In this note the well-posedness of the Dirichlet problem (1.2) below is proved in the class $H^{1,p}_0(\Omega)$ for all $1
@article{119121, author = {Maria Alessandra Ragusa}, title = {Elliptic boundary value problem in Vanishing Mean Oscillation hypothesis}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {40}, year = {1999}, pages = {651-663}, zbl = {1010.46032}, mrnumber = {1756544}, language = {en}, url = {http://dml.mathdoc.fr/item/119121} }
Ragusa, Maria Alessandra. Elliptic boundary value problem in Vanishing Mean Oscillation hypothesis. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) pp. 651-663. http://gdmltest.u-ga.fr/item/119121/
Lectures on Elliptic boundary Value Problems, Van Nostrand, N.J., 1965. | MR 0178246 | Zbl 0142.37401
Strongly elliptic systems of differential equations, in: Contributions to the Theory of Partial Differential Equations, Princeton University Press, Princeton N.J., 1954, pp.15-51. | MR 0067306 | Zbl 0057.32901
Elliptic Equations In Partial Differential Equations, Interscience, New York, 1964, pp.131-299. | MR 0165224
Interior $W^{2,p}$ estimates for non-divergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), 149-168. (1991) | MR 1191890
$W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with $VMO$ coefficients, Trans. Amer. Math. Soc. 336 (1993), 841-853. (1993) | MR 1088476
Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. (1976) | MR 0412721 | Zbl 0326.32011
Sulla differenziabilita' e l'analiticita' delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. 3 (1957), 25-43. (1957) | MR 0093649 | Zbl 0084.31901
$L^p$ Estimates for divergence form elliptic equations with discontinuous coefficients, Boll. U.M.I. (7) 10-A (1996), 409-420. (1996) | MR 1405255
Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. | MR 0445088 | Zbl 1121.46001
The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc. 55 (1944), 132-151. (1944) | MR 0009701 | Zbl 0061.26201
On the differentiability of the solutions of linear elliptic equations, Comm. Pure Appl. Math. 6 (1953), 299-326. (1953) | MR 0058828
Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, 1983. | MR 0737190 | Zbl 1042.35002
Dirichlet's problem for linear elliptic partial differential equations, Math. Scand. 1 (1953), 55-72. (1953) | MR 0064979 | Zbl 0053.39101
On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. (1961) | MR 0131498 | Zbl 0102.04302
Linear and Quasilinear Elliptic Equations, English Translation: Academic Press, New York, 2nd Russian ed., 1973.
On Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math. 8 (1955), 615-633. (1955) | MR 0078558 | Zbl 0067.07502
Partial Differential Equations of Elliptic Type, second revised edition, Springer Verlag, 1970. | MR 0284700 | Zbl 0198.14101
Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui, Annali di Matematica 63 (1963), 353-386. (1963) | MR 0170090 | Zbl 0156.34001
Alcune osservazioni sulla maggiorazione in $L^{\nu}$ delle soluzioni deboli delle equazioni ellittiche del secondo ordine, Annali di Matematica 61 (1963), 151-170. (1963) | MR 0177187
Some properties of bounded mean oscillation, Studia Math. 41 (1977), 63-75. (1977) | MR 0445210
On nonlinear elliptic partial differential equations and Hölder continuity, Comm. Pure Appl. Math. 6 (1953), 103-156. (1953) | MR 0064986 | Zbl 0050.09801
Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 5 (1955), 649-675. (1955) | MR 0075415 | Zbl 0067.07602
On functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. (1975) | MR 0377518
Contributi alla regolarizzazione delle soluzioni dei problemi al contorno per equazioni del secondo ordine ellittiche, Ann. Scuola Norm. Sup. Pisa (3) 12 (1958), 223-245. (1958) | MR 0125313 | Zbl 0082.09701
I problemi al contorno per le equazioni differentiali di tipo ellittico, Atti VI Congr. Un. Mat. Ital., Napoli Edizioni Cremonese, 1960, pp.21-44. | MR 0123807
Le probléme de Dirichlet pour les équations elliptiques du second ordre á coefficients discontinuous, Ann. Inst. Fourier 15 (1965), 189-258. (1965) | MR 0192177
Equations elliptiques du second ordre á coefficients discontinues, Séminaire de Mathématiques Supérieures 16, Les Presses de l'Université de Montréal, 1966. | MR 0251373
Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa (3) 27 (1973), 265-308. (1973) | MR 0369884 | Zbl 0279.35025