Matrix triangulation of hypoelliptic boundary value problems
Annales de l'Institut Fourier, Tome 42 (1992) no. 4, p. 805-824
Étant donné un problème aux limites dans Ω×[0,T] avec Ω ouvert de R n , (n>1), nous réduisons, par le procédé de triangulation des matrices, le problème donné à deux systèmes du premier ordre, et étudions des majorations des valeurs propres des matrices correspondantes. L’hypoellipticité jusqu’à la frontière est donc caractérisée en termes de l’opérateur de Calderon associé au problème donné.
Given a hypoelliptic boundary value problem on ω×[0,T) with ω an open set in R n , (n>1), we show by matrix triangulation how to reduce it to two uncoupled first order systems, and how to estimate the eigenvalues of the corresponding matrices. Parametrices for the first order systems are constructed. We then characterize hypoellipticity up to the boundary in terms of the Calderon operator corresponding to the boundary value problem.
@article{AIF_1992__42_4_805_0,
     author = {Artino, R. A. and Barros-Neto, J.},
     title = {Matrix triangulation of hypoelliptic boundary value problems},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {42},
     number = {4},
     year = {1992},
     pages = {805-824},
     doi = {10.5802/aif.1310},
     zbl = {0758.35098},
     mrnumber = {93k:35287},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1992__42_4_805_0}
}
Artino, R. A.; Barros-Neto, J. Matrix triangulation of hypoelliptic boundary value problems. Annales de l'Institut Fourier, Tome 42 (1992) no. 4, pp. 805-824. doi : 10.5802/aif.1310. http://www.numdam.org/item/AIF_1992__42_4_805_0/

[1] S. Agmon, A. Douglis, L. Nirenberg, Estimates Near the Boundary for Solutions of Elliptic Partial Differential Operators Satisfying General Boundary Conditions, I Comm. Pure Appl. Math., 12 (1959), 623-727 ; II Comm. Pure Appl. Math., 17 (1964), 35-92. | Zbl 0123.28706

[2] M.S. Agranovich, Boundary Value Problems for Systems of First Order Pseudodifferential Operators, Russian Math. Surveys, 24 (1969), 59-126. | MR 91h:35234 | Zbl 0193.06605

[3] R. A. Artino & J. Barros-Neto, The Factorization of Hypoelliptic Pseudodifferential Operators, to appear. | Zbl 0802.35174

[4] R. A. Artino & J. Barros-Neto, Hypoelliptic Boundary Value Problems, Lecture Notes in Pure and Applied Mathematics, vol. 53, Marcel Dekker, Inc., New York, (1980). | MR 81k:35031 | Zbl 0422.35021

[5] R. A. Artino & J. Barros-Neto, A Construction of a Parametrix for an Elliptic Boundary Value Problem, to appear in Portugaliae Mathematica. | Zbl 0824.35152

[6] J. Barros-Neto, The Parametrix of Regular Hypoelliptic Boundary Value Problems, Ann. Sc. Norm. Sup. di Pisa, vol. XXVI, Fasc. I (1972), 247-268. | Numdam | MR 57 #16952 | Zbl 0235.35026

[7] L. Boutet De Monvel, Boundary Value Problems for Elliptic Pseudodifferential Equations, Acta Math., 126 (1971), 11-51. | MR 53 #11674 | Zbl 0206.39401

[8] A. P. Calderon, Boundary Value Problems for Elliptic Equations, Proceedings of the Joint Soviet - American Symposium on Partial Differential Equations, Novosibirsk Acad. Sci. USSR, 1-4 (1963).

[9] I.G. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, Nauka, Moscow, 1973.

[10] L. Hörmander, On the Regularity of the Solutions of Boundary Value Problems, Acta Math., 99 (1958), 225-264. | Zbl 0083.09201

[11] L. Hörmander, The Analysis of Linear Partial Differential Operators II, Springer-Verlag, Berlin (1983). | Zbl 0521.35001

[12] L. Hörmander, On the Interior Regularity of Solutions of Partial Differential Operators, Comm. Pure and Appl. Math., vol XI (1958). | Zbl 0081.31501

[13] L. Hörmander, Pseudodifferential Operators and Hypoelliptic Equations, Proc. of Symposia in Pure Mathematics, vol. X (1967), 138-183. | Zbl 0167.09603

[14] H. Kumano-Go, Pseudodifferential Operators, The MIT Press, Cambridge, Mass. and London, England, 1981. | MR 84c:35113 | Zbl 0453.47026

[15] Y. Lopatinski, On a Method of Reducing Boundary Problems for a System of Differential Equations of Elliptic Type to Regular Equations, Ukrain. Math. Z., 5 (1953), 123-151.

[16] B. Malgrange, Sur une Classe d'Opérateurs Différentiels Hypoelliptiques, Bull. Soc. Math. France, 85 (1957), 282-306. | Numdam | MR 21 #5063 | Zbl 0082.09303

[17] X. J. Ping, The Factorization of Hypoelliptic Operators with Constant Strength (unpublished).

[18] M. E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, N.J. (1981). | MR 82i:35172 | Zbl 0453.47026

[19] F. Treves, Introduction to Pseudodifferential Operators and Fourier Integral Operators (2 vols.), Plenum, New York, 1980. | Zbl 0453.47027

[20] M.I. Visik, G.I. Eskin, Convolution Equations in a Bounded Region, Russian Math. Surveys, 20-3 (1965), 86-152. | Zbl 0152.34202