Soit un domaine borné à frontière lipschitzienne. On montre que est le compactifié de Martin pour une classe assez étendue d’opérateurs uniformément elliptiques aux dérivées partielles d’ordre deux.
Soient une variété riemannienne ouverte et un domaine relativement compact à frontière lipschitzienne. On a alors que est le compactifié de Martin défini par la restriction au domaine de l’opérateur de Laplace-Beltrami sur . Par conséquent, à chaque variété riemannienne ouverte on peut associer au plus une variété riemannienne compact à bord dont est l’intérieur.
The Martin compactification of a bounded Lipschitz domain is shown to be for a large class of uniformly elliptic second order partial differential operators on .
Let be an open Riemannian manifold and let be open relatively compact, connected, with Lipschitz boundary. Then is the Martin compactification of associated with the restriction to of the Laplace-Beltrami operator on . Consequently an open Riemannian manifold has at most one compactification which is a compact Riemannian manifold with boundary whose interior is .
@article{AIF_1978__28_2_25_0, author = {Taylor, John C.}, 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}, address = {Grenoble}, volume = {28}, number = {2}, year = {1978}, doi = {10.5802/aif.688}, mrnumber = {58 #6302}, zbl = {0363.31010}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.688/} }
TY - JOUR AU - Taylor, John C. TI - On the Martin compactification of a bounded Lipschitz domain in a riemannian manifold JO - Annales de l'Institut Fourier PY - 1978 SP - 25 EP - 52 VL - 28 IS - 2 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.688/ DO - 10.5802/aif.688 LA - en ID - AIF_1978__28_2_25_0 ER -
%0 Journal Article %A Taylor, John C. %T On the Martin compactification of a bounded Lipschitz domain in a riemannian manifold %J Annales de l'Institut Fourier %D 1978 %P 25-52 %V 28 %N 2 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/aif.688/ %R 10.5802/aif.688 %G en %F AIF_1978__28_2_25_0
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. http://archive.numdam.org/articles/10.5802/aif.688/
[1] 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] On topologies and boundaries in potential theory, Lecture Notes in Mathematics 175 Springer-Verlag, Berlin-Heidelberg, New York, 1971. | MR | Zbl
,[3] On the existence of boundary values for harmonic functions in several variables, Arkiv för Math., 4 (1962), 393-399. | MR | Zbl
,[4] Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier, 12 (1962), 415-571. | Numdam | MR | Zbl
,[5] Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc., 147 (1970), 507-527. | MR | Zbl
and ,[6] An axiomatic treatment of pairs of elliptic differential equations, Ann. Inst. Fourier, 16 (2) (1966), 167-208. | Numdam | Zbl
,[7] Minimal positive harmonic functions, Trans. Amer. Math. Soc., 49 (1941), 137-172. | JFM | MR | Zbl
,[8] Probability and potentials, Blaisdell Publishing Company, Waltham, Mass. 1966. | MR | Zbl
,[9] Barriers on cones for uniformly elliptic operators, Ann. di Mat. pura ed appl. (IV), 76 (1967), 93-106. | MR | Zbl
,[10] Elementary differential Topology, Annals of Math. Studies 54, Princeton University Press, Princeton N.J., 1963. | Zbl
,[11] Functionals of Finite Riemann Surfaces, Princeton University Press, Princeton, N.J., 1954. | MR | Zbl
and ,[12] On the Harnack inequality for linear elliptie equations, Jour. d'Anal. Math., 4 (1955-1956), 297-308. | Zbl
,[13] 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] The Topology of Fibre Bundles, Princeton University Press, Princeton, N.J., 1951. | MR | Zbl
,[15] The Martin boundary of equivalent sheaves, Ann. Inst. Fourier, XX (1) (1970) 433-456. | Numdam | MR | Zbl
,[16] Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, to appear.
,[17] Remarques sur la variation des fonctions sous-harmoniques, Ann. Inst. Fourier, 2 (1950), 101-112. | Numdam | MR | Zbl
,[18] Martin boundary for linear elliptic differential operators of second order in a manifold, J. Math. Soc. Japan, 16 (1964), 307-334. | MR | Zbl
,Cité par Sources :