On the Dirichlet Problem for Vectorial Hamilton--Jacobi Equations
Abstract
We give sufficient conditions for the existence of solutions to the Hamilton--Jacobi equations with Dirichlet boundary condition: $$ \cases{ g(x,{\hbox{\rm det}}Du(x))=0, \ & for a.e. $x\in\Omega,$\cr u(x)=\varphi(x), & for $x\in\partial\Omega,$} $$ obtaining, in addition, an application to the theory of existence of minimizers for a class of nonconvex variational problems.
MSC codes
Keywords
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Alberto Bressan, The most likely path of a differential inclusion, J. Differential Equations, 88 (1990), 155–174
2.
Arrigo Cellina, On the differential inclusion , Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 69 (1980), 1–6
3.
A. Cellina, On minima of a functional of the gradient: Sufficient condition, Nonlinear Anal., 20 (1993), pp. 343–347.
4.
Arrigo Cellina, Sandro Zagatti, A version of Olech’s lemma in a problem of the calculus of variations, SIAM J. Control Optim., 32 (1994), 1114–1127
5.
Arrigo Cellina, Sandro Zagatti, An existence result in a problem of the vectorial case of the calculus of variations, SIAM J. Control Optim., 33 (1995), 960–970
6.
Enrico Giusti, Metodi diretti nel calcolo delle variazioni, Unione Matematica Italiana, Bologna, 1994vi+422
7.
Bernard Dacorogna, Paolo Marcellini, General existence theorems for Hamilton‐Jacobi equations in the scalar and vectorial cases, Acta Math., 178 (1997), 1–37
8.
Bernard Dacorogna, Jürgen Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 1–26
9.
F. S. De Blasi and G. Pianigiani, A Baire category approach to the existence of solutions of multivalued differental equations in Banach spaces, Funkcial Evac., 25 (1982), pp. 153–162.
10.
Lawrence Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics, Vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1990viii+80
11.
Ivar Ekeland, Roger Témam, Convex analysis and variational problems, Classics in Applied Mathematics, Vol. 28, Society for Industrial and Applied Mathematics (SIAM), 1999xiv+402, Translated from the French
Information & Authors
Information
Published In
SIAM Journal on Mathematical Analysis
Pages: 1481 - 1491
ISSN (online): 1095-7154
Copyright
Copyright © 1998 Society for Industrial and Applied Mathematics.
History
Published online: 1 August 2006
MSC codes
Keywords
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- On the minimum problem for non-quasiconvex vectorial functionalsNonlinear Differential Equations and Applications NoDEA, Vol. 29, No. 6 | 25 September 2022
- Strong compactness in Sobolev spacesmanuscripta mathematica, Vol. 156, No. 3-4 | 11 September 2017
- Solutions of vectorial Hamilton–Jacobi equations and minimizers of nonquasiconvex functionalsJournal of Mathematical Analysis and Applications, Vol. 335, No. 2 | 1 Nov 2007
- Existence of Minimizers for Polyconvex and Nonpolyconvex ProblemsSIAM Journal on Control and Optimization, Vol. 44, No. 4 | 26 July 2006
- Minimization of Functionals of the Gradient by Baire's TheoremSIAM Journal on Control and Optimization, Vol. 38, No. 2 | 26 July 2006
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].