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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-1993-1088476-1
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 }$ .

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  • [AT] A. Alvino and G. Trombetti, Second order elliptic equations whose coefficients have their first derivatives weakly-$ {L^n}$ , Ann. Mat. Pura Appl. (4) 138 (1984), 331-340. MR 779550 (86e:35040)
  • [Cf] L. Caffarelli, Interior a priori estimates for solutions of fully non-linear equations, Ann. of Math. 130 (1989), 189-213. MR 1005611 (90i:35046)
  • [Cl] A. P. Calderon, Integrales singulares $ y$ sus aplicaciones a ecuaciones diferenciales hyperbolicas, Cursos y Seminarios de Matematicas 3, Univ. de Buenos Aires, 1960. MR 0123834 (23:A1156)
  • [Cn] A. Canfora, Extension of an integral inequality of C. Miranda and applications to the elliptic equations with discontinuous coefficients, Czechoslovak Math. J. 39 (1989), 385-421. 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] F. Chiarenza, M. Frasca, and P. Longo, Interior $ {W^{2,p}}$, estimates for non divergence elliptic equations with discontinuous coefficients, Ricerche Mat. (to appear). MR 1191890 (93k:35051)
  • [CRW] R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976). MR 0412721 (54:843)
  • [GR] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. Stud., vol. 116, North-Holland, Amsterdam, 1985. MR 807149 (87d:42023)
  • [G] J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981. MR 628971 (83g:30037)
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
  • [Gr] D. 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. MR 0082030 (18:486d)
  • [K] A. I. Koshelev, On boundedness in $ {L^p}$ of solutions of elliptic differential equations, Mat. Sb. 38 (1956), 359-372. MR 0078563 (17:1213e)
  • [M1] C. Miranda, Sulle equazioni ellittiche del secondo ordine a coefficienti discontinui, Ann. Mat. Pura Appl. 63 (1963). MR 0170090 (30:331)
  • [M2] -, Partial differential equations of elliptic type, Springer-Verlag, Berlin, 1970. MR 0284700 (44:1924)
  • [S] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. MR 0377518 (51:13690)
  • [T] A. Torchinsky, Real-variable methods in harmonic analysis, Academic Press, New York, 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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DOI: https://doi.org/10.1090/S0002-9947-1993-1088476-1
Article copyright: © Copyright 1993 American Mathematical Society

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