On the Martin compactification of a bounded Lipschitz domain in a riemannian manifold
Annales de l'Institut Fourier, Tome 28 (1978) no. 2, pp. 25-52.

Soit DR n un domaine borné à frontière lipschitzienne. On montre que D ¯ est le compactifié de Martin pour une classe assez étendue d’opérateurs uniformément elliptiques aux dérivées partielles d’ordre deux.

Soient X une variété riemannienne ouverte et MX un domaine relativement compact à frontière lipschitzienne. On a alors que M ¯ est le compactifié de Martin défini par la restriction au domaine D de l’opérateur de Laplace-Beltrami sur X. Par conséquent, à chaque variété riemannienne ouverte X on peut associer au plus une variété riemannienne compact à bord dont X est l’intérieur.

The Martin compactification of a bounded Lipschitz domain DR n is shown to be D ¯ for a large class of uniformly elliptic second order partial differential operators on D.

Let X be an open Riemannian manifold and let MX be open relatively compact, connected, with Lipschitz boundary. Then M ¯ is the Martin compactification of M associated with the restriction to M of the Laplace-Beltrami operator on X. Consequently an open Riemannian manifold X has at most one compactification which is a compact Riemannian manifold with boundary whose interior is X.

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     title = {On the {Martin} compactification of a bounded {Lipschitz} domain in a riemannian manifold},
     journal = {Annales de l'Institut Fourier},
     pages = {25--52},
     publisher = {Institut Fourier},
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     year = {1978},
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Taylor, John C. On the Martin compactification of a bounded Lipschitz domain in a riemannian manifold. Annales de l'Institut Fourier, Tome 28 (1978) no. 2, pp. 25-52. doi : 10.5802/aif.688. https://aif.centre-mersenne.org/articles/10.5802/aif.688/

[1] J. M. Bony, Majorations a priori et problèmes frontières elliptiques du second ordre, Sem. Choquet (Initiation à l'Analyse) 5e année 1965-1966 exposée 3. | Numdam | Zbl

[2] M. Brelot, On topologies and boundaries in potential theory, Lecture Notes in Mathematics 175 Springer-Verlag, Berlin-Heidelberg, New York, 1971. | MR | Zbl

[3] L. Carleson, On the existence of boundary values for harmonic functions in several variables, Arkiv för Math., 4 (1962), 393-399. | MR | Zbl

[4] Mme R.-M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier, 12 (1962), 415-571. | Numdam | MR | Zbl

[5] R. Hunt and R. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc., 147 (1970), 507-527. | MR | Zbl

[6] P. Lœb, An axiomatic treatment of pairs of elliptic differential equations, Ann. Inst. Fourier, 16 (2) (1966), 167-208. | Numdam | Zbl

[7] R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc., 49 (1941), 137-172. | JFM | MR | Zbl

[8] P.-A. Meyer, Probability and potentials, Blaisdell Publishing Company, Waltham, Mass. 1966. | MR | Zbl

[9] K. Miller, Barriers on cones for uniformly elliptic operators, Ann. di Mat. pura ed appl. (IV), 76 (1967), 93-106. | MR | Zbl

[10] J. R. Munkres, Elementary differential Topology, Annals of Math. Studies 54, Princeton University Press, Princeton N.J., 1963. | Zbl

[11] M. Schiffer and D. C. Spencer, Functionals of Finite Riemann Surfaces, Princeton University Press, Princeton, N.J., 1954. | MR | Zbl

[12] J. Serrin, On the Harnack inequality for linear elliptie equations, Jour. d'Anal. Math., 4 (1955-1956), 297-308. | Zbl

[13] J. Stallings, On infinite processes leading to differentiability in the complement of a point, publ. in Differential and Combinatorial Topology ed., by S. S. Cairns, Princeton University Press, Princeton, N.J., 1965. | MR | Zbl

[14] N. Steenrod, The Topology of Fibre Bundles, Princeton University Press, Princeton, N.J., 1951. | MR | Zbl

[15] J. C. Taylor, The Martin boundary of equivalent sheaves, Ann. Inst. Fourier, XX (1) (1970) 433-456. | Numdam | MR | Zbl

[16] A. Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, to appear.

[17] M. Brelot, Remarques sur la variation des fonctions sous-harmoniques, Ann. Inst. Fourier, 2 (1950), 101-112. | Numdam | MR | Zbl

[18] S. Ito, Martin boundary for linear elliptic differential operators of second order in a manifold, J. Math. Soc. Japan, 16 (1964), 307-334. | MR | Zbl

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