A problem with an obstacle that goes out to the boundary of the domain for a class of quadratic functionals on ℝ^{ℕ}

Arkhipova, A.

doi:10.1090/S1061-0022-2011-01172-0

A problem with an obstacle that goes out to the boundary of the domain for a class of quadratic functionals on $\mathbb {R}^N$
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by A. A. Arkhipova
Translated by: S. Kislyakov

St. Petersburg Math. J. 22 (2011), 847-875

DOI: https://doi.org/10.1090/S1061-0022-2011-01172-0

Published electronically: August 18, 2011

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Abstract:

A variational problem with obstacle is studied for a quadratic functional defined on vector-valued functions $u : \Omega \to \mathbb {R}^N,\ N>1$. It is assumed that the nondiagonal matrix that determines the quadratic form of the integrand depends on the solution and is “split”. The role of the obstacle is played by a closed (possibly, noncompact) set $\mathcal {K}$ in $\mathbb {R}^N$ or a smooth hypersurface $S$. It is assumed that $u(x)\in \mathcal {K}$ or $u(x)\in S$ a.e. on $\Omega$. This is a generalization of a scalar problem with an obstacle that goes out to the boundary of the domain. It is proved that the solutions of the variational problems in question are partially smooth in $\bar {\Omega }$ and that the singular set $\Sigma$ of the solution satisfies $H_{n-2}(\Sigma )=0$.

References

Mariano Giaquinta and Enrico Giusti, On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31–46. MR 666107, DOI 10.1007/BF02392725
Mariano Giaquinta and Enrico Giusti, The singular set of the minima of certain quadratic functionals, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 1, 45–55. MR 752579
J. Jost and M. Meier, Boundary regularity for minima of certain quadratic functionals, Math. Ann. 262 (1983), no. 4, 549–561. MR 696525, DOI 10.1007/BF01456068
Michael Wiegner, Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme, Math. Z. 147 (1976), no. 1, 21–28. MR 407430, DOI 10.1007/BF01214271
Stefan Hildebrandt and Kjell-Ove Widman, On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 1, 145–178. MR 457936
Frank Duzaar, Variational inequalities and harmonic mappings, J. Reine Angew. Math. 374 (1987), 39–60. MR 876220, DOI 10.1515/crll.1987.374.39
F. Duzaar and M. Fuchs, Optimal regularity theorems for variational problems with obstacles, Manuscripta Math. 56 (1986), no. 2, 209–234. MR 850371, DOI 10.1007/BF01172157
Martin Fuchs, A regularity theorem for energy minimizing maps of Riemannian manifolds, Comm. Partial Differential Equations 12 (1987), no. 11, 1309–1321. MR 888462, DOI 10.1080/03605308708820528
Martin Fuchs, Some remarks on the boundary regularity for minima of variational problems with obstacles, Manuscripta Math. 54 (1985), no. 1-2, 107–119. MR 808683, DOI 10.1007/BF01171702
Martin Fuchs and Nicola Fusco, Partial regularity results for vector-valued functions which minimize certain functionals having nonquadratic growth under smooth side conditions, J. Reine Angew. Math. 390 (1988), 67–78. MR 953677
Martin Fuchs and Michael Wiegner, The regularity of minima of variational problems with graph obstacles, Arch. Math. (Basel) 53 (1989), no. 1, 75–81. MR 1005172, DOI 10.1007/BF01194875
Stefan Hildebrandt, Harmonic mappings of Riemannian manifolds, Harmonic mappings and minimal immersions (Montecatini, 1984) Lecture Notes in Math., vol. 1161, Springer, Berlin, 1985, pp. 1–117. MR 821968, DOI 10.1007/BFb0075136
Stéfan Hildebrandt, Helmut Kaul, and Kjell-Ove Widman, An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math. 138 (1977), no. 1-2, 1–16. MR 433502, DOI 10.1007/BF02392311
S. Hildebrandt and K.-O. Widman, Variational inequalities for vector-valued functions, J. Reine Angew. Math. 309 (1979), 191–220. MR 542048
A. Arkhipova, Variational problem for an obstacle in $\Bbb R^N$ for a class of quadratic functionals, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 362 (2008), no. Kraevye Zacachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 39, 15–47, 364 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 159 (2009), no. 4, 391–410. MR 2760549, DOI 10.1007/s10958-009-9452-9
Hugo Beirão da Veiga and Franco Conti, Equazioni ellittiche non lineari con ostacoli sottili. Applicazioni allo studio dei punti regolari, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 26 (1972), 533–562 (Italian, with English summary). MR 364862
L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations 4 (1979), no. 9, 1067–1075. MR 542512, DOI 10.1080/03605307908820119
Jens Frehse, On Signorini’s problem and variational problems with thin obstacles, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 2, 343–362. MR 509085
David Kinderlehrer, The smoothness of the solution of the boundary obstacle problem, J. Math. Pures Appl. (9) 60 (1981), no. 2, 193–212. MR 620584
N. N. Ural′ceva, A problem with one-sided conditions on the boundary for a quasilinear elliptic equation, Problems in mathematical analysis, No. 6: Spectral theory, boundary value problems (Russian), Izdat. Leningrad. Univ., Leningrad, 1977, pp. 172–189, 204 (Russian). MR 0509596
N. N. Ural′tseva, On the regularity of solutions of variational inequalities, Uspekhi Mat. Nauk 42 (1987), no. 6(258), 151–174, 248 (Russian). MR 933999
I. Athanasopoulos and L. A. Caffarelli, Optimal regularity of lower dimensional obstacle problems, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34], 49–66, 226 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 132 (2006), no. 3, 274–284. MR 2120184, DOI 10.1007/s10958-005-0496-1
G. Fichera, Existence theorems in elasticity, Handbuch der Physik, Bd. 6a/2, Springer-Verlag, Berlin, 1972.
Jindřich Nečas, On regularity of solutions to nonlinear variational inequalities for second-order elliptic systems, Rend. Mat. (6) 8 (1975), no. 2, 481–498 (English, with Italian summary). MR 382827
N. N. Ural′ceva, Strong solutions of the generalized Signorini problem, Sibirsk. Mat. Zh. 19 (1978), no. 5, 1204–1212, 1216 (Russian). MR 508511
R. Schumann, Zur Regularität einer Kontakt-Randwertaufgabe, Z. Anal. Anwendungen 9 (1990), no. 5, 455–465 (German, with English and Russian summaries). MR 1119544, DOI 10.4171/ZAA/415
A. A. Arkhipova and N. N. Ural′tseva, Regularity of solutions of diagonal elliptic systems under convex constraints on the boundary of the domain, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 152 (1986), no. Kraev. Zadachi Mat. Fiz. i Smezhnye Vopr. Teor. Funktsiĭ18, 5–17, 181 (Russian, with English summary); English transl., J. Soviet Math. 40 (1988), no. 5, 591–598. MR 869237, DOI 10.1007/BF01094182
A. A. Arkhipova and N. N. Ural′tseva, Regularity of the solutions of variational inequalities with convex constraints on the boundary of the domain for nonlinear operators with a diagonal principal part, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 3 (1987), 13–19, 127 (Russian, with English summary). MR 928154
A. A. Arkhipova and N. N. Ural′tseva, Limit smoothness of the solutions of variational inequalities under convex constraints on the boundary of the domain, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 163 (1987), no. Kraev. Zadachi Mat. Fiz. i Smezhn. Vopr. Teor. Funktsiĭ 19, 5–16, 186 (Russian, with English summary); English transl., J. Soviet Math. 49 (1990), no. 5, 1121–1128. MR 918937, DOI 10.1007/BF02208707
Arina Arkhipova, Signorini-type problem in $\Bbb R^N$ for a class of quadratic functionals, Nonlinear partial differential equations and related topics, Amer. Math. Soc. Transl. Ser. 2, vol. 229, Amer. Math. Soc., Providence, RI, 2010, pp. 15–38. MR 2667630, DOI 10.1090/trans2/229/02
Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
O. A. Ladyzhenskaya and N. N. Ural′tseva, Lineĭnye i kvazilineĭnye uravneniya èllipticheskogo tipa, Izdat. “Nauka”, Moscow, 1973 (Russian). Second edition, revised. MR 0509265
Mariano Giaquinta and Enrico Giusti, Nonlinear elliptic systems with quadratic growth, Manuscripta Math. 24 (1978), no. 3, 323–349. MR 481490, DOI 10.1007/BF01167835
A. A. Arkhipova, On the regularity of solutions of boundary-value problem for quasilinear elliptic systems with quadratic nonlinearity, J. Math. Sci. 80 (1996), no. 6, 2208–2225. Nonlinear boundary-value problems and some questions of function theory. MR 1420674, DOI 10.1007/BF02362383
Yun Mei Chen and Michael Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), no. 1, 83–103. MR 990191, DOI 10.1007/BF01161997
K.-O. Widman, Inequalities for the Green functions of second order elliptic operators, Univ. Linköping, Inst. Math., 1972.
Kjell-Ove Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand. 21 (1967), 17–37 (1968). MR 239264, DOI 10.7146/math.scand.a-10841
Lawrence C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal. 116 (1991), no. 2, 101–113. MR 1143435, DOI 10.1007/BF00375587
N. N. Ural′tseva, Estimation on the boundary of the domain of derivatives of solutions of variational inequalities, Linear and nonlinear boundary value problems. Spectral theory (Russian), Probl. Mat. Anal., vol. 10, Leningrad. Univ., Leningrad, 1986, pp. 92–105, 213 (Russian). Translated in J. Soviet Math. 45 (1989), no. 3, 1181–1191. MR 860572