𝑊^{2,𝑝}-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients

Chiarenza, Filippo; Frasca, Michele; Longo, Placido

doi:10.1090/S0002-9947-1993-1088476-1

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$W\sp {2,p}$ -solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients

Authors: Filippo Chiarenza, Michele Frasca and Placido Longo
Journal: Trans. Amer. Math. Soc. 336 (1993), 841-853
MSC: Primary 35R05; Secondary 35J25
MathSciNet review: 1088476
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Abstract: We prove a well-posedness result in the class ${W^{2,p}} \cap W_0^{1,p}$ for the Dirichlet problem

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {Lu = f} & {{\text{a.e.}}\;{\tex... ...Omega }, \\ {u = 0} & {{\text{on}}\;\partial \Omega }. \\ \end{array} } \right.$

We assume the coefficients of the elliptic nondivergence form equation that we study are in ${\text{VMO}} \cap {L^\infty }$ .

[AT] Angelo Alvino and Guido Trombetti, Second order elliptic equations whose coefficients have their first derivatives weakly-𝐿ⁿ, Ann. Mat. Pura Appl. (4) 138 (1984), 331–340 (English, with Italian summary). MR 779550 (86e:35040), http://dx.doi.org/10.1007/BF01762551
[Cf] Luis A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189–213. MR 1005611 (90i:35046), http://dx.doi.org/10.2307/1971480
[Cl] Alberto-P. Calderón, Integrales singulares y sus aplicaciones a ecuaciones diferenciales hiperbolicas, Cursos y Seminarios de Matemática, Fasc. 3. Universidad de Buenos Aires, Buenos Aires, 1960 (Spanish). MR 0123834 (23 #A1156)
[Cn] Albino Canfora, Extension of an integral inequality of C. Miranda and applications to the elliptic equations with discontinuous coefficients, Czechoslovak Math. J. 39(114) (1989), no. 3, 385–422. MR 1006306 (91b:35030)
[CZ] A. Canfora and P. Zecca, Su di una equazione ellittica in ${\mathbb{R}^n}$ a coefficienti discontinui e non derivabili, Ann. Mat. Pura Appl. (4) 141 (1985), 369-384.
[CFL] Filippo Chiarenza, Michele Frasca, and Placido Longo, Interior 𝑊^{2,𝑝} estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), no. 1, 149–168. MR 1191890 (93k:35051)
[CRW] R. R. Coifman, R. Rochberg, and Guido Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611–635. MR 0412721 (54 #843)
[GR] José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149 (87d:42023)
[G] John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1981. MR 628971 (83g:30037)
[GT] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
[Gr] Donato Greco, Nuove formole integrali di maggiorazione per le soluzioni di un’equazione lineare di tipo ellittico ed applicazioni alla teoria del potenziale, Ricerche Mat. 5 (1956), 126–149 (Italian). MR 0082030 (18,486d)
[K] A. I. Košelev, On boundedness of 𝐿_{𝑝} of derivatives of solutions of elliptic differential equations, Mat. Sb. N.S. 38(80) (1956), 359–372 (Russian). MR 0078563 (17,1213e)
[M1] Carlo Miranda, Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui, Ann. Mat. Pura Appl. (4) 63 (1963), 353–386 (Italian). MR 0170090 (30 #331)
[M2] Carlo Miranda, Partial differential equations of elliptic type, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York, 1970. Second revised edition. Translated from the Italian by Zane C. Motteler. MR 0284700 (44 #1924)
[S] Donald Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. MR 0377518 (51 #13690), http://dx.doi.org/10.1090/S0002-9947-1975-0377518-3
[T] Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press Inc., Orlando, FL, 1986. MR 869816 (88e:42001)
[Z1] P. Zecca, Teoremi di esistenza e di unicita' di soluzioni periodiche per una equazione ellittica a coefficienti discontinui e non derivabili, Ricerche Mat. 35 (1986), 61-74.
[Z2] -, Un nuovo teorema di esistenza per un'equazione ellittica a coefficienti discontinui e non derivabili, Ricerche Mat. 35 (1986), 215-229.

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