Maximal generalized solutions of Hamilton–Jacobi equations

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• 作者：Sandro Zagatti
• 关键词：Hamilton–Jacobi equations ; Viscosity solution ; Maximality ; Strong convergence ; Fully nonlinear PDE’s
• 刊名：Annali dell'Universita di Ferrara
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：62
• 期：2
• 页码：373-398
• 全文大小：565 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Geometry
  History of Mathematics
  Numerical Analysis
  Algebraic Geometry
  
• 出版者：Springer Milan
• ISSN：1827-1510
• 卷排序：62

文摘

We study the Dirichlet problem for Hamilton–Jacobi equations of the form $$\begin{aligned} {\left\{ \begin{array}{ll} H(x, \nabla u(x)) = 0 &{}\quad \text {in} \ \Omega \\ u(x)=\varphi (x) &{}\quad \text {on} \ \partial \Omega , \end{array}\right. } \end{aligned}$$without continuity assumptions on the hamiltonian H with respect to the variable x. We find a class of Caratheodory functions H for which the problem admits a (maximal) generalized solution which, in the continuous case, coincides with the classical viscosity solution.KeywordsHamilton–Jacobi equationsViscosity solutionMaximalityStrong convergenceFully nonlinear PDE’sMathematics Subject Classification35F2135F2049L2546B5035F30References1.Camilli, F., Siconolfi, A.: Hamilton–Jacobi equations with measurable dependence in the state variable. Adv. Diff. Equ. 8, 733–768 (2003)MathSciNetMATHGoogle Scholar2.Camilli, F., Siconolfi, A.: Time-dependent measurable Hamilton–Jacobi equations. Comm. Part. Diff. Equ. 30, 813–847 (2005)MathSciNetCrossRefMATHGoogle Scholar3.Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equations and Optimal Control. Birkhäuser, Basel (2004)MATHGoogle Scholar4.De Zan, C., Soravia, P.: Cauchy problems for noncoercive Hamilton–Jacobi–Isaacs equations with discontinuous coefficients. Interface Free Bound. 12, 347–368 (2010)MathSciNetCrossRefMATHGoogle Scholar5.Dacorogna, B., Marcellini, P.: Implicit Partial Differential Equations. Birkhäuser, Basel (1999)CrossRefMATHGoogle Scholar6.Garavello, M., Soravia, P.: Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running costs. NoDEA 11(3), 271–298 (2004)MathSciNetCrossRefMATHGoogle Scholar7.Lucchetti, R.: Convexity and Well Posed Problems. Springer-Verlag, New York (2006)CrossRefMATHGoogle Scholar8.Lions, P.L.: Generalized solutions of Hamilton–Jacobi equations. Research notes in Math. 69, Pitman, London (1982)9.Ostrov, D.N.: Solutions of Hamilton–Jacobi equations and scalar conservation laws with discontinuous space-time dependence. J. Diff. Equ. 182, 51–77 (2002)MathSciNetCrossRefMATHGoogle Scholar10.Soravia, P.: Boundary value problems for Hamilton–Jacobi equations with dscontinuous lagrangian. Indiana Univ. Math. J. 51(2), 451–477 (2002)MathSciNetCrossRefMATHGoogle Scholar11.Soravia, P.: Degenerate eikonal equations with discontinuous refraction index. ESAIM Control Optim. Calc. Var. 12(2), 216–230 (2006)MathSciNetCrossRefMATHGoogle Scholar12.Zagatti, S.: On the minimum problem for non convex scalar functionals. SIAM J. Math. Anal. 37, 982–995 (2005)MathSciNetCrossRefMATHGoogle Scholar13.Zagatti, S.: Uniqueness and continuous dependence on boundary data for integro-extremal minimizer of the functional of the gradient. J. Convex Anal. 14, 705–727 (2007)MathSciNetMATHGoogle Scholar14.Zagatti, S.: Solutions of Hamilton–Jacobi equations and minimizers of non quasiconvex functionals. J. Math. Anal. Appl. 335, 1143–1160 (2007)MathSciNetCrossRefMATHGoogle Scholar15.Zagatti, S.: Minimizers of non convex scalar functionals and viscosity solutions of Hamilton–Jacobi equations. Calc. Var. PDE’s 31(4), 511–519 (2008)MathSciNetCrossRefMATHGoogle Scholar16.Zagatti, S.: Qualitative properties of integro-esxtremal minimizers of non-homogeneous scalar functionals. Int. J. Pure Appl. Math. 51(1), 103–116 (2009)MathSciNetMATHGoogle Scholar17.Zagatti, S.: Minimization of non quasiconvex functionals by integro-extremization method. DCDS-A 21(2), 625–641 (2008)MathSciNetCrossRefMATHGoogle Scholar18.Zagatti, S.: On viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 361(1), 41–59 (2009)MathSciNetCrossRefMATHGoogle Scholar19.Zagatti, S.: An integro-extremization approach for non coercive and evolution Hamilton–Jacobi equations. J. Convex Anal. 18(4), 1141–1170 (2011)20.Zagatti, S.: Maximal generalized solution of eikonal equation. J. Diff. Equ. 257, 231–263 (2014)MathSciNetCrossRefMATHGoogle Scholar21.Zagatti, S.: Maximal generalized solution for a class of Hamilton–Jacobi equations. Adv. Pure Appl. Math. ISSN (Online) 1869–6090. ISSN (Print) 1867–1152, 2015 (2015). doi:10.1515/apam-2015-0014 Copyright information© Università degli Studi di Ferrara 2016Authors and AffiliationsSandro Zagatti1Email author1.SISSATriesteItaly About this article CrossMark Print ISSN 0430-3202 Online ISSN 1827-1510 Publisher Name Springer Milan About this journal Reprints and Permissions Article actions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips