Ngô Quốc Anh

March 23, 2011

A proof of the uniqueness of the solution of the prescribing Gaussian curvature problem

Filed under: Uncategorized — Ngô Quốc Anh @ 22:58

Let us continue the problem of prescribing Gaussian curvature. Our PDE reads as the follows

\displaystyle -\Delta u +K_0(x)=K(x)e^{2u}, \quad x \in M

where M is a compact manifold without the boundary. Today we show that if

\displaystyle K(x) \leqslant 0

then our PDE has unique solution.

Assume that u_1 and u_2 are solutions to the PDE, that is

\displaystyle\begin{gathered} - \Delta {u_1} + {K_0}(x) = K(x){e^{2{u_1}}}, \hfill \\ - \Delta {u_2} + {K_0}(x) = K(x){e^{2{u_2}}}, \hfill \\ \end{gathered}

By subtracting, we have

\displaystyle - \Delta ({u_1} - {u_2}) = K(x)({e^{2{u_1}}} - {e^{2{u_2}}}).

Multiplying both sides by u_1-u_2, integrating over M, and the using the integration by parts we arrive at

\displaystyle\int_M {{{\left| {\nabla ({u_1} - {u_2})} \right|}^2}dv} = \int_M {K(x)\frac{{{e^{2{u_1}}} - {e^{2{u_2}}}}}{{{u_1} - {u_2}}}{{\left| {{u_1} - {u_2}} \right|}^2}dv} .

Since K(x) \leqslant 0, it follows that

\displaystyle\int_M {{{\left| {\nabla ({u_1} - {u_2})} \right|}^2}dv} \leqslant 0.

In particular, u_1 \equiv u_2.

2 Comments »

  1. If i put in right hand side a space-time white noise on a compact Riemannian manifold, then the uniqueless of the solution of this semi-linear elliptic SPDE may be true in the sense of distribution. Try to get some suitable contruction is hard, but the result will be new.

    Comment by Tuan Minh — April 5, 2011 @ 23:54

    • Could you talk your idea in detail here?

      Comment by Ngô Quốc Anh — April 6, 2011 @ 0:02


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