Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Variational inequalities and free boundary problems


Author: David Kinderlehrer
Journal: Bull. Amer. Math. Soc. 84 (1978), 7-26
MSC (1970): Primary 35-XX, 49-XX
DOI: https://doi.org/10.1090/S0002-9904-1978-14397-3
MathSciNet review: 0466885
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • [ADNl] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727. MR 125307
  • [ADN2] 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. 22 (1964), 35-92. MR 162050
  • [A1] H. Alt, A free boundary problem associated with the flow of groundwater, Arch. Rational Mech. Anal. 64 (1977), 111-126. MR 428901
  • [An-F] R. F. Anderson and A. Friedman, A quality control problem and quasi-variational inequalities, Arch. Rational Mech. Anal. 63 (1976), 205-252. MR 456857
  • [Ba1] C. Baiocchi, Su un problema a frontiera libera connesso a questioni di idraulica, Ann. Mat. Pura Appl. 92 (1972), 107-127. MR 408443
  • [Ba2] C. Baiocchi, Free boundary problems in the theory of fluid flow through porous media, Proc. Internat. Congr. Math. (Vancouver, 1974), Vol. II, Canad. Math. Congress, 237-343. MR 421331
  • [Ba3] C. Baiocchi, Problèmes à frontière libre et inéquations variationnelles, C. R. Acad. Sci. Paris 283 (1976), 29-32. MR 427816
  • [Ben-F] A. Bensoussan and A. Friedman, Nonzero-sum stochastic differential games with stopping times and free boundary problems, Trans. Amer. Math. Soc. 231 (1977), 275-327. MR 453082
  • [Ben-Li1] A. Bensoussan and J. L. Lions, Problèmes de temps d'arrêt optimal et inéquations variationelles paraboliques, J. Appl. Anal. 3 (1973), 267-294. MR 449843
  • [Ben-Li2] A. Bensoussan and J. L. Lions, Temps d'arrêt optimal et contrôle impulsionnel. Vol. I. (to appear).
  • [Be-Br] H. Berestycki and H. Brezis, Sur certains problèmes de frontière libre, C. R. Acad. Sci. Paris 283 (1976), 1091-1094. MR 442342
  • [Br1] H. Brezis, Problèmes unilatéraux, J. Math. Pures Appl. 51 (1972), 1-168. MR 428137
  • [Br2] H. Brezis, Multiplicateur de Lagrange en torsion elastroplastique, Arch. Rational Mech. Anal. 41 (1971), 254-265. MR 346345
  • [Br-D] H. Brezis and G. Duvaut, Ecoulment avec sillage autour d'un profile symétrique sans incidence, C. R. Acad. Sci. Paris 276 (1973), 875-878.
  • [Br-K] H. Brezis and D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana J. Math. 23 (1974), 831-844. MR 361436
  • [BR-S1] H. Brezis and G. Stampacchia, Sur la régularité de la solution d'inéquations elliptiques, Bull. Soc. Math. France 96 (1968), 153-180. MR 239302
  • [Br-S2] H. Brezis and G. Stampacchia, The hodograph method in fluid-dynamics in the light of variational inequalities, Arch. Rational Mech. Anal. 61 (1976), 1-18. MR 477505
  • [C1] L. A. Caffarelli, The smoothness of a free boundary in a filtration problem, Arch. Rational Mech. Anal. 63 (1976), 77-86. MR 425743
  • [C2] L. A. Caffarelli, The regularity of elliptic and parabolic free boundaries, Acta Math.
  • [C3] L. A. Caffarelli, The local regularity of a phreatic surface, Arch. Rational Mech. Anal. (to appear).
  • [C-R1] L. A. Caffarelli and N. M. Riviere, Smoothness and analyticity of free boundaries in variational inequalities, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 3 (1976), 289-310. MR 412940
  • [C-N2] L. A. Caffarelli and N. M. Riviere, Asymptotic behavior of free boundaries at their singular points, Ann. of Math, (to appear). MR 463690
  • [C-N3] L. A. Caffarelli and N. M. Riviere, On the lipschitz character of the stress tensor when twisting an elastic plastic bar, Arch. Rational Mech. Anal. (to appear). MR 513957
  • [D] G. Duvaut, Resolution d'un probleme de Stefan (Fusion d'un bloc de glace a zero degré), C. R. Acad. Sci. Paris 276 (1973), 1461-1463. MR 328346
  • [Fr-Ber] L. Fraenkel and M. Berger, A global theory of steady vortex rings in an ideal fluid, Acta Math. 132 (1974), 13-51. MR 422916
  • [Frel] J. Frehse, On the regularity of the solution of a second order variational inequality, Boll. Un. Mat. Ital. 6 (1972), 312-315. MR 47 #7197. MR 318650
  • [Fre2] J. Frehse, Two dimensional variational problems with thin obstacles, Math. Z. 143 (1975), 279-288. MR 380550
  • [Fre3] J. Frehse, On Signorini's problem and variational problems with thin obstacles, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 4 (1977), 343-362. MR 509085
  • [F1] A. Friedman, On the regularity of the solutions of non-linear elliptic and parabolic systems of partial differential equations, J. Math. Mech. 7 (1958), 43-60. MR 118970
  • [F2] A. Friedman, The Stefan problem in several space variables, Trans. Amer. Math. Soc. 133 (1968), 51-87. MR 227625
  • [F3] A. Friedman, Analyticity of the free boundary for the Stefan problem, Arch Rational Mech. Anal. 61 (1976), 97-125. MR 407452
  • [F4] A. Friedman, On the free boundary of a quasi variational inequality arising in a problem of quality control, Trans. Amer. Math. Soc. (to appear).
  • [F-J1] A. Friedman and R. Jensen, A parabolic quasi-variational inequality arising in hydraulics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1975), 421-468. MR 380106
  • [F-J2] A. Friedman and R. Jensen, Elliptic quasi-variational inequalities and application to a non-stationary problem in hydraulics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1976), 47-88. MR 402266
  • [F-K] A. Friedman and D. Kinderlehrer, A one phase Stefan problem, Indiana Univ. Math. J. 24 (1975), 1005-1035. MR 385326
  • [G] P. Garabedian, Partial differential equations, Wiley, New York, 1964. MR 162045
  • [Ge] C. Gerhardt, Regularity of solutions of nonlinear variational inequalities, Arch. Rational Mech. Anal. 52 (1973), 389-393. MR 377262
  • [Gia-P] M. Giaquinta and L. Pepe, Esistenza e regolarità per il problema dell’ area minima con ostacoli in n variabili, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 25 (1971), 481-507. MR 46 #4335.
  • [Giu] E. Giusti, Minimal surfaces with obstacles, CIME course on Geometric Measure Theory and Minimal Surfaces, Edizioni Cremonese, Rome, 1973, pp. 119-153. MR 407701
  • [J.-Ser] H. Jenkins and J. Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew Math. 229 (1968), 170-187. MR 222467
  • [Ka] C. Kahane, Analyticity of solutions of mildly singular integral equations, Comm. Pure Appl. Math. 18 (1965), 593-626. MR 206649
  • [K1] D. Kinderlehrer, Variational inequalities with lower dimensional obstacles, Israel J. Math. 10 (1971), 339-348. MR 304999
  • [K2] D. Kinderlehrer, The coincidence set of solutions of certain variational inequalities, Arch. Rational Mech. Anal. 40 (1971), 231-250. MR 271799
  • [K3] D. Kinderlehrer, How a minimal surface leaves an obstacle, Acta Math. 130 (1973), 221-242. MR 419997
  • [K4] D. Kinderlehrer, The free boundary determined by the solution as a differential equation, Indiana Univ. Math. J. 25 (1976), 195-208. MR 393807
  • [K-N1] D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1977), 373-391. MR 440187
  • [K-N2] D. Kinderlehrer and L. Nirenberg, The smoothness of the free boundary in the one phase Stefan problem, Comm. Pure Appl. Math. (to appear). MR 480348
  • [K-N2] D. Kinderlehrer and L. Nirenberg, Analyticity at the boundary of solutions of nonlinear second order parabolic equations, Comm. Pure Appl. Math. (to appear). MR 460897
  • [K-N-Sp] D. Kinderlehrer, L. Nirenberg and J. Spruck, (to appear).
  • [K-Sp] D. Kinderlehrer and J. Spruck, The shape and smoothness of stable plasma configurations, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. (to appear). MR 481511
  • [K-S-1] D. Kinderlehrer and G. Stampacchia, A free boundary problem in potential theory, Ann. Inst. Fourier (Grenoble) 25 (1975), 323-344. MR 507997
  • [K-S2] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications,
  • [L1] H. Lewy, A note on harmonic functions and a hydrodynamical application, Proc. Amer. Math. Soc. 3 (1952), 11-113. MR 49399
  • [L2] H. Lewy, On the boundary behavior of minimal surfaces, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 103-110. MR 14, 168. MR 49398
  • [L3] H. Lewy, On minimal surfaces with partially free boundary, Comm. Pure Appl. Math. 4 (1952), 1-13. MR 14, 662. MR 52711
  • [L4] H. Lewy, On the reflection laws of second order differential equations in two independent variables, Bull Amer. Math. Soc. 65 (1959), 37-58. MR 104048
  • [L5] H. Lewy, On a minimum problem for superharmonic functions, Proc. Internat. Conf. on Functional Analysis, Tokyo, 1969.
  • [L6] H. Lewy, On the nature of the boundary separating two domains with different regimes, Accad. Naz. Lincei, 1975.
  • [L-S1] H. Lewy and G. Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math. 22 (1969), 153-188. MR 40 #816. MR 247551
  • [L-S2] H. Lewy and G. Stampacchia, On existence and smoothness of solutions of some noncoercive variational inequalities, Arch. Rational Mech. Anal. 41 (1971), 241-253. MR 346313
  • [Li-S] J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493-519. MR 35 #7178. MR 216344
  • [Mag] E. Magenes, Topics in parabolic equations: some typical free boundary problems, Proc. of NATO Advanced Study Institute, Liege, 1976. MR 470497
  • [Ma1] S. Mazzone, Existence and regularity of the solution of certain nonlinear variational inequalities with an obstacle, Arch. Rational Mech. Anal. 57 (1974), 115-127. MR 367439
  • [Ma2] S. Mazzone, Un problema di disequazioni variazionali per superficie di curvatura media assegnata, Boll. Un. Mat. Ital. 7 (1973), 318-329. MR 47 #4109. MR 315560
  • [Mi] M. Miranda, Frontieri minimali con ostacoli, Ann. Univ. Ferrara 21 (1971), 29-37. MR 301617
  • [Mo] C. B. Morrey, Jr., Multiple integrals in the calculus of variations, Springer, New York and Berlin, 1966. MR 202511
  • [Mu-S] M. K. V. Murthy and G. Stampacchia, A variational inequality with mixed boundary conditions, Israel J. Math. 13 (1972), 188-224. MR 333430
  • [P] J. Puel, Un probleme de valeurs propres non lineare et de frontiére libre, C. R. Acad. Sci. Paris 284 (1977), 861-864.
  • [Sc1] D. G. Schaeffer, Some examples of singularities in a free boundary, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1 (1977), 133-144. MR 516201
  • [Sc2] D. G. Schaeffer, An example of the plasma problem with infinitely many solutions, Comm. in P. D. E. 2 (1977), 587-600. MR 602535
  • [Sh] V. D. Shafranov, On magnetohydrodynamical equilibrium configurations, J. Exp. Theoret. Phys. (USSR) 33 (1957), 710-722. MR 93282
  • [S-1] G. Stampacchia, Le problème de Dirichlet pur les équations elliptique du second order à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), 189-258. MR 33 #404. MR 192177
  • [S-2] G. Stampacchia, Variational inequalities, theory and applications of monotone operators, Proc. of NATO Advanced Study Institute, Oderisi-Gubbio, 1969. MR 425699
  • [S-3] G. Stampacchia, Variational inequalities, Proc. Internat. Congr. of Math., Nice, 1970, 877-883. MR 513209
  • [S-4] G. Stampacchia, On the filtration of a liquid through a porous medium, Uspehi Mat. Nauk 29 (178) (1974), 89-101. MR 413739
  • [S-5] G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413-4416. MR 166591
  • [Te1] R. Temam, Variational principles related to the equilibrium shape of a plasma in an axisymmetric torus, J. Plasma Physics (1976).
  • [Te2] R. Temam, A non-linear eigenvalue problem: the shape at equilibrium of a confined plasma, Arch. Rational Mech. Anal. 60 (1975), 51-73. MR 412637
  • [Te3] R. Temam, Remarks on a free boundary problem arising in plasma physics, Comm. in P. D. E. 2 (1977), 563-585. MR 602544
  • [Ti] T. W. Ting, Elastic-plastic torsion problem, Arch. Rational Mech. Anal. 24 (1969), 228-244. MR 264889

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1970): 35-XX, 49-XX

Retrieve articles in all journals with MSC (1970): 35-XX, 49-XX


Additional Information

DOI: https://doi.org/10.1090/S0002-9904-1978-14397-3

American Mathematical Society