Abstract
The paper studies the existence of non-tangential limit for solutions of the Dirichlet problem with L2-boundary data. We also prove the mutual absolute continuity of the associated harmonic measure and the Lebesgue surface measure.
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CAFFARELLI, L., FABES, E., KENIG, C., Completely singular elliptic-harmonic measures, Indiana Univ. Math. J, 30, 917–924 (1981)
CAFFARELLI, L., FABES, E., MORTOLA, S., SALSA, S., Boundary behavior of non-negative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30, 621–640 (1981)
CHABROWSKI, J., THOMPSON, B., On the “boundary values in L2 of the solutions of linear elliptic equations, University of Queensland - preprint
DAHLBERG, B.E.J., On the Poisson integral for Lipschitz and C1-domains, Studia Math. 64, 13–23 (1979)
GILBARG, D., TRUDINGER, N.S., Elliptic partial differential equations of second order, Berlin-Heidelberg-New York, Springer 1977
GUŠČIN, A.K., MIKHAILOV, V.P., On the boundary values in Lp(p>l) of solutions of elliptic equations, Mat. Sb., 108, 1–19 (1979)
JERISON, D.S., KENIG, C.E., Boundary behavior of harmonic functions in non-tangentially accessible domains, Princeton University - preprint
JERISON, D.S., KENIG, C.E., Boundary value problems in Lipschitz domains, Princeton University -preprint
JERISON, D.S., KENIG, C.E., The Dirichlet problem in non-smooth domains, Ann. of Math. 113, 367–382,(1981)
KAPANADZE, J.V., On the boundary values of a second order linear elliptic equation, Bul. Acad. Sci. Georgian S.S.R., 93, 285–288, (1979)
LADYZHENSKAJA, O.A., URAL'TSEVA, N.N., Linear and quasi-linear elliptic equations, New York-London, Academic Press (1969)
LITTMAN, W., STAMPACCHIA, G., WEINBERGER, H.F., Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa, 17, 45–79, (1963).
MIKHAILOV, V.P., Boundary values of solutions of elliptic equations in the ball, Mat. Sb. 100, 1–13 (1972)
MIKHAILOV, V.P., Boundary values of elliptic equations in a domain with smooth boundary, Mat.Sb. 101, 163–188 (1973)
MIKHAILOV, V.P., Dirichlet's problem for a second order elliptic equation, Differencial'nye Uravnenija 12, 1877–1891 (1976)
MIKHAILOV, V.P., On the boundary values of the solutions of elliptic equations, Appl. Mat. Optim. 6, 193–199 (1980)
MIRANDA, C., Sulle equazioni ellitiche del secondo ordine di tipo non variazionale a coefficienti discontinui, Ann. Mat. Pura Appl. 63, 353–386 (1963)
MIRANDA, C., Alcune limitazioni integrali per le soluzioni delle equazioni lineari ellittiche del secondo ordine, Ann. Mat. Pura. Appl.49, 375–384 (1960)
MODICA, L., MORTOLA, S., Construction of a singular elliptic-harmonic measure, Manuscripta Math. 33, 81–88 (1980)
NEČAS, J., On the regularity of second order elliptic partial differential equations with unbounded Dirichlet integral, Arch. Rational Mech.Anal. 9, 134–144 (1962)
NEČAS, J., Les méthodes directes en théorie des équations elliptiques, Prague, Academia Editeurs (1967)
STAMPACCHIA, G., Le probléme de Dirichlet pour les équations elliptiques du second ordre á coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15,189–258 (1965)
STAMPACCHIA, G. Équations elliptiques du second ordre á coefficients discontinus, Les Presses de l'Université de Montréal, Seminaire de Mathematiques Superieurs (1965)
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Chabrowski, J. Note on the Dirichlet problem with L2-boundary data. Manuscripta Math 40, 91–108 (1982). https://doi.org/10.1007/BF01168238
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DOI: https://doi.org/10.1007/BF01168238