# zbMATH — the first resource for mathematics

The Neumann problem for elliptic equations with non-smooth coefficients. (English) Zbl 0807.35030
The so-called Neumann $$(N_ p)$$ and regularity $$(R_ p)$$ problems for a divergence form symmetric elliptic operator $$L = \text{div} A \nabla$$ are considered. Here $$A = (a_{ij} (x))$$ is a matrix of bounded measurable functions. The problem is considered in the unit ball $$B$$, but the methods of proof are sufficiently general to allow $$B$$ to be a bounded starlike Lipschitz domain.
The results in this work are of two types. First, the meaning of and consequences of solvability of the Neumann and regularity problems for general operator $$L$$ are considered. A Neumann function for $$L$$ is introduced. It is proved that the same sort of estimates known for the Green’s function of $$L$$ are satisfied. This leads to a potential representation for solutions of the Neumann problem and to uniqueness results for weak solutions. Further nontangential convergence results for the Neumann and regularity problem with data in $$L^ p (\partial B)$$ are considered. These results are utilized to investigate the general consequences of solvability of $$(N_ p)$$ and $$(R_ p)$$. An example which show that the Neumann problem for data in $$L^ p (d \sigma)$$ need not be solvable for any arbitrary elliptic operator is given.
The results of the second type concern solvability of $$(N_ p)$$ and $$(R_ p)$$ for certain classes of operators. Specifically, if $$A(\theta)$$ is an elliptic symmetric matrix $$a_{ij} \equiv a_{ij} (\theta)$$, $$\theta \in S^{n-1}$$ (radial independence) then $$(N_ 2)$$ and $$(R_ 2)$$ are solvable for $$L = \text{div} A \nabla$$ in the unit ball $$B$$. And if $$A(r, \theta)$$ is a small perturbation of $$A(\theta) = A(1, \theta)$$ then $$(N_ 2)$$ and $$(R_ 2)$$ are solvable for $$L = \text{div} A(r, \theta) \nabla$$. At the end of the paper the authors indicate a list of several questions which remain open and some directions for further research.

##### MSC:
 35J25 Boundary value problems for second-order elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000) 35J67 Boundary values of solutions to elliptic equations and elliptic systems
##### Keywords:
non-smooth coefficients; Green’s function
Full Text:
##### References:
 [1] [B-A] Beurling, A., Ahlfors, L.: The boundary correspondence under quasiconformal mappings. Acta Math.96, 125-142 (1956) · Zbl 0072.29602 · doi:10.1007/BF02392360 [2] [C-F-K] Caffarelli, L., Fabes, E., Kenig, C.: Completely singular elliptic-harmonic measures. Indiana Univ. Math. J.30, 917-924 (1981) · Zbl 0482.35020 · doi:10.1512/iumj.1981.30.30067 [3] [C-F-M-S] Caffarelli, L., Fabes, E., Mortola, S., Salsa, S.: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana Univ. Math. J.30, 621-640 (1981) · Zbl 0512.35038 · doi:10.1512/iumj.1981.30.30049 [4] [C-Sc] Calderón, A., Scott, R.: Sobolev type inequalities forp>0. Stud. Math.62, 75-92 (1978) [5] [C-F] Coifman, R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Stud. Math.51, 241-250 (1974) · Zbl 0291.44007 [6] [D1] Dahlberg, B.: On the Poisson integral for Lipschitz andC 1 domains. Stud. Math.66, 7-24 (1979) [7] [D2] Dahlberg, B.: On the absolute continuity of elliptic measures. Am. J. Math.108, 1119-1138 (1986) · Zbl 0644.35032 · doi:10.2307/2374598 [8] [D-J-K] Dahlberg, B., Jerison, D., Kening, C.: Area integral estimates for elliptic differential operators with non-smooth coefficients. Ark. Mat.22, 97-108 (1984) · Zbl 0537.35025 · doi:10.1007/BF02384374 [9] [D-K] Dahlberg, B., Kenig, C.: Hardy spaces and the Neumann problem inL p for Laplace’s equation in Lipschitz domains. Ann. Math.125, 437-465 (1987) · Zbl 0658.35027 · doi:10.2307/1971407 [10] [DeG] DeGiorgi, E.: Sulle differenziabilita e analiticita delle estremali degli integrali multipli regulari, Mem. Accad. Sci. Torino3, 25-43 (1957) [11] [D] Duong, X.T.:H ? functional calculus of elliptic partial differential operators inL p spaces. Ph.D. Thesis, Macquarie University, Sidney, Australia (1990) [12] [F-J-K] Fabes, E., Jerison, D., Kenig C.: Necessary and sufficient conditions for absolute continuity of elliptic harmonic measure. Ann. Math.119, 121-141 (1984) · Zbl 0551.35024 · doi:10.2307/2006966 [13] [F-S] Fefferman, C., Stein, E.:H p spaces of several variables. Acta Math.129, 137-192 (1972) · Zbl 0257.46078 · doi:10.1007/BF02392215 [14] [RF] Fefferman, R.: A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator. J. Am. Math. Soc.2, 127-135 (1989) · Zbl 0694.35050 · doi:10.1090/S0894-0347-1989-0955604-8 [15] [F-K-P] Fefferman, R., Kenig, C., Pipher, J.: The theory of weights and the Dirichlet problem for elliptic equations. Ann. Math.134, 65-124 (1991) · Zbl 0770.35014 · doi:10.2307/2944333 [16] [G] Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Non-linear Elliptic Systems. (Ann. Math. Stud., vol. 105) 1983 Princeton University Press Princeton: NJ · Zbl 0516.49003 [17] [G-W] Grüter, M., Widman, K.-O.: The Green function for uniformly elliptic equations. Man. Math.37, 303-342 (1982) · Zbl 0485.35031 · doi:10.1007/BF01166225 [18] [H-M-W] Hunt, R., Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Am. Math. Soc.176, 227-251 (1973) · Zbl 0262.44004 · doi:10.1090/S0002-9947-1973-0312139-8 [19] [H-W1] Hunt, R., Wheeden, R.: On the boundary values of harmonic functions. Trans. Am. Math. Soc.132, 307-322 (1986) · Zbl 0159.40501 · doi:10.1090/S0002-9947-1968-0226044-7 [20] [H-W2] Hunt, R., Wheeden, R.: Positive harmonic functions on Lipschitz domains. Trans. Am. Math. Soc.147, 507-527 (1970) · Zbl 0193.39601 · doi:10.1090/S0002-9947-1970-0274787-0 [21] [J-K1] Jerison, D., Kenig, C.: The Dirichlet problem in non-smooth domains. Ann. Math.113, 367-382 (1981) · Zbl 0453.35036 · doi:10.2307/2006988 [22] [J-K2] Jerison, D., Kenig, C.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc.4, 203-207 (1981) · Zbl 0471.35026 · doi:10.1090/S0273-0979-1981-14884-9 [23] [J] Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math.147, 71-88 (1981) · Zbl 0489.30017 · doi:10.1007/BF02392869 [24] [K-P] Kenig, C., Pipher, J.: The oblique derivative on Lipschitz domains withL p data. Am. J. Math.110, 715-737 (1988) · Zbl 0676.35019 · doi:10.2307/2374647 [25] [K-S] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. (Pure Appl. Math., vol. 88) New York London: Academic Press 1980 · Zbl 0457.35001 [26] [L-S-W] Littman, W., Stampacchia, G., Weinberger, H.: Regular points for elliptic equations with discontinuous coefficients. Ann. Sc. Norm. Super. Pisa17, 43-77 (1983) · Zbl 0116.30302 [27] [Mo] Moser, J.: On Harnack’s theorem for elliptic differential equations, Commun. Pure Appl. Math.14, 577-591 (1961) · Zbl 0111.09302 · doi:10.1002/cpa.3160140329 [28] [Mu] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc.165, 207-226 (1972) · Zbl 0236.26016 · doi:10.1090/S0002-9947-1972-0293384-6 [29] [N] Nash, J.: Continuity of the solutions of parabolic and elliptic equations. Am. J. Math.80, 931-954 (1957) · Zbl 0096.06902 · doi:10.2307/2372841 [30] [S-W] Serrin, J., Weinberger, H.: Isolated singularities of solutions of linear elliptic equations. Am. J. Math.88, 158-272 (1966) · Zbl 0137.07001 · doi:10.2307/2373060 [31] [S] Stein, E.: Singular Integrals and Differentiability Properties of Funcitons. Princeton: Princeton University Press 1970 [32] [V] Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal.59, 572-611 (1984) · Zbl 0589.31005 · doi:10.1016/0022-1236(84)90066-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.