Variational inequalities and free boundary problems
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- Bull. Amer. Math. Soc. 84 (1978), 7-26
References
- 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, DOI 10.1002/cpa.3160120405
- 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 162050, DOI 10.1002/cpa.3160170104
- Hans Wilhelm Alt, A free boundary problem associated with the flow of ground water, Arch. Rational Mech. Anal. 64 (1977), no. 2, 111–126. MR 428901, DOI 10.1007/BF00280093
- Robert F. Anderson and Avner Friedman, A quality control problem and quasi-variational inequalities, Arch. Rational Mech. Anal. 63 (1976/77), no. 3, 205–252. MR 456857, DOI 10.1007/BF00251581
- Claudio Baiocchi, Su un problema di frontiera libera connesso a questioni di idraulica, Ann. Mat. Pura Appl. (4) 92 (1972), 107–127. MR 408443, DOI 10.1007/BF02417940
- Claudio Baiocchi, Free boundary problems in the theory of fluid flow through porous media, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 237–243. MR 0421331
- Claudio Baiocchi, Problèmes à frontière libre et inéquations variationnelles, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 2, Ai, A29–A32. MR 427816
- Alain Bensoussan and Avner Friedman, Nonzero-sum stochastic differential games with stopping times and free boundary problems, Trans. Amer. Math. Soc. 231 (1977), no. 2, 275–327. MR 453082, DOI 10.1090/S0002-9947-1977-0453082-7
- A. Bensoussan and J.-L. Lions, Problèmes de temps d’arrêt optimal et inéquations variationnelles paraboliques, Applicable Anal. 3 (1973), 267–294 (French). MR 449843, DOI 10.1080/00036817308839070 [Ben-Li2] A. Bensoussan and J. L. Lions, Temps d’arrêt optimal et contrôle impulsionnel. Vol. I. (to appear).
- Henri Berestycki, Sur certains problèmes de valeurs propres non linéaires, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 4, Aii, A171–A173 (French, with English summary). MR 442342
- Haïm Brézis, Problèmes unilatéraux, J. Math. Pures Appl. (9) 51 (1972), 1–168. MR 428137
- H. Brézis and M. Sibony, Équivalence de deux inéquations variationnelles et applications, Arch. Rational Mech. Anal. 41 (1971), 254–265 (French). MR 346345, DOI 10.1007/BF00250529 [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.
- Haïm Brézis and David Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J. 23 (1973/74), 831–844. MR 361436, DOI 10.1512/iumj.1974.23.23069
- Haïm R. Brezis and Guido Stampacchia, Sur la régularité de la solution d’inéquations elliptiques, Bull. Soc. Math. France 96 (1968), 153–180 (French). MR 239302, DOI 10.24033/bsmf.1663
- Haïm Brézis and Guido Stampacchia, The hodograph method in fluid-dynamics in the light of variational inequalities, Arch. Rational Mech. Anal. 61 (1976), no. 1, 1–18. MR 477505, DOI 10.1007/BF00251859
- Luis A. Caffarelli, The smoothness of the free surface in a filtration problem, Arch. Rational Mech. Anal. 63 (1976), no. 1, 77–86. MR 425743, DOI 10.1007/BF00280143 [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).
- L. A. Caffarelli and N. M. Rivière, Smoothness and analyticity of free boundaries in variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 2, 289–310. MR 412940
- L. A. Caffarelli and N. M. Rivière, Asymptotic behaviour of free boundaries at their singular points, Ann. of Math. (2) 106 (1977), no. 2, 309–317. MR 463690, DOI 10.2307/1971098
- L. A. Caffarelli and N. M. Rivière, The Lipschitz character of the stress tensor, when twisting an elastic plastic bar, Arch. Rational Mech. Anal. 69 (1979), no. 1, 31–36. MR 513957, DOI 10.1007/BF00248408
- Georges Duvaut, Résolution d’un problème de Stefan (fusion d’un bloc de glace à zéro degré), C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1461–A1463 (French). MR 328346
- L. E. Fraenkel and M. S. Berger, A global theory of steady vortex rings in an ideal fluid, Acta Math. 132 (1974), 13–51. MR 422916, DOI 10.1007/BF02392107
- Jens Frehse, On the regularity of the solution of a second order variational inequality, Boll. Un. Mat. Ital. (4) 6 (1972), 312–315 (English, with Italian summary). MR 0318650
- Jens Frehse, Two dimensional variational problems with thin obstacles, Math. Z. 143 (1975), no. 3, 279–288. MR 380550, DOI 10.1007/BF01214380
- 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
- Avner Friedman, On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations, J. Math. Mech. 7 (1958), 43–59. MR 0118970, DOI 10.1512/iumj.1958.7.57004
- Avner Friedman, The Stefan problem in several space variables, Trans. Amer. Math. Soc. 133 (1968), 51–87. MR 227625, DOI 10.1090/S0002-9947-1968-0227625-7
- Avner Friedman, Analyticity of the free boundary for the Stefan problem, Arch. Rational Mech. Anal. 61 (1976), no. 2, 97–125. MR 407452, DOI 10.1007/BF00249700 [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).
- Avner Friedman and Robert Jensen, A parabolic quasi-variational inequality arising in hydraulics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 3, 421–468. MR 380106
- Avner Friedman and Robert Jensen, Elliptic quasi-variational inequalities and application to a non-stationary problem in hydraulics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 1, 47–88. MR 402266
- Avner Friedman and David Kinderlehrer, A one phase Stefan problem, Indiana Univ. Math. J. 24 (1974/75), no. 11, 1005–1035. MR 385326, DOI 10.1512/iumj.1975.24.24086
- P. R. Garabedian, Partial differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0162045
- Claus Gerhardt, Regularity of solutions of nonlinear variational inequalities, Arch. Rational Mech. Anal. 52 (1973), 389–393. MR 377262, DOI 10.1007/BF00247471 [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.
- E. Giusti, Minimal surfaces with obstacles, Geometric measure theory and minimal surfaces (Centro Internaz. Mat. Estivo (C.I.M.E.), III Ciclo, Varenna, 1972) Edizioni Cremonese, Rome, 1973, pp. 119–153. MR 0407701
- Howard Jenkins and James Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math. 229 (1968), 170–187. MR 222467, DOI 10.1515/crll.1968.229.170
- Charles Kahane, Analyticity of solutions of mildly singular integral equations, Comm. Pure Appl. Math. 18 (1965), 593–626. MR 206649, DOI 10.1002/cpa.3160180404
- David Kinderlehrer, Variational inequalities with lower dimensional obstacles, Israel J. Math. 10 (1971), 339–348. MR 304999, DOI 10.1007/BF02771651
- David Kinderlehrer, The coincidence set of solutions of certain variational inequalities, Arch. Rational Mech. Anal. 40 (1970/71), 231–250. MR 271799, DOI 10.1007/BF00281484
- David Kinderlehrer, How a minimal surface leaves an obstacle, Acta Math. 130 (1973), 221–242. MR 419997, DOI 10.1007/BF02392266
- David Kinderlehrer, The free boundary determined by the solution to a differential equation, Indiana Univ. Math. J. 25 (1976), no. 2, 195–208. MR 393807, DOI 10.1512/iumj.1976.25.25016
- D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 2, 373–391. MR 440187
- David Kinderlehrer and Louis Nirenberg, The smoothness of the free boundary in the one phase Stefan problem, Comm. Pure Appl. Math. 31 (1978), no. 3, 257–282. MR 480348, DOI 10.1002/cpa.3160310302
- David Kinderlehrer and Louis Nirenberg, Analyticity at the boundary of solutions of nonlinear second-order parabolic equations, Comm. Pure Appl. Math. 31 (1978), no. 3, 283–338. MR 460897, DOI 10.1002/cpa.3160310303 [K-N-Sp] D. Kinderlehrer, L. Nirenberg and J. Spruck, (to appear).
- David Kinderlehrer and Joel Spruck, The shape and smoothness of stable plasma configurations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 1, 131–148. MR 481511
- David Kinderlehrer and Guido Stampacchia, A free boundary value problem in potential theory, Ann. Inst. Fourier (Grenoble) 25 (1975), no. 3-4, xvii, 323–344 (English, with French summary). MR 507997 [K-S2] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications,
- Hans Lewy, A note on harmonic functions and a hydrodynamical application, Proc. Amer. Math. Soc. 3 (1952), 111–113. MR 49399, DOI 10.1090/S0002-9939-1952-0049399-9
- Hans Lewy, On the boundary behavior of minimal surfaces, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 103–110. MR 49398, DOI 10.1073/pnas.37.2.103
- Hans Lewy, On mimimal surfaces with partially free boundary, Comm. Pure Appl. Math. 4 (1951), 1–13. MR 52711, DOI 10.1002/cpa.3160040102
- Hans Lewy, On the reflection laws of second order differential equations in two independent variables, Bull. Amer. Math. Soc. 65 (1959), 37–58. MR 104048, DOI 10.1090/S0002-9904-1959-10270-6 [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.
- Hans Lewy and Guido Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math. 22 (1969), 153–188. MR 247551, DOI 10.1002/cpa.3160220203
- Hans Lewy and Guido Stampacchia, On existence and smoothness of solutions of some non-coercive variational inequalities, Arch. Rational Mech. Anal. 41 (1971), 241–253. MR 346313, DOI 10.1007/BF00250528
- J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493–519. MR 216344, DOI 10.1002/cpa.3160200302
- E. Magenes, Topics in parabolic equations: some typical free boundary problems, Boundary value problems for linear evolution: partial differential equations (Proc. NATO Advanced Study Inst., Liège, 1976) NATO Advanced Study Inst. Ser., Ser. C: Math. and Phys. Sci., Vol. 29, Reidel, Dordrecht, 1977, pp. 239–312. MR 0470497
- Silvia Mazzone, Existence and reularity of the solution of certain nonlinear variational inequalities with an obstacle, Arch. Rational Mech. Anal. 57 (1975), 115–127. MR 367439, DOI 10.1007/BF00248413
- Silvia Mazzone, Un problema di disequazioni variazionali per superficie di curvatura media assegnata, Boll. Un. Mat. Ital. (4) 7 (1973), 318–329 (Italian, with English summary). MR 0315560
- M. Miranda, Frontiere minimali con ostacoli, Ann. Univ. Ferrara Sez. VII (N.S.) 16 (1971), 29–37 (Italian, with French summary). MR 301617
- Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511, DOI 10.1007/978-3-540-69952-1
- M. K. V. Murthy and G. Stampacchia, A variational inequality with mixed boundary conditions, Israel J. Math. 13 (1972), 188–224 (1973). MR 333430, DOI 10.1007/BF02760237 [P] J. Puel, Un probleme de valeurs propres non lineare et de frontiére libre, C. R. Acad. Sci. Paris 284 (1977), 861-864.
- David G. Schaeffer, Some examples of singularities in a free boundary, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 1, 133–144. MR 516201
- David G. Schaeffer, Non-uniqueness in the equilibrium shape of a confined plasma, Comm. Partial Differential Equations 2 (1977), no. 6, 587–600. MR 602535, DOI 10.1080/03605307708820040
- V. D. Shafranov, On magnetohydrodynamical equilibrium configurations, Soviet Physics JETP 6 (1958), 545–554. MR 0093282
- Guido Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 189–258 (French). MR 192177, DOI 10.5802/aif.204
- Guido Stampacchia, Variational inequalities, Theory and Applications of Monotone Operators (Proc. NATO Advanced Study Inst., Venice, 1968) Edizioni “Oderisi”, Gubbio, 1969, pp. 101–192. MR 0425699
- Guido Stampacchia, Variational inequalities, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 877–883. MR 0513209
- G. Stampakk′ja, On the filtration of fluid through a porous medium with variable cross section, Uspehi Mat. Nauk 29 (1974), no. 4(178), 89–101 (Russian). Translated from the English by T. D. Ventcel′; Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ, II. MR 0413739
- Guido Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413–4416 (French). MR 166591 [Te1] R. Temam, Variational principles related to the equilibrium shape of a plasma in an axisymmetric torus, J. Plasma Physics (1976).
- R. Temam, A non-linear eigenvalue problem: the shape at equilibrium of a confined plasma, Arch. Rational Mech. Anal. 60 (1975/76), no. 1, 51–73. MR 412637, DOI 10.1007/BF00281469
- R. Temam, Remarks on a free boundary value problem arising in plasma physics, Comm. Partial Differential Equations 2 (1977), no. 6, 563–585. MR 602544, DOI 10.1080/03605307708820039
- Tsuan Wu Ting, Elastic-plastic torsion problem. III, Arch. Rational Mech. Anal. 34 (1969), 228–244. MR 264889, DOI 10.1007/BF00281140
Additional Information
- 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