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Wiener criterion for a class of degenerate elliptic operators. (English) Zbl 0633.35018
The authors give some geometric criteria (analogous to Wiener’s Poincaré’s and Zaremba’s criteria for the Laplacian) for the regularity of boundary points for the Dirichlet problem relative to a class of partial differential operators of the form $$\sum^{n}_{j=1}X^ 2_ j$$, fulfilling Hörmander’s condition.
Reviewer: W.Wendt

##### MSC:
 35J25 Boundary value problems for second-order elliptic equations 35J67 Boundary values of solutions to elliptic equations and elliptic systems 35B65 Smoothness and regularity of solutions to PDEs 35A30 Geometric theory, characteristics, transformations in context of PDEs 35J70 Degenerate elliptic equations
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##### References:
 [1] Bauer, H, Harmonische Räume and ihre potentialtheorie, () · Zbl 0142.38402 [2] Bony, J.M, Principe du maximum, inegalité de Harnack et unicité du probleme de Cauchy pour LES operateurs elliptiques dégénérés, Ann. inst. Fourier (Grenoble), 19, No. 1, 277-304, (1969) · Zbl 0176.09703 [3] Constantinescu, C; Cornea, A, Potential theory on harmonic spaces, (1972), Springer-Verlag Berlin · Zbl 0248.31011 [4] Fefferman, C; Phong, D.H, Subelliptic eigenvalue problems, (), 590-606 · Zbl 0503.35071 [5] Franchi, B; Lanconelli, E, Une metrique associée a une classe d’operateurs elliptiques dégénérés, (), 105-114 · Zbl 0553.35033 [6] Franchi, B; Lanconelli, E, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. scuola norm. sup. Pisa cl. sci., 10, 4, 523-541, (1983) · Zbl 0552.35032 [7] Hermann, R, Differential geometry and the calculus of variations, (1968), Academic Press New York/London · Zbl 0219.49023 [8] Hervé, R.M, Recherches axiomatiques sur la theorie des fonctions surharmoniques et du potential, Ann. inst. Fourier (Grenoble), 12, 415-571, (1962) · Zbl 0101.08103 [9] Hervé, R.M; Hervé, M, LES fonctions surharmoniques dans l’axiomatique de M. Brelot associées a un operateur elliptique dégénéré, Ann. inst. Fourier (Grenoble), 22, 2, 131-145, (1972) · Zbl 0224.31014 [10] Hörmander, L, Hypoelliptic second order differential equations, Acta math., 119, 147-171, (1967) · Zbl 0156.10701 [11] Kellogg, O.D, Foundations of potential theory, (1967), Springer-Verlag Berlin · Zbl 0152.31301 [12] \scA. Nagel, E. M. Stein, and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math., in press. · Zbl 0578.32044 [13] Negrini, P; Scornazzani, V, Superharmonic functions and regularity of boundary point for a class of elliptic-parabolic partial differential operators, Boll. un. mat. ital. C, 3, No. 1, 85-107, (1984) · Zbl 0552.35035 [14] Sanchez-Calle, A, Fundamental solutions and geometry of the sum of squares of vector fields, Invent. math., 78, 143-160, (1984) · Zbl 0582.58004
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