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The conormal derivative problem for equations of variational type in nonsmooth domains


Author: Gary M. Lieberman
Journal: Trans. Amer. Math. Soc. 330 (1992), 41-67
MSC: Primary 35J65; Secondary 49Q20
DOI: https://doi.org/10.1090/S0002-9947-1992-1116317-1
MathSciNet review: 1116317
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Abstract: It is well known that elliptic boundary value problems in smooth domains have smooth solutions, but if the domain is, say, $ {C^1}$, the solutions need not be Lipschitz. Recently Korevaar has identified a class of Lipschitz domains, in which solutions of the capillary problem are Lipschitz assuming the contact angle relates correctly to the geometry of the domain. Lipschitz bounds for more general boundary value problems in the same class of domains are proved. Applications to variational inequalities are also considered.


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  • [1] R. Finn, Equilibrium capillary surfaces, Springer-Verlag, Berlin and New York, 1986. MR 816345 (88f:49001)
  • [2] M. Giaquinta and E. Giusti, Global $ {C^{1,\alpha }}$-regularity for second order elliptic equations in divergence form, J. Reine Angew. Math. 351 (1984), 55-65. MR 749677 (85k:35077)
  • [3] C. Gerhardt, Global regularity of solutions to the capillarity problem, Ann. Scuola Norm Sup. Pisa Cl. Sci. (3) 4 (1976), 157-176. MR 0602007 (58:29199)
  • [4] D. Gilbarg and L. Hormander, Intermediate Schauder estimates, Arch. Rational Mech. Anal 74 (1980), 297-318. MR 588031 (82a:35038)
  • [5] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd Ed., Springer-Verlag, Berlin and New York, 1983. MR 737190 (86c:35035)
  • [6] B. Huisken, Second order boundary regularity for quasilinear variational inequalities, Manuscripta Math. 63 (1989), 333-342. MR 986188 (90d:35116)
  • [7] G. Huisken, Capillarity surfaces over obstacles, Pacific J. Math. 117 (1985), 121-141. MR 777440 (87c:49047)
  • [8] L. I. Kamynin and B. N. Khimchenko, The principle of the maximum and boundary Lipschitz bounds for the solution of a second order elliptic-parabolic equation, Siberian Math. J. 15 (1974), 242-260.
  • [9] N. J. Korevaar, Maximum principle gradient estimates for the capillary problem, Comm. Partial Differential Equations 13 (1988), 1-31. MR 914812 (89d:35061)
  • [10] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations, Izdat. "Nauka", Moscow, 1964; English transl., Academic Press, New York, 1968. MR 0244627 (39:5941)
  • [11] G. M. Lieberman, The quasilinear Dirichlet problem with decreased regularity at the boundary, Comm. Partial Differential Equations 6 (1981), 437-467. MR 612553 (83g:35032)
  • [12] -, The conormal derivative problem for elliptic equations of variational type, J. Differential Equations 49 (1983), 218-257. MR 708644 (85j:35073)
  • [13] -, Interior gradient bounds for non-uniformly parabolic equations, Indiana Univ. Math. J. 32 (1983), 579-601. MR 703286 (85i:35079)
  • [14] -, Regularized distance and its applications, Pacific J. Math. 117 (1985), 329-353. MR 779924 (87j:35101)
  • [15] -, Quasilinear elliptic equations with nonlinear boundary conditions, Proc. Sympos. Pure Math., vol. 45, part 2, Amer. Math. Soc., Providence, R.I., 1986, pp. 113-117. MR 843601 (87g:35089)
  • [16] -, Intermediate Schauder theory for oblique derivative problems, Arch. Rational Mech. Anal 93 (1986), 129-134. MR 823115 (87j:35112)
  • [17] -, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data, Comm. Partial Differential Equations 11 (1986), 167-229. MR 818099 (87f:35073)
  • [18] -, Local estimates for subsolutions and supersolutions of oblique derivative problems for general second-order elliptic equations, Trans. Amer. Math. Soc. 304 (1987), 343-353. MR 906819 (88j:35061)
  • [19] -, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl. (4) 148 (1987), 77-99, 397-398. MR 932759 (90k:35053a)
  • [20] -, Hölder continuity of the gradient at a corner for the capillary problem and related results, Pacific J. Math. 133 (1988), 115-135. MR 936359 (89h:35050)
  • [21] -, The conormal derivative problem for non-uniformly parabolic equations, Indiana Univ. Math. J. 37 (1988), 23-72. MR 942094 (89k:35117)
  • [22] -, Oblique derivative problems in Lipschitz domains. II. Discontinuous boundary data, J. Reine Angew. Math. 389 (1988), 1-24. MR 953664 (89h:35094)
  • [23] -, Boundary regularity for linear and quasilinear variational inequalities, Proc. Roy. Soc. Edinburgh 112 (1989), 319-326. MR 1014660 (91a:35035)
  • [24] N. G. Meyers, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand. 26 (1970), 255-292. MR 0277741 (43:3474)
  • [25] J. H. Michael and L. Simon, Sobolev and mean value inequalities on generalized submanifolds of $ {R^n}$, Comm. Pure Appl. Math. 26 (1973), 361-379. MR 0344978 (49:9717)
  • [26] J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math. 111 (1964), 247-302. MR 0170096 (30:337)
  • [27] -, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, Contributions to Nonlinear Functional Analysis, Academic Press, 1971, pp. 565-602. MR 0402274 (53:6095)
  • [28] L. Simon, Interior gradient bounds for non-uniformly elliptic equations, Indiana Univ. Math. J. 25 (1976), 821-855. MR 0412605 (54:727)
  • [29] N. N. Ural'tseva, Solvability of the capillary problem, Vestnik Leningrad. Univ. 19 (1973), 54-64; 1 (1975), 143-149; English transl., Vestnik Leningrad Univ. Math. 6 (1979), 363-375; 8 (1980), 151-158.
  • [30] -, Estimates for the maxima of the moduli of the gradients for solutions of capillarity problems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 115 (1982), 274-284; English transl., J. Soviet Math. 28 (1985), 806-813. MR 660089 (84d:53008)
  • [31] G. M. Lieberman, The natural generalization of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), 311-361. MR 1104103 (92c:35041)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1116317-1
Keywords: Quasilinear elliptic equations, conormal derivative, boundary value problems, variational inequalities
Article copyright: © Copyright 1992 American Mathematical Society

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