Classical, viscosity and average solutions for PDE’s with nonnegative characteristic form
Gutiérrez, Cristian E. ; Lanconelli, Ermanno
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 15 (2004), p. 17-28 / Harvested from Biblioteca Digitale Italiana di Matematica
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We compare several definitions of weak solutions to second order partial differential equations with nonnegative characteristic form.

In questa Nota confrontiamo alcune nozioni di soluzione per equazioni alle derivate parziali del secondo ordine con forma caratteristica semidefinita positiva.

Publié le : 2004-03-01
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     author = {Cristian E. Guti\'errez and Ermanno Lanconelli},
     title = {Classical, viscosity and average solutions for PDE's with nonnegative characteristic form},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {15},
     year = {2004},
     pages = {17-28},
     zbl = {1098.35052},
     mrnumber = {2102746},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_2004_9_15_1_17_0}
}

Gutiérrez, Cristian E.; Lanconelli, Ermanno. Classical, viscosity and average solutions for PDE’s with nonnegative characteristic form. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 15 (2004) pp. 17-28. http://gdmltest.u-ga.fr/item/RLIN_2004_9_15_1_17_0/

[1] Bony, J.M., Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptic dégénérés. Ann. Inst. Fourier (Grenoble), 19, 1969, 277-304. | MR 262881 | Zbl 0176.09703

[2] M. Brelot et al. (eds.), Séminaire de théorie du potentiel, LNM 713. Berlin-Heidelberg-New York 1979.

[3] Citti, G. - Garofalo, N. - Lanconelli, E., Harnack’s inequality for sum of squares of vector fields plus a potential. Amer. J. Math., 115(3), 1993, 699-734. | MR 1221840 | Zbl 0795.35018

[4] Fulks, W., An approximate Gauss mean value theorem. Pacific J. Math., 14, 1964, 513-516. | MR 162047 | Zbl 0163.34503

[5] Fulks, W., A mean value theorem for the heat equation. Proc. Amer. Math. Soc., 17, 1966, 6-11. | MR 192200 | Zbl 0152.10503

[6] Garofalo, N. - Lanconelli, E., Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type. Trans. Amer. Math. Soc., 321(2), 1990, 775-792. | MR 998126 | Zbl 0719.35007

[7] Hansson, M., Harmonicity of functions satisfying a weak form of the mean value property. Arch. Math. (Basel), 76, 2001, 283-291. | MR 1825008 | Zbl 0992.31003

[8] Kupcov, L.P., The mean value property and the maximum principle for second order parabolic equations. Dokl. Akad. Nauk SSSR, 242(3), 1978, 529-532. | MR 507137 | Zbl 0436.35040

[9] Lanconelli, E. - Polidoro, S., On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Politec. Torino, 52(1), 1994, 29-63. | MR 1289901 | Zbl 0811.35018

[10] Lanconelli, E. - Pascucci, A., Superparabolic functions related to second order hypoelliptic operators. Potential Anal., 11(3), 1999, 303-323. | MR 1717108 | Zbl 0940.35054

[11] Pagani, C.D., Approximation of the solutions to the first boundary value problem for parabolic equations. Confer. Sem. Mat. Univ. Bari, 154, 1978, 26 pp. | MR 516134 | Zbl 0423.35046

[12] Pini, B., Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico. Rend. Sem. Mat. Univ. Padova, 23, 1954, 422-434. | MR 65794 | Zbl 0057.32801

[13] Pucci, C. - Talenti, G., Elliptic (second-order) partial differential equations with measurable coefficients and approximating integral equations. Advances in Math., 19(1), 1976, 48-105. | MR 419989

[14] Ramaswamy, S., Maximum principle for viscosity sub solutions and viscosity sub solutions of the Laplacian. Rend. Mat. Acc. Lincei, s. 9, v. 4, 1993, 213-217. | MR 1250500 | Zbl 0822.35019

[15] Uguzzoni, F., A note on a generalized form of the Laplacian and of sub-Laplacians. Arch. Math. (Basel), 80, 2003, 516-524. | MR 1995632 | Zbl 1083.31006

[16] Watson, N.A., A theory of subtemperatures in several variables. Proc. London Math. Soc., 26(3), 1973, 385-417. | MR 315289 | Zbl 0253.35045