# zbMATH — the first resource for mathematics

Solutions positives et mesure harmonique pour des operateurs paraboliques dans des ouverts “Lipschitziens”. (Positive solutions and harmonic measure for parabolic operators in “Lipschitz” domains). (French) Zbl 0734.35040
Let L be a parabolic operator on $${\mathbb{R}}^{n+1}$$ written in divergence form and with Lipschitz coefficients relatively to an adapted metric. We compare, near the boundary, the relative behavior of positive L-solutions on a Lipschitz domain. We first establish a so-called weak boundary Harnack principle. We then establish a uniform Harnack principle for certain particular positive L-solutions. This principle then allows us to prove another strong boundary Harnack principle for certain pairs of positive L-solutions. Then, we can generalize to L-operators some of J. T. Kemper results: we characterize the Martin boundary for “Lipschitz” domains and we show that the positive L-solutions on such domains admit non tangential limits except for a negligible set with respect to harmonic measure. Finally, in the last part, and for slightly more regular domains, we establish the equivalence between harmonic measure, adjoint harmonic measure and surface measure thus developing some results of J. M. Wu and R. Kaufman.

##### MSC:
 35K10 Second-order parabolic equations 31B25 Boundary behavior of harmonic functions in higher dimensions 31C35 Martin boundary theory
Full Text:
##### References:
 [1] [1] , Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier, 28-4 (1978), 162-213. · Zbl 0377.31001 [2] [2] , Une propriété de la compactification de Martin d’un domaine euclidien, Ann. Inst. Fourier, 29-4 (1979), 71-90. · Zbl 0589.31002 [3] [3] , Régularité d’accés des bouts et frontière de Martin d’un domaine euclidien, J. Math. Pures & Appl., 63 (1984), 215-260. · Zbl 0509.31006 [4] [4] , Comparaison des mesures harmoniques et des fonctions de Green pour des opérateurs elliptiques sur un domaine lipschitzien, C. R. Acad. Sc., Paris 294, série 1 (1982), 505-508. · Zbl 0504.35037 [5] [5] , Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., 73 (1967), 890-896. · Zbl 0153.42002 [6] [6] , On the existence of boundary values for harmonic functions of several variables, Ark. för Math., 4 (1962).. · Zbl 0107.08402 [7] The analysis of linear partial differential operators IV (1984) · Zbl 0406.28009 [8] [8] , A relative Fatou theorem, Proc. Nat. Acad. Sci., USA, 45 (1959), 215-222. · Zbl 0106.07801 [9] [9] , Classical potential theory and its probabilistic counterpart, New York, Springer-Verlag, 1984. · Zbl 0549.31001 [10] [10] & , A new proof of Moser’s parabolic Harnack inequality via the old idea of Nash. Arch. Rat. Mech. and Anal., 96 (1986), 326-338. · Zbl 0652.35052 [11] [11] , & , Comparison Theorems for Temperatures in non-cylindrical domains, Atti della Accademia Nazionale dei Lincei, 77 (1984), 1-12. · Zbl 0625.35007 [12] [12] , & , A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations, Illin. J. of Maths., 30 n°4 (1986), 536-565. · Zbl 0625.35006 [13] [13] , & , Wiener’s criterion for divergence form parabolic operators with C1-Dini continious coefficients, Duke Math. Journal, 59-1 (1989), 191-232. · Zbl 0705.35057 [14] [14] , Partial differential equations of parabolic type, Prentice-Hall, Englewood cliffs, N.J., 1964. · Zbl 0144.34903 [15] [15] , Recherches sur la théorie axiomatique des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier, 12 (1962), 415-471. · Zbl 0101.08103 [16] [16] , Inégalités de Harnack à la frontière pour des opérateurs paraboliques, thèse Paris 11, France, 1989. · Zbl 0661.47042 [17] [17] , Inégalités de Harnack à la frontière pour des opérateurs paraboliques, C.R. Acad. Sci., Paris, 308, série 1 (1989), 401-404. · Zbl 0661.47042 [18] [18] , Inégalités de Harnack à la frontière pour des opérateurs paraboliques (2). Estimations de la mesure harmonique de certains ouverts de Rn+1, C.R. Acad. Sci., Paris, 308, série 1 (1989), 441-444. · Zbl 0661.47042 [19] [19] & , On the boundary values of harmonic functions, Trans. Amer. Math. Soc., 32 (1968), 307-322. · Zbl 0159.40501 [20] [20] & , Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc., 147 (1970), 507-528. · Zbl 0193.39601 [21] [21] & , Singularity of parabolic measures, Compositio Mathematica, 40 n°2 (1980), 243-250. · Zbl 0387.31001 [22] [22] , Temperatures in several variables: kernel functions, representations and parabolic boundary values, Trans. Amer. Math. Soc., 167 (1972), 243-262. · Zbl 0238.35039 [23] [23] & , Parabolic measure and the Dirichlet problem for the heat equation in two dimensions, Ind. U. Maths. Journal, 37, n°3 (1988). · Zbl 0698.35068 [24] [24] , Minimal positive harmonic functions, Trans. Amer. Math. Soc., (1941), 137-172. · JFM 67.0343.03 [25] [25] , A Harnack inequality for parabolic differential equations, Comm. Pure & Appl. Math., 17 (1964), 101-134. · Zbl 0149.06902 [26] [26] , Real and complex analysis, 2nd. ed., Mc Graw-Hill, 1974. · Zbl 0278.26001 [27] [27] , On the Harnack inequality for linear elliptic equations, J. Anal. Math., 4 (1956), 292-308. · Zbl 0070.32302 [28] [28] , Théorème de limites fines et problème de Dirichlet, Ann. Inst. Fourier, 18-2 (1968), 121-134. · Zbl 0187.35902 [29] [29] , On the boundary behavior of solutions to a class of elliptic partial differential equations, Ark. för Math., 6 (1967), 485-533. · Zbl 0166.37702 [30] [30] , On parabolic measures and subparabolic functions, Trans. Amer. Math. Soc., 251 (1979), 171-186. · Zbl 0426.35044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.