For the second installment of this series of posts (which started here) on the first Bourbaki seminar of 2015, we will discuss Gilles Carron… 2,384 more words

## Tags » Maximum Principle

#### A strong maximum principle due to Tolksdorf and application to the prescribed scalar curvature equation

In this note, we consider a strong maximum principle due to Tolksdorf and application to the prescribed scalar curvature equation. In the context of closed and compact Riemannian manifolds of dimension , it can be stated as follows… 397 more words

#### The generalized maximum principle

Let us start our notes with a very fundamental maximum principle for any strongly second order elliptic operator. We have

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Theorem(Maximum principle). Let satisfy the differential inequality…

#### Monge-Ampere equation and bounday behavior of domain

Suppose is a bounded domain and is . If there exists a **convex** function satisfies

Then is uniformly convex. In other words, the principle curvature of every point on , namely , are positive. 151 more words

#### Improvement of Aleksandrov maximum principle

For we have

where .

There is an easy proof when the RHS is . We will adjust the original proof.

By replacing with , it suffices to assume on . 92 more words

#### Strong barrier function and generalized linear elliptic equation

Suppose is a domain in . satisfies the exterior sphere condition. That is there exists a ball such that . Then the function

is a barrier function for at . 111 more words

#### One simple maximum principle

If is elliptic with in a bounded domain , and satisfies in , then

Let

Then and on . By the maximum principle, we conclude, … 12 more words