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A remark on regularization in Hilbert spaces. (English) Zbl 0631.49018
For u, a uniformly continuous function on a Hilbert space H and for \(\epsilon >0\) define \[ \underline u_{\epsilon}(x)=\sup_{z\in H}\inf_{y\in H}[u(y)+(1/(2\epsilon))\| z-y\|^ 2- (1/\epsilon)\| z-x\|^ 2]\quad and \] \[ \bar u_{\epsilon}(x)=\inf_{z\in H}\sup_{y\in H}[u(y)- (1/(2\epsilon))\| z-y\|^ 2+(1/\epsilon)\| z-x\|^ 2]. \] It is shown that \(u_{\epsilon}=\bar u_{\epsilon}\) and \(u_{\epsilon}=\underline u_{\epsilon}\) are in \(C^{1,1}\), that they approximate u uniformly as \(\epsilon\downarrow 0\) and have the properties \(\| \nabla u_{\epsilon}\|_{\infty}\leq c_{\epsilon}\| u\|_{\infty}\), \(\| (\nabla u_{\epsilon}(x)-\nabla u_{\epsilon}(y))/| x-y| \|_{\infty}\leq c_{\epsilon}\| u\|_{\infty}\), \(\inf_{H}u_{\epsilon}\leq u_{\epsilon}\leq \sup_{H}u\), \(\| \nabla u_{\epsilon}\|_{\infty}\leq \| (u(x)-u(y))/| x-y| \|_{\infty}\leq \infty\). The relations of the approximants to viscosity solutions of Hamilton Jacobi equations \(u_ t+()| \nabla u|^ 2=0\) and \(u_ t-()| \nabla u|^ 2=0\) are discussed. Of course in the infinite dimensional case approximants with the desired properties can be constructed by means of mollifiers.
Reviewer: P.Szeptycki

49L99 Hamilton-Jacobi theories
35D05 Existence of generalized solutions of PDE (MSC2000)
49J45 Methods involving semicontinuity and convergence; relaxation
Full Text: DOI
[1] V. Barbu and G. Da Prato,Hamilton-Jacobi Equations in Hilbert Spaces, Pitman, London, 1983.
[2] M. G. Crandall and P. L. Lions,Viscosity solutions of Hamilton-Jacobi equations, Trans. Am. Math. Soc.277 (1983), 1–42. · Zbl 0599.35024 · doi:10.1090/S0002-9947-1983-0690039-8
[3] M. G. Crandall and P. L. Lions, in preparation.
[4] M. G. Crandall, L. C. Evans and P. L. Lions,Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Am. Math. Soc.282 (1984), 487–502. · Zbl 0543.35011 · doi:10.1090/S0002-9947-1984-0732102-X
[5] I. Ekeland and J. M. Lasry,On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface. Ann. Math.112 (1980), 238–319. · Zbl 0449.70014 · doi:10.2307/1971148
[6] P. L. Lions,Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982.
[7] A. S. Nemirovski and S. M. Semenov,The polynomial approximation of functions in Hilbert spaces, Mat. Sb. (N.S.)92 (134) (1973), 257–281.
[8] A. Pommelet,Transformée de Toland et la théorie de Morse pour certaines fonctions non différentiables, Thèse de 3e cycle, Université Paris – Dauphine, 1984.
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