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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

##### MSC:
 49L99 Hamilton-Jacobi theories 35D05 Existence of generalized solutions of PDE (MSC2000) 49J45 Methods involving semicontinuity and convergence; relaxation
##### Keywords:
viscosity solutions; Hamilton Jacobi equations
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##### References:
  V. Barbu and G. Da Prato,Hamilton-Jacobi Equations in Hilbert Spaces, Pitman, London, 1983.  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  M. G. Crandall and P. L. Lions, in preparation.  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  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  P. L. Lions,Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982.  A. S. Nemirovski and S. M. Semenov,The polynomial approximation of functions in Hilbert spaces, Mat. Sb. (N.S.)92 (134) (1973), 257–281.  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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