A remark on regularization in Hilbert spaces. (English) Zbl 0631.49018

Lasry, J. M.; Lions, P. L.

Isr. J. Math. 55, 257-266 (1986).

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

References:

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