×

Gårding’s inequality for elliptic operators with degeneracy. (English. Russian original) Zbl 1205.47045

Math. Notes 87, No. 2, 189-203 (2010); translation from Mat. Zametki 87, No. 2, 201-216 (2010).
In the interesting paper under review, the author derives a Gårding-type weighted inequality for degenerate elliptic operators in an arbitrary (bounded or unbounded) domain of the \(n\)-dimensional Euclidean space \(\mathbb R^n.\) That inequality is used in the study of unique solvability of a specific variational problem of Dirichlet type. The lower coefficients of the considered degenerate elliptic operators are assumed to belong to suitable weighted \(L_p\)-spaces.

MSC:

47F05 General theory of partial differential operators
35J70 Degenerate elliptic equations
35J99 Elliptic equations and elliptic systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] L. D. Kudryavtsev, ”Direct and inverse embedding theorems: Applications to the solution of elliptic equations by variational methods,” in Trudy Mat. Inst. Steklov (Izd. AN SSSR, Moscow, 1959), Vol. 55, pp. 3–182 [in Russian].
[2] S. M. Nikol’skii, P. I. Lizorkin, and N. V. Miroshin, ”Weighted functional spaces and their application to investigation of boundary value problems for degenerate elliptic equations,” Izv. Vyssh.Uchebn. Zaved. Mat., No. 8, 4–30 (1988) [Soviet Math. (Iz. VUZ) 32 (8), 1–40 (1988)].
[3] N. V. Miroshin, ”Dirichlet variational problem for an elliptic operator singular on the boundary,” Differ. Uravn. 24(3), 455–464 (1988) [Differ. Equations 24 (3), 323–329 (1988)].
[4] S. A. Iskhokov, ”On the smoothness of solutions of degenerating differential equations,” Differ. Uravn. 31(4), 641–653 (1995) [Differ. Equations 31 (4), 594–606 (1995)]. · Zbl 0855.35047
[5] S. A. Iskhokov, ”On solvability and smoothness of a solution to the variational Dirichlet problem for degenerate elliptic equations in the half-space,” Mat. Zamet. YAGU 5(2), 85–106 (1998).
[6] S. A. Iskhokov, ”Smoothness of the generalized solution of an elliptic equation with nonpower degeneracy,” Differ. Uravn. 39(11), 1536–1542 (2003) [Differ. Equations 39 (11), 1618–1625 (2003)]. · Zbl 1329.35137
[7] L. Nirenberg, ”On elliptic partial differential equations,” Ann. Scuola Norm. Sup. Pisa (3) 13(3), 115–162 (1959). · Zbl 0088.07601
[8] V. I. Burenkov, Sobolev Spaces on Domains, in Teubner-Texte Math. (B. G. Teubner, Stuttgart, 1998), Vol. 137. · Zbl 0893.46024
[9] K. Yosida, Functional Analysis (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965; Mir, Moscow, 1967). · Zbl 0126.11504
[10] Yu. V. Egorov, Lectures on Partial Differential Equations: Supplementary Chapters (Izd. Moskov. Univ., Moscow, 1985) [in Russian].
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.