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General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases. (English) Zbl 0901.49027
Consider the following Dirichlet problem for the Hamilton-Jacobi equation: \[ F(Du(x))= 0,\qquad\text{a.e. }x\in\Omega, \] \[ u(x)= \varphi(x),\qquad x\in\partial\Omega, \] where \(\Omega\) is an open bounded set of \(\mathbb{R}^n\), \(F: \mathbb{R}^{n\times N}\to \mathbb{R}\) and \(\varphi\in W^{1,\infty}(\Omega,\mathbb{R}^N)\), with \(n,N\geq 1\).
In this paper, the authors introduce some new and interesting conditions on \(F\) and \(\varphi\) and a new method to prove existence of solutions.
In the scalar case, \(N= 1\), they prove that if \(E= \{z\in\mathbb{R}^n: F(z)= 0\}\) is closed and \[ D\varphi\in E\cup\text{int co }E, \] where \(\text{int co }E\) denotes the interior of convex hull of \(E\), the problem has solutions in \(W^{1,\infty}\).
In the vectorial case \(N>1\), the existence proof is achieved provided that \(F\) is quasiconvex and satisfies some additional assumptions.
Finally, the prescribed singular values case is also studied. The proofs use the Baire category methods and weak lower semicontinuity arguments.

MSC:
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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[1] Acerbi, E. &Fusco, N., Semicontinuity problems in the calculus of variations.Arch. Rational Mech. Anal., 86 (1984), 125–145. · Zbl 0565.49010 · doi:10.1007/BF00275731
[2] Alibert, J. J. &Dacorogna, B., An example of a quasiconvex function that is not polyconvex in two dimensions.Arch. Rational Mech. Anal., 117 (1992), 155–166. · Zbl 0761.26009 · doi:10.1007/BF00387763
[3] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity.Arch. Rational Mech. Anal., 63 (1977), 337–403. · Zbl 0368.73040 · doi:10.1007/BF00279992
[4] Ball, J. M. &James, R. D., Fine phase mixtures as minimizers of energy.Arch. Rational Mech. Anal., 100 (1987), 15–52. · Zbl 0629.49020 · doi:10.1007/BF00281246
[5] Bardi, M. & Capuzzo Dolcetta, I.,Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, 1997. · Zbl 0890.49011
[6] Barles, G.,Solutions de viscosité des équations de Hamilton-Jacobi. Mathématiques et Applications, 17. Springer-Verlag, Berlin, 1994. · Zbl 0819.35002
[7] Benton, S. H.,The Hamilton-Jacobi Equation. A Global Approach. Academic Press, New York, 1977. · Zbl 0418.49001
[8] Capuzzo Dolcetta, I. &Evans, L. C., Optimal switching for ordinary differential equations.SIAM J. Control Optim., 22 (1988), 1133–1148.
[9] Capuzzo Dolcetta, I. &Lions, P. L., Viscosity solutions of Hamilton-Jacobi equations and state-constraint problem.Trans. Amer. Math. Soc., 318 (1990), 643–683. · Zbl 0702.49019 · doi:10.2307/2001324
[10] Carathéodory, C.,Calculus of Variations and Partial Differential Equations of the First Order, Part 1. Holden-Day, San Francisco, CA, 1965. · Zbl 0134.31004
[11] Cellina, A., On the differential inclusionx’, 1].Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 69 (1980), 1–6. · Zbl 0922.34009
[12] –, On minima of a functional of the gradient: sufficient conditions.Nonlinear Anal., 20 (1993), 343–347. · Zbl 0784.49022 · doi:10.1016/0362-546X(93)90138-I
[13] Cellina, A. & Perrotta, S., On a problem of potential wells. Preprint. · Zbl 0880.49005
[14] Crandall, M. G., Evans, L. C. &Lions, P. L., Some properties of viscosity solutions of Hamilton-Jacobi equations.Trans. Amer. Math. Soc., 282 (1984), 487–502. · Zbl 0543.35011 · doi:10.1090/S0002-9947-1984-0732102-X
[15] Crandall, M. G., Ishii, H. &Lions, P. L., User’s guide to viscosity solutions of second order partial differential equations.Bull. Amer. Math. Soc., 27 (1992), 1–67. · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[16] Crandall, M. G. &Lions, P. L., Viscosity solutions of Hamilton-Jacobi equations.Trans. Amer. Math. Soc., 277 (1983), 1–42. · Zbl 0599.35024 · doi:10.1090/S0002-9947-1983-0690039-8
[17] Dacorogna, B.,Direct Methods in the Calculus of Variations. Applied Math. Sci., 78. Springer-Verlag, Berlin, 1989. · Zbl 0703.49001
[18] Dacorogna, B. &Marcellini, P., A counterexample in the vectorial calculus of variations, inMaterial Instabilities in Continuun Mechanics (J.M. Ball, ed.), pp. 77–83. Oxford Sci. Publ., Oxford, 1988.
[19] –, Existence of minimizers for non quasiconvex integrals.Arch. Rational Mech. Anal., 131 (1995), 359–399. · Zbl 0837.49002 · doi:10.1007/BF00380915
[20] Dacorogna, B. &Moser, J., On a partial differential equation involving the Jacobian determinant.Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 1–26. · Zbl 0707.35041
[21] De Blasi, F. S. &Pianigiani, G., A Baire category approach to the existence of solutions of multivalued differential equations in Banach spaces.Funkcial. Ekvac., 25 (1982), 153–162. · Zbl 0535.34009
[22] –, Non-convex-valued differential inclusions in Banach spaces.J. Math. Anal. Appl., 157 (1991), 469–494. · Zbl 0728.34013 · doi:10.1016/0022-247X(91)90101-5
[23] Douglis, A., The continuous dependence of generalized solutions of non-linear partial differential equations upon initial data.Comm. Pure Appl. Math., 14 (1961), 267–284. · Zbl 0117.31102 · doi:10.1002/cpa.3160140307
[24] Ekeland, I. &Témam, R.,Analyse convexe et problèmes variationnels. Dunod Gauthier-Villars, Paris, 1974.
[25] Fleming, W. H. &Soner, H. M.,Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics, 25. Springer-Verlag, New York, 1993. · Zbl 0773.60070
[26] Kohn, R. V. &Strang, G., Optimal design and relaxation of variational problems, I.Comm. Pure Appl. Math., 39 (1986), 113–137. · Zbl 0609.49008 · doi:10.1002/cpa.3160390107
[27] Kružkov, S. N., Generalized solutions of Hamilton-Jacobi equation of eikonal type.Math. USSR-Sb., 27 (1975), 406–446. · Zbl 0369.35012 · doi:10.1070/SM1975v027n03ABEH002522
[28] Lax, P. D., Hyperbolic systems of conservation laws, II.Comm. Pure Appl. Math., 10 (1957), 537–566. · Zbl 0081.08803 · doi:10.1002/cpa.3160100406
[29] Lions, P. L.,Generalized Solutions of Hamilton-Jacobi Equations. Res. Notes in Math., 69. Pitman, Boston, MA-London, 1982. · Zbl 0497.35001
[30] Marcellini, P., Approximation of quasiconvex functions and lower semicontinuity of multiple integrals.Manuscripta Math., 51 (1985), 1–28. · Zbl 0573.49010 · doi:10.1007/BF01168345
[31] Mascolo, E. &Schianchi, R., Existence theorems for nonconvex problems.J. Math. Pures Appl., 62 (1983), 349–359. · Zbl 0522.49001
[32] Meyers, N. G., Quasiconvexity and lower semicontinuity of multiple variational integrals of any order.Trans. Amer. Math. Soc., 119 (1965), 125–149. · Zbl 0166.38501 · doi:10.1090/S0002-9947-1965-0188838-3
[33] Morrey, C. B.,Multiple Integrals in the Calculus of Variations. Grundlehren Math. Wiss., 130. Springer-Verlag, Berlin, 1966. · Zbl 0142.38701
[34] Pianigiani, G., Differential inclusions. The Baire category method, inMethods of Nonconvex Analysis (A. Cellina, ed.), pp. 104–136. Lecture Notes in Math., 144. Springer-Verlag, Berlin-New York, 1990.
[35] Rockafellar, R. T.,Convex Analysis. Princeton Univ. Press, Princeton, NJ, 1970. · Zbl 0193.18401
[36] Rund, H.,The Hamilton-Jacobi Theory in the Calculus of Variations. Van Nostrand, Princeton, NJ, 1966. · Zbl 0141.10602
[37] Šverák, V., Rank-one convexity does not imply quasiconvexity.Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 185–189. · Zbl 0777.49015
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