×

Almost classical solutions of Hamilton-Jacobi equations. (English) Zbl 1161.26005

The authors study the existence of almost classical solutions of fairly general Hamilton-Jacobi equations on open subsets of \(\mathbb R^d\) and on \(d\)-dimensional smooth manifolds for \(d \geq 2\).
Let \(d \geq 2\) and let \(\Omega\) be an open subset of \(\mathbb R^d\). Let \(F : \mathbb R \times \Omega \times \mathbb R^{d} \rightarrow \mathbb R\) and \(u_{0} : \partial \Omega \rightarrow \mathbb R\) be continuous functions. We say that \(u : \overline \Omega \rightarrow \mathbb R\) is an almost classical solution of \(F(u(x),x,\nabla u(x))=0\) with Dirichlet condition \(u|_{ \partial \Omega} = u_{0}\) if
\(\bullet\)
\(u(x) = u_{0}(x)\), \(x \in \partial \Omega\);
\(\bullet\)
for every \(x \in \Omega\), \(u\) is differentiable at \(x\) and \(F(u(x),x,\nabla u(x)) \leq 0\);
\(\bullet\)
\(u\) satisfies \(F(u(x),x,\nabla u(x))=0\) for almost every \(x \in \Omega\) in the sense of Lebesgue measure on \(\mathbb R^d\).
The authors obtain the following.
Theorem. Let \(d \geq 2\), let \(\Omega\) be an open subset of \(\mathbb R^d\) and let \(F : \mathbb R \times \Omega \times \mathbb R^{d} \rightarrow \mathbb R\) be a continuous function. Suppose that
(A)
there exists a continuous function \(u_{0} : \overline \Omega \rightarrow \mathbb R\) such that \(u_{0}\) is \(C^{1}\)-smooth on \(\Omega\) and \(F(u_{0}(x),x,\nabla u_{0}(x)) \leq 0\) \((x \in \Omega)\);
(B)
for each compact set \(K \subseteq \Omega\) there exist an \(M_{K}>0\) and an \(\alpha_{K} > 0\) such that for every \(x \in K\), \(u \in [0,\alpha_{K}]\) and \(p \in \mathbb R ^{d}\) satisfying \(\|p\| \geq M_{K}\) we have \(F(u_{0}(x)+u,x,p) > 0\).
Then there exists an almost classical solution \(u\) of \(F(u(x),x,\nabla u(x))=0\) with Dirichlet condition \(u|_{ \partial \Omega} = u_{0}\). Moreover, if \(u_{0}\) is \(C^{1}\)-smooth on \(\mathbb R^d\) then \(u\) can be extended to a differentiable function on \(\mathbb R^d\).
Analogous results are obtained for similarly general Hamilton-Jacobi equations on \(d\)-dimensional smooth manifolds for \(d \geq 2\). As a very special corollary of their results, for every Riemannian manifold \(M\) of dimension \(\geq 2\) the authors prove the existence of a differentiable function \(u\) on \(M\) which satisfies the eikonal equation \(\| \nabla u(x) \|_{x}=1\) almost everywhere on \(M\).

MSC:

26B05 Continuity and differentiation questions
35B65 Smoothness and regularity of solutions to PDEs
58J32 Boundary value problems on manifolds
PDFBibTeX XMLCite
Full Text: DOI Euclid EuDML

References:

[1] Azagra, D., Ferrera, J. and López-Mesas, F.: Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220 (2005), no. 2, 304-361. · Zbl 1067.49010 · doi:10.1016/j.jfa.2004.10.008
[2] Barles, G.: Solutions de viscosité des équations de Hamilton-Jacobi. Mathématiques el Applications 17 . Springer-Verlag, Paris, 1994. · Zbl 0819.35002
[3] Benameur, M. T.: Triangulations and the stability theorem for foliations. Pacific J. Math. 179 (1997), no. 2, 221-239. · Zbl 0871.57029
[4] Buczolich, Z.: Solution to the gradient problem of C. E. Weil. Rev. Mat. Iberoamericana 21 (2005), no. 3, 889-910. · Zbl 1116.26007 · doi:10.4171/RMI/439
[5] Crandall, M. G., Ishii, H. and Lions, P. L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1-67. · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[6] Deville, R. and Matheron, É.: Infinite games, Banach space geometry and the eikonal equation. Proc. Lond. Math. Soc. (3) 95 (2007), no. 1, 49-68. · Zbl 1163.91007 · doi:10.1112/plms/pdm005
[7] Fathi, A. and Siconolfi, A.: Existence of \(C^1\) critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155 (2004), no. 2, 363-388. · Zbl 1061.58008 · doi:10.1007/s00222-003-0323-6
[8] Malý, J. and Zelený, M.: A note on Buczolich’s solution of the Weil gradient problem: a construction based on an infinite game. Acta Math. Hungar. 113 (2006), no. 1-2, 145-158. · Zbl 1127.26006 · doi:10.1007/s10474-006-0096-7
[9] Mantegazza, C. and Mennucci, A. C.: Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds. Appl. Math. Optim. 47 (2003), no. 1, 1-25. · Zbl 1048.49021 · doi:10.1007/s00245-002-0736-4
[10] Weil, C. E.: On properties of derivatives. Trans. Amer. Math. Soc. 114 (1965), 363-376. JSTOR: · Zbl 0163.29604 · doi:10.2307/1994180
[11] Whitney, H.: Geometric integration theory. Princeton University Press. Princeton, N.J., 1957. · Zbl 0083.28204
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.