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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:
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