Let satisfy the elliptic equation

Assume is jointly concave with respect to and

Then

can not achieve positive maximum in the interior of .

See the paper Korevaar 1983

when time goes to infinity

Let satisfy the elliptic equation

Assume is jointly concave with respect to and

Then

can not achieve positive maximum in the interior of .

See the paper Korevaar 1983

In the last nonlinear analysis seminar, Professor Espinar talked about the overdetermined elliptic problem(OEP) which looks like the following

There is a BCN conjecture related to this

**BCN: **If is Lipschitz, is a smooth(in fact, Lipschitz) connected domain with connected where OEP admits a bounded solution, then must be either a ball, a half space, a generalized cylinder or the complement of one of them.

BCN is false in . Epsinar wih Mazet proved BCN when . This implies the Shiffer conjecture in dimension 2. In higher dimension of Shiffer conjecture, if we know the domain is contained in one hemisphere of , then one can use the equator or the great circle to perform the moving plane.

**Strong maximum principle**: Suppose with , where is connected and open in . If achieves maximum over at an interior point, then must be a constant.

**Proof:** Suppose . Set . If achieves maximum at an interior pt and not constant over all , then . choose a point such that . Consider the largest ball with radius centering at . Suppose , then and , because is the maximum. However, applying the hopf lemma to in , we know where is the outer normal vector of . Contradiction.

**Remark:** If and achieves nonngative maximum in the interior of . Then must be a constant.

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. Moreover, is connect.

For any boundary point of , we may suppose the point is origin. Since , there exists a neighborhood of 0 such that is represented by with . Choosing the principle coordinate system, poositive axis is interior normal at origin and

here .

Since near the origin, it following, on differentiation,

for .

So at origin, we get

which means

So on and for all . However, is continuous function on , which means does not change sign on .

Since is convex and on , then in . Also we know that is positive definite matrix. Thus in , by the hopf lemma,

at

So near origin is almost , which is positive by the property of . So is positive.

For the argument is connect, see the paper in the following remark.

Caffarellli, Nirenberg and Spruck.* The Dirichlet problem for nonlinear second order elliptic equations,III:Functions of the eigenvalues of the Hessian*.

Also refer to the formula [GT, p471].

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 . Suppose and achieves maximum at 0.

Firstly, . Let be the function whose graph is the cone with vertex and base . Since , there exists a ball such that . Let be the function whose graph is cone with vertex and base . Then

Define is the length of the ray in direction tha lies in ,

is defined as the region bounded by , . This is because such that means that

Suppose a bounded region in with boundary , , then the volume of . With this fact in hand,

and

By the holder inequality

.

So

Gilbarg Trudinger’s book chapter 9. 9.1

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

Let

Then and on . By the maximum principle, we conclude,

So

We also have satisfies the above inequality, so the conclusion holds.