学术报告

【online】The Lax-Oleinik representation in non-compact setting

发布人：发布时间： 2020-06-12

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**题目：**The Lax-Oleinik representation in non-compact setting

**时间：**2020年06月18日 21:00-22:00

**地点：**Zoom APP **会议 ID: **993 2636 1438 **密码: **144541

**报告人：**Albert Fathi（Affiliation Georgia Tech, USA）

**摘要：**We will be interested in viscosity solutions of the evolution Hamilton-Jacobi equation ∂t U + H (x, ∂x U ) = 0.

Here we think of the case where U : [0, +∞[×M → R, with M is a manifold.

If M is compact, as has been known for a long time, the maximum principle yields uniqueness for a given initial condition U|{0}×M. This in turn implies the representation by a Lax-Oleinik type formula.

When M is not compact, the global maximum principle does not immediately hold.

Hitoshi Ishii and his coworkers obtained results about 10 years ago under some restrictions when M = Rn. Basically the restrictions are about controlled growth at infinity.

We will explain that under the hypothesis that H is Tonelli, all continuous solutions of the evolution Hamilton-Jacobi equation above satisfy the Lax-Oleinik representation even for non-compact M. This of course will imply uniqueness for a given initial condition.

Moreover, we will also show that if any pointwise finite U is given by the Lax-Oleinik representation is automatically continuous and therefore a viscosity solution.

**邀请人****：** 程伟 老师