Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The $L^p$ Dirichlet problem and nondivergence harmonic measure


Author: Cristian Rios
Journal: Trans. Amer. Math. Soc. 355 (2003), 665-687
MSC (2000): Primary 35J25; Secondary 35B20, 31B35
DOI: https://doi.org/10.1090/S0002-9947-02-03145-8
Published electronically: October 1, 2002
MathSciNet review: 1932720
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Dirichlet problem

\begin{displaymath}\left\{ \begin{array}{rcl} \mathcal{L} u & = & 0\quad\text{ in }D,\\ u & = & g\quad\text{ on }\partial D \end{array}\right.\end{displaymath}

for two second-order elliptic operators $\mathcal{L}_k u=\sum_{i,j=1}^na_k^{i,j}(x)\,\partial_{ij} u(x)$, $k=0,1$, in a bounded Lipschitz domain $D\subset\mathbb{R} ^n$. The coefficients $a_k^{i,j}$ belong to the space of bounded mean oscillation ${{BMO}}$ with a suitable small ${{BMO}}$ modulus. We assume that ${\mathcal{L}}_0$ is regular in $L^p(\partial D, d\sigma)$ for some $p$, $1<p<\infty$, that is, $\Vert Nu\Vert _{L^p}\le C\,\Vert g\Vert _{L^p}$ for all continuous boundary data $g$. Here $\sigma$ is the surface measure on $\partial D$ and $Nu$ is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients $\varepsilon^{i,j}(x)=a^{i,j}_1(x)-a^{i,j}_0(x)$ that will assure the perturbed operator $\mathcal{L}_1$ to be regular in $L^q(\partial D,d\sigma)$ for some $q$, $1<q<\infty$.


References [Enhancements On Off] (What's this?)

  • 1. F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336, $2$ (1993), 841-853. MR 93f:35232
  • 2. R. Fefferman, C. Kenig and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Annals of Math. 134 (1991), 65-124. MR 93h:31010
  • 3. B. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), no. 5, 1119-1138. MR 88i:35061
  • 4. R. Fefferman, A Criterion for the absolute continuity of the harmonic measure associated with an elliptic operator, J. Amer. Math. Soc. 2 (1989), 127-135. MR 90b:35068
  • 5. C. Rios, Sufficient conditions for the absolute continuity of the nondivergence harmonic measure, Ph.D. thesis, University of Minnesota, Minneapolis, Minnesota (2001).
  • 6. C. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, 83, American Math. Society, (1994). MR 96a:35040
  • 7. P. Bauman, Properties of nonnegative solutions of second-order elliptic equations and their adjoints, Ph.D. thesis, University of Minnesota, Minneapolis, Minnesota (1982).
  • 8. P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints, Arkiv fur Matematik, 22 (1984), 153-173. MR 86m:35008
  • 9. E. Fabes and D. Stroock, The $L^p$-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J. 51(1984), 997-1016. MR 86g:35057
  • 10. L. Escauriaza and C. Kenig, Area integral estimates for solutions and normalized adjoint solutions to nondivergence form elliptic equations, Ark. Mat. 31(1993), 275-296. MR 95f:35059
  • 11. B. Muckenhoupt, The equivalence of two conditions for weight functions, Studia Math. 49 (1974), 101-106. MR 50:2790
  • 12. J. García-Cuerva and J.L. Rubio de Francia, Weighted norm inequalities and related topics, Math. Studies 116, North Holland, 1985. MR 87d:42023
  • 13. L. Escauriaza, Weak type-$(1,1)$ inequalities and regularity properties of adjoint and normalized adjoint solutions to linear nondivergence form operators with VMO coefficients, Duke Math. J. 74 (1994), no. 1, 177-201. MR 95h:42018
  • 14. F. Chiarenza, M. Frasca and P. Longo, Interior $W^{2,p}$estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche di Matematica XL, fasc. $1^o$ (1991), 149-168. MR 93k:35051
  • 15. E. Fabes, N. Garofalo, S. Marín-Malave and S. Salsa, Fatou theorems for some nonlinear elliptic equations, Rev. Mat. Ib. 4(1988), 227-251. MR 91e:35092
  • 16. E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7, no. 1, 77-116 (1982). MR 84i:35070
  • 17. B. Dahlberg, Estimates of harmonic measure, Arch. Rat. Mech. Anal. 65 (1977), 275-288. MR 57:6470
  • 18. B. Dahlberg, On the Poisson integral for Lipschitz and $C^1$domains, Studia Math. 66 (1979), 13-24. MR 81g:31007
  • 19. L. Modica and S. Mortola, Construction of a singular elliptic-harmonic-measure, Manuscrita Math. 33 (1980), 81-98. MR 81m:31001
  • 20. D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag (1998); Reprint of 1998 original, 2001. MR 2001k:35004
  • 21. D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. MR 51:13690
  • 22. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51(1974), 241-250. MR 50:10670
  • 23. E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, (1993).
  • 24. M. Bramanti and M.C. Cerutti, Commutators of singular integrals on homogeneous spaces, Boll. Un. Mat. Ital. B (7) 10 (1996), no 4, 843-883. MR 99c:42026
  • 25. P. Bauman, Equivalence of the Green's function for diffusion operators in ${\mathbb{R} ^n}$: a counterexample, Proc. Amer. Mat. Soc., 91 (1984), 64-68. MR 85d:35026
  • 26. N. Krylov and M. Safonov, An estimate of the probability that a diffusion process hits a set of positive measure, Dokl. Acad. Nauk. S.S.S.R. 245 (1979), 253-255 (in Russian). English translation in Soviet. Mat. Dokl. 20 (1979), 253-255.
  • 27. M. Safonov, Harnack's inequality for elliptic equations and the Hölder property of their solutions, J. Soviet. Math. (1983), 851-863.
  • 28. B. Dahlberg, D. Jerison and C. Kenig, Area integral estimates for elliptic differential operators with nonsmooth coefficients, Arkiv. Mat. 22 (1984), 97-108. MR 85h:35021

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J25, 35B20, 31B35

Retrieve articles in all journals with MSC (2000): 35J25, 35B20, 31B35


Additional Information

Cristian Rios
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Address at time of publication: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8R-B19 Canada
Email: riosc@math.mcmaster.ca

DOI: https://doi.org/10.1090/S0002-9947-02-03145-8
Keywords: Nondivergence elliptic equations, Dirichlet problem, harmonic measure
Received by editor(s): April 5, 2002
Received by editor(s) in revised form: May 17, 2002
Published electronically: October 1, 2002
Dedicated: In memory of E. Fabes
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society