Geodesic
A class of degenerate elliptic operators
Sbornik. Mathematics, Tome 8 (1969) no. 3, pp. 357-382 
A. V. Fursikov. A class of degenerate elliptic operators. Sbornik. Mathematics, Tome 8 (1969) no. 3, pp. 357-382. http://geodesic.mathdoc.fr/item/SM_1969_8_3_a3/
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     title = {A~class of degenerate elliptic operators},
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     url = {http://geodesic.mathdoc.fr/item/SM_1969_8_3_a3/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In a bounded region $G\subset R^n$ we consider an operator $A$ which is elliptic inside the region and degenerate on its boundary $\Gamma$. More precisely, the operator $A$ has the following form in the local coordinate system $(x',x_n)$, in which the boundary $\Gamma$ is given by the equation $x_n=0$ and $x_n>0$ for points in the region $G$: $$ Au=\sum_{|l'|+l_n+\beta\leqslant2m}a_{l',l_n,\beta}(x',x_n)q^\beta x_n^{l_n}D_{x'}^{l'}D_{x_n}^{l_n}u $$ where $q$ is a parameter, and $$ \sum_{|l'|+l_n+\beta=2m}a_{l',l_n,\beta}(x',0)q^\beta{\xi'}^{l'}{\xi_n}^{l^n}\ne0\quad\text{for}\quad|\xi|+|q|\ne0. $$ The operator $A$ will be proved Noetherian in certain spaces under the condition that $|q|$ is sufficiently large. In addition, some results will be obtained relating to how the smoothness of the solution of the equation $Au=f$ depends on the magnitude of the parameter. A theorem is formulated concerning unique solvability in approperiate spaces for a class of degenerate parabolic operators. Bibliography: 8 titles.

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[2] M. I. Vishik, G. I. Eskin, “Uravneniya v svertkakh v ogranichennoi oblasti”, Uspekhi matem. nauk, XX:3(123) (1965), 89–152

[3] A. S. Dynin, “Mnogomernye ellipticheskie kraevye zadachi s odnoi neizvestnoi funktsiei”, DAN SSSR, 141:2 (1961), 285–287 | MR | Zbl

[4] V. A. Kondratev, “Kraevye zadachi dlya parabolicheskikh uravnenii v zamknutykh oblastyakh”, Trudy Mosk. matem. ob-va, XV, 1966, 400–451

[5] L. N. Slobodetskii, “Otsenka v $L_2$ reshenii lineinykh ellipticheskikh i parabolicheskikh sistem”, Vestnik LGU, seriya mekh., matem. i astr., 1960, no. 7, 28–47

[6] L. Nirenberg, “Remarks on strongly elliptic partial differential equations”, Comm. Pure Appl. Math., 8 (1965), 648–674 | DOI | MR

[7] I. Peetre, “Another approach, to elliptic boundary problems”, Comm. Pure Appl. Math., 14 (1961), 711–731 ; Matematika, 7:1 (1963), 43–65 | DOI | MR | Zbl

[8] M. Schechter, “General boundary value problems for elliptic partial differential equations”, Comm. Pure Appl. Math., 12 (1959), 457–486 ; Matematika, 4:5 (1960), 93–122 | DOI | MR | Zbl