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

Lieberman, Gary M.

doi:10.1090/S0002-9947-1992-1116317-1

The conormal derivative problem for equations of variational type in nonsmooth domains
HTML articles powered by AMS MathViewer

by Gary M. Lieberman

Trans. Amer. Math. Soc. 330 (1992), 41-67

DOI: https://doi.org/10.1090/S0002-9947-1992-1116317-1

PDF | Request permission

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.

References

Robert Finn, Equilibrium capillary surfaces, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 284, Springer-Verlag, New York, 1986. MR 816345, DOI 10.1007/978-1-4613-8584-4
M. Giaquinta and E. Giusti, Global $C^{1,\alpha }$-regularity for second order quasilinear elliptic equations in divergence form, J. Reine Angew. Math. 351 (1984), 55–65. MR 749677
Claus Gerhardt, Global regularity of the solutions to the capillarity problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 1, 157–175. MR 602007
David Gilbarg and Lars Hörmander, Intermediate Schauder estimates, Arch. Rational Mech. Anal. 74 (1980), no. 4, 297–318. MR 588031, DOI 10.1007/BF00249677
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Barbara Huisken, Second-order boundary regularity for quasilinear variational inequalities, Manuscripta Math. 63 (1989), no. 3, 333–342. MR 986188, DOI 10.1007/BF01168375
Gerhard Huisken, Capillary surfaces over obstacles, Pacific J. Math. 117 (1985), no. 1, 121–141. MR 777440

The principle of the maximum and boundary Lipschitz bounds for the solution of a second order elliptic-parabolic equation

15

Nicholas J. Korevaar, Maximum principle gradient estimates for the capillary problem, Comm. Partial Differential Equations 13 (1988), no. 1, 1–31. MR 914812, DOI 10.1080/03605308808820536
Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
Gary M. Lieberman, The quasilinear Dirichlet problem with decreased regularity at the boundary, Comm. Partial Differential Equations 6 (1981), no. 4, 437–497. MR 612553, DOI 10.1080/03605308108820184
Gary M. Lieberman, The conormal derivative problem for elliptic equations of variational type, J. Differential Equations 49 (1983), no. 2, 218–257. MR 708644, DOI 10.1016/0022-0396(83)90013-X
Gary M. Lieberman, Interior gradient bounds for nonuniformly parabolic equations, Indiana Univ. Math. J. 32 (1983), no. 4, 579–601. MR 703286, DOI 10.1512/iumj.1983.32.32041
Gary M. Lieberman, Regularized distance and its applications, Pacific J. Math. 117 (1985), no. 2, 329–352. MR 779924
Gary M. Lieberman, Quasilinear elliptic equations with nonlinear boundary conditions, Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 113–117. MR 843601, DOI 10.1090/pspum/045.2/843601
Gary M. Lieberman, Intermediate Schauder estimates for oblique derivative problems, Arch. Rational Mech. Anal. 93 (1986), no. 2, 129–134. MR 823115, DOI 10.1007/BF00279956
Gary M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data, Comm. Partial Differential Equations 11 (1986), no. 2, 167–229. MR 818099, DOI 10.1080/03605308608820422
Gary M. Lieberman, Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations, Trans. Amer. Math. Soc. 304 (1987), no. 1, 343–353. MR 906819, DOI 10.1090/S0002-9947-1987-0906819-0
Gary M. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl. (4) 148 (1987), 77–99. MR 932759, DOI 10.1007/BF01774284
Gary M. Lieberman, Hölder continuity of the gradient at a corner for the capillary problem and related results, Pacific J. Math. 133 (1988), no. 1, 115–135. MR 936359
Gary M. Lieberman, The conormal derivative problem for nonuniformly parabolic equations, Indiana Univ. Math. J. 37 (1988), no. 1, 23–72. MR 942094, DOI 10.1512/iumj.1988.37.37002
Gary M. Lieberman, Oblique derivative problems in Lipschitz domains. II. Discontinuous boundary data, J. Reine Angew. Math. 389 (1988), 1–21. MR 953664, DOI 10.1515/crll.1988.389.1
Gary M. Lieberman, Boundary regularity for linear and quasilinear variational inequalities, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), no. 3-4, 319–326. MR 1014660, DOI 10.1017/S0308210500018771
Norman G. Meyers, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand. 26 (1970), 255–292 (1971). MR 277741, DOI 10.7146/math.scand.a-10981
J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$, Comm. Pure Appl. Math. 26 (1973), 361–379. MR 344978, DOI 10.1002/cpa.3160260305
James Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302. MR 170096, DOI 10.1007/BF02391014
James Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Publ. Math. Res. Center Univ. Wisconsin, No. 27, Academic Press, New York, 1971, pp. 565–601. MR 0402274
Leon Simon, Interior gradient bounds for non-uniformly elliptic equations, Indiana Univ. Math. J. 25 (1976), no. 9, 821–855. MR 412605, DOI 10.1512/iumj.1976.25.25066

Solvability of the capillary problem

19

1

6

8

N. N. Ural′tseva, Estimates of the maximum moduli of gradients for solutions of capillarity problems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115 (1982), 274–284, 312 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 14. MR 660089
Gary M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural′tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), no. 2-3, 311–361. MR 1104103, DOI 10.1080/03605309108820761