second order differential operators', Harmonic Analysis, Lecture Notes in Math. 1277, Springer-Verlag, Berlin, Heidelberg, New York, 1987, pp. 46-77.

    Google Scholar 

  29. Mal`y, J.: 'Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations atirregular points', Comm. Math. Univ. Carolinae 37(1996), 23-42.

    Google Scholar 

  30. Mal`y, J. and Ziemer, P.W.: Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs 51, American Mathematical Soc., 1997.

  31. Mosco, U.: 'Wiener criterion and potential estimates for the obstacle problem', Indiana Un. Math. J. 36(3) (1987), 455-494.

    Google Scholar 

  32. Nagel, A., Stein, E. and Weinger, S.: 'Balls and metrics defined by vector fields I: Basic properties', Acta Math. 155(1985), 103-147.

    Google Scholar 

  33. Rakotoson, J.M. and Ziemer, P.W.: 'Local behavior of solutions of quasilinear elliptic equations with general structure', Trans. of A.M.S. 319(1990), 747-764.

    Google Scholar 

  34. Sartori, E.: 'Local regularity for solutions to Schrödinger equations relative to Dirichlet forms', Rend. Acc. Naz. Sc. detta dei XL, Mem. di Mat. e Appl. 19(1) (1995), 169-189.

    Google Scholar 

  35. Sanchez Calle, A.: 'Fundamental solutions and geometry of square of vector field', Inv. Math. 78(1984), 143-160.

    Google Scholar 

  36. Vodop'yanov, S.K.: 'Potential theory on homogeneous groups', Math. USSR Sbornik 66(1990), 59-81.

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  1. Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

    Marco Biroli

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  1. Marco Biroli
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Biroli, M. Schröndiger Type and Relaxed Dirichlet Problems for the Subelliptic p-Laplacian. Potential Analysis 15, 1–16 (2001). https://doi.org/10.1023/A:1011204609821

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