A uniform result in dimension 2

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A uniform result in dimension 2

Samy Skander Bahoura

To cite this version:

Samy Skander Bahoura. A uniform result in dimension 2. 2018. �hal-01823210v2�

A uniform result in dimension 2.

Samy Skander Bahoura ^∗

Université Pierre et Marie Curie, 75005, Paris, France.

Abstract

We give a uniform result in dimension 2 for the solutions to an equation on compact Riemannian surface without boundary.

Keywords: dimension 2, Riemannian surface without boundary. Uniform result.

1 Introduction and Main result

We set ∆ = −∇ i (∇ ⁱ ) the Laplace-Beltrami operator. We are on compact Rie- mannian surface (M, g) without boundary.

We start with the following example: for all ǫ > 0 the constant functions z ǫ = log ǫ

a with a > 0, are solutions to ∆z ǫ +ǫ = ae ^z

and tend to −∞ uniformly on M .

Question: What’s about the solutions u ǫ to the following equation

∆u ǫ + ǫ = V ǫ e ^u

, (E ǫ ) with 0 < a ≤ V ǫ (x) ≤ b < +∞ on M ?

Next, we assume V ǫ Hölderian and V ǫ → V in L

The equation (E ǫ ) is of prescribed scalar curvature type equation. The term ǫ replace the scalar curvature.

Theorem 1.1 . If ǫ → 0, the solutions u ǫ to (E ǫ ) satisfy:

sup

M

u ǫ → −∞.

By using the same arguments of the next theorem, we have:

∗

E-mails: samybahoura@yahoo.fr, samybahoura@gmail.com

Theorem 1.2 . If ǫ → 0, the solutions u ǫ to (E ǫ ) satisfy:

u ǫ − log ǫ → k ∈ R.

uniformly on M .

Thus, we have a unifrom bound for the solutions:

k

+ log ǫ ≤ u ǫ ≤ log ǫ + k

.

We also have another proof of the uniqueness result which appear in [2].

This proof uses Brezis Merle arguments.

Theorem 1.3 . If ǫ → 0, the solutions u ǫ to (E ǫ ) with V ǫ ≡ 1, are such:

u ǫ ≡ log ǫ.

2 Proof of the theorems 1,2,3.

Proof of theorem 1:

We have:

Z

M

V ǫ e ^u

→ 0. (∗)

Let’s consider x ǫ a point such that max M u ǫ = u ǫ (x ǫ ), then x ǫ → x

. We consider a neighborhood of x

and we use isothermal coordinates around x

(see [6]), there exists α > 0 and a regular function φ such that:

∆

u ǫ + ǫe ^φ = V ǫ e ^φ e ^u

in B(0, α).

The metric g of M satisfies g = e ^φ (dx

+ dy

).

Let’s consider u

such that:

∆

u

= e ^φ in B(0, α).

(with Dirichlet condition for example).

The function v ǫ = u ǫ + ǫu

satisfies:

∆

v ǫ = ˜ V ǫ e ^v

, with V ˜ ǫ = V ǫ e ^φ−ǫu

. We use (∗) to have:

Z

B(0,α)

e ^v

→ 0 and 0 < a ˜ ≤ V ˜ ǫ ≤ ˜ b. (∗∗)

with α > 0.

The sequence v ǫ satisfies all the conditions of the theorem of Brezis and Merle, see [4].

As v ǫ satisfy (∗∗), the last condtion of the theorem of [4] is not possible.

Now, suppose that the first assertion of the theorem of Brezis and Merle is true. We have the local boundedness result. We can say that u ǫ converge uniformly on M to a fonction u and in C

topology by the elliptic estimates.

If we tend ǫ to 0 we get that u satisfies in the sense of distributions:

∆u = V e ^u .

If we integrate the equation, we have a contradiction (since 0 < a ≤ V ≤ b < +∞.

Thus, u ǫ satisfies the second assertion of the theorem of [4] and thus u ǫ

diverge uniformly to −∞ on M . Proof of Theoreme 2,3:

We set,

w ǫ = u ǫ − log ǫ.

Then, w ǫ is solution to:

∆w ǫ + ǫ = ǫV ǫ e ^w

.

We use Brezis and Merle’s theorem and the previous arguments of theoerm 1, to have a convergence to a constant:

w ǫ → w

= ct, uniformly on M .

In isothermal coordinates around x

= lim x ǫ with x ǫ such that, w ǫ (x ǫ ) = max M w ǫ , as in the previous case

∆

w ǫ + ǫe ^φ = ǫe ^φ V ǫ e ^w

in B(0, α).

The metric g of M satisfies g = e ^φ (dx

+ dy

). Let’s consider u

such that:

∆

u

= e ^φ in B(0, α).

(with Dirichlet condition for example).

The function v ǫ = w ǫ + ǫu

satisfies:

∆

v ǫ = ˜ V ǫ e ^v

, with V ˜ ǫ = ǫV ǫ e ^φ−ǫu

.

One can apply the theorem of Brezis and Merle, see [4].

First we have,

Z

M

V ǫ e ^w

= |M |,

which imply,

w ǫ 6→ −∞, and,

Z

M

ǫV ǫ e ^w

= ǫ|M | → 0, which imply the non-concentration.

And,

w ǫ → w

, in the C

topology with,

∆w

= 0 ⇒ w

≡ k ∈ R.

For the third theorem we have:

Z

M

e ^w

= |M | ⇒ k = 0.

We write:

w ǫ = ¯ w ǫ + f i , with the fact that,

Z

M

f i = 0,

and, f i is solution to:

∆f i = ǫ i (e ^w

− 1) = ǫ i (e ^w

(1 + f i + O(f _i

)) − 1), We multiply the equation by f i and we integrate, we obtain:

||∇f i ||

_L

= o(||f i ||

_L

).

This is in contradiction with the Poincaré inequality if f i 6≡ 0.

Thus,

f i ≡ 0, w i ≡ w ¯ i = 0.

References

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[2] Bartolucci, D. Uniqueness and bifurcation for seminilinear elliptic equations

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3,4, pp 503-519.

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Wiss.,224, 1983.

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