Nonsymmetric systems

on nonsmooth planar domains

Authors:
G. C. Verchota and A. L. Vogel

Journal:
Trans. Amer. Math. Soc. **349** (1997), 4501-4535

MSC (1991):
Primary 35J55, 31A25

DOI:
https://doi.org/10.1090/S0002-9947-97-02047-3

MathSciNet review:
1443894

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study boundary value problems, in the sense of Dahlberg, for second order constant coefficient strongly elliptic systems. In this class are systems without a variational formulation, viz. the nonsymmetric systems. Various similarities and differences between this subclass and the symmetrizable systems are examined in nonsmooth domains.

**[ADN64]**S. Agmon, A. Douglis, and L. Nirenberg,*Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. ii.*, Comm. Pure Appl. Math**17**(1964), 35-92. MR**28:5252****[BG93]**Heinrich Begehr and Robert P Gilbert,*Transformations, transmutations, and kernel functions Volume 2*, Pitman Monographs and Surveys in Pure and Applied Mathematics 59, Longman Scientific & Technical, Copublished by John Wiley & Sons Inc., New York, 1993. MR**95b:35001****[Bit48]**A. V. Bitsadze,*On the uniqueness of the solutions of the Dirichlet problem for elliptic partial differential equations*, Uspehi Mat. Nauk**3**(1948), no. 6, 211-212. (Russian) MR**10:300f****[Cal77]**A. P. Calderon,*Cauchy integrals on Lipschitz curves and related operators*, Proc. Nat. Acad. Sci. U.S.A.**74**(1977), 1324-1327. MR**57:6445****[CD93]**M. Costabel and M. Dauge,*Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems*, Math. Nachr.**162**(1993), 209-237. MR**94k:35090****[CMM82]**R. R. Coifman, A. McIntosh, and Y. Meyer,*L'intégrale de Cauchy définit un opérateur borné sur pour les courbes Lipschitziennes*, Ann. of Math.**116**(1982), 361-387. MR**84m:42027****[CW77]**R. R. Coifman and G. Weiss,*Extensions of Hardy spaces and their use in analysis*, Bull. Amer. Math. Soc.**83**(1977), 569-646. MR**56:6264****[Dah77]**B. E. J. Dahlberg,*On estimates of harmonic measure*, Arch. Rational Mech. Anal.**65**(1977), 272-288. MR**57:6470****[Dah79]**B. E. J. Dahlberg,*On the Poisson integral for Lipschitz and domains*, Studia Math.**66**(1979), 13-24. MR**81g:31007****[DK]**B. E. J. Dahlberg, C. E. Kenig, J. Pipher, and G. C. Verchota,*Area integral estimates and maximum principles for higher order elliptic equations and systems*, in preparation.**[DK87]**B. E. J. Dahlberg and C. E. Kenig,*Hardy spaces and the Neumann problem in for Laplace's equation in Lipschitz domains*, Annals of Mathematics**125**(1987), 437-465. MR**88d:35044****[DK90]**B. E. J. Dahlberg and C. E. Kenig,*estimates for the 3-dimensional systems of elastostatics on Lipschitz domains*, Lecture Notes in Pure and Applied Math., vol. 122, Dekker, New York, 1990, pp. 621-634. MR**91h:35053****[DKV86]**B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota,*The Dirichlet problem for the biharmonic equation in a Lipschitz domain*, Ann. Inst. Fourier(Grenoble)**36**(1986), 109-135. MR**88a:35070****[DKV88]**B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota,*Boundary value problems for the system of elastostatics in Lipschitz domains*, Duke Math Journal**37**(1988), 795-818. MR**90d:35259****[DL92]**L. Diomeda and B. Lisena,*The Dirichlet problem for elliptic systems in piecewise plane domains*, Indiana University Math Journal**41**(1992), no. 3, 649-670. MR**94a:35035****[FJL76]**E. B. Fabes, M. Jodeit, and J. E. Lewis,*On the spectra of a Hardy kernel*, J. Funct. Anal.**21**(1976), 187-194. MR**52:15114****[FJL77]**E. B. Fabes, M. Jodeit, and J. E. Lewis,*Double layer potentials for domains with corners and edges*, Indiana Univ. Math. J.**26**(1977), 95-114. MR**55:5879****[FJR78]**E. B. Fabes, M. Jodeit, and N. M. Riviere,*Potential techniques for boundary value problems on domains*, Acta Math.**141**(1978), 165-186. MR**80b:31006****[FK81]**E. B. Fabes and C. E. Kenig,*On the Hardy space of a domain*, Arkiv Mat.**19**(1981), 1-22. MR**84a:42029****[Gar81]**John B. Garnett,*Bounded analytic functions*, Academic Press, New York, 1981. MR**83g:30037****[JK81]**D. Jerison and C. E. Kenig,*The Neumann problem on Lipschitz domains*, Bull. Amer. Math. Soc.**4**(1981), 203-207. MR**84a:35064****[Joh55]**Fritz John,*Plane waves and spherical means applied to partial differential equations*, Interscience, New York, 1955. MR**17:746d****[Ken80]**C. E. Kenig,*Weighted spaces on Lipschitz domains*, American Journal of Mathematics**102**(1980), no. 1, 129-163. MR**81d:30060****[Koz90]**V. A. Kozlov,*On the singularities of solutions of the Dirichlet problem for elliptic equations in the neighborhood of corner points*, Leningrad Math. J.**1**(1990), 967-982. MR**91c:35048****[KWCQ85]**Hua Loo Keng, Lin Wei, and Wu Ci-Quian,*Second-order systems of partial differential equations in the plane*, Pitman Advanced Publishing Program, Boston London Melbourne, 1985. MR**88m:35002****[LP80]**J. E. Lewis and C. Parenti,*Pseudodifferential operators and Hardy kernels on*, Annali Sc. Norm. Sup. Pisa Ser IV**7**(1980), 481-503. MR**83d:47052****[LP83]**J. E. Lewis and C. Parenti,*Pseudodifferential operators of Mellin type*, Comm. PDE**8**(1983), 477-544. MR**86f:35185****[PV93]**J. Pipher and G. C. Verchota,*The maximum principle for biharmonic functions in Lipschitz and domains*, Commentarii Math. Helvetici**68**(1993), 385-414. MR**94j:35030****[PV95a]**J. Pipher and G. C. Verchota,*Dilation invariant estimates and the boundary Garding inequality for higher order elliptic operators*, Ann. Of Math.**142**(1995), 1-38. MR**96g:35052****[PV95b]**J. Pipher and G. C. Verchota,*Maximum principles for the polyharmonic equation on Lipschitz domains*, Potential Analysis**4**(1995), 615-636. MR**96i:35021****[SK60]**Ding Shia-Kuai et al.,*On the definition of the second order elliptic system of partial differential equations with constant coefficients*, Acta Math. Sinica**10**(1960), 276-287; English translation, Chinese Math.**1**(1960), 288-299. MR**27:2713****[Var77]**N. T. Varopoulos,*BMO functions and the -equation*, Pacific J. Math.**71**(1977), 221-273. MR**58:22639a****[Ver84]**G. C. Verchota,*Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains*, J. Funct. Anal.**59**(1984), no. 3, 572-611. MR**86e:35038****[Ver87]**G. C. Verchota,*The Dirichlet problem for the biharmonic equation in domains*, Ind. Univ. Math J.**36**(1987), 867-895. MR**88m:35051****[Ver96]**G. C. Verchota,*Potentials for the Dirichlet problem in Lipschitz domains*, Potential Theory-ICPT 94 (Kouty, 1994), de Gruyter, Berlin, 1996, pp. 167-187. CMP**96:17**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
35J55,
31A25

Retrieve articles in all journals with MSC (1991): 35J55, 31A25

Additional Information

**G. C. Verchota**

Affiliation:
Department of Mathematics, Syracuse University, Syracuse, New York 13244

**A. L. Vogel**

Affiliation:
Department of Mathematics, Syracuse University, Syracuse, New York 13244

Email:
alvogel@izebug.syr.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-02047-3

Keywords:
Elliptic,
bianalytic,
weak maximum principle,
Rellich identity,
boundary value problems,
nonvariational

Received by editor(s):
November 10, 1995

Additional Notes:
The first author was partially supported by NSF Grant DMS-9401354.

Article copyright:
© Copyright 1997
American Mathematical Society