Ngô Quốc Anh

August 9, 2011

Stereographic projection, 5

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 14:39

I put here several common formulas that are needed when we work on stereographic projection. First we try to calculate

\displaystyle\Delta \left( {{{\left( {\frac{2}{{1 + |x{|^2}}}} \right)}^{n - 2}}} \right).

We do step by step.

Step 1. A direct computation leads us to

\displaystyle\begin{gathered} {\partial _i}\left( {{{\left( {\frac{2}{{1 + |x{|^2}}}} \right)}^{n - 2}}} \right) = (n - 2){\left( {\frac{2}{{1 + |x{|^2}}}} \right)^{n - 3}}{\partial _i}\left( {\frac{2}{{1 + |x{|^2}}}} \right) \hfill \\ \qquad= (n - 2){\left( {\frac{2}{{1 + |x{|^2}}}} \right)^{n - 3}}\frac{{ - 4{x_i}}}{{{{(1 + |x{|^2})}^2}}}. \hfill \\ \end{gathered}

Therefore,

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August 6, 2011

Definition of coercivity

Filed under: PDEs — Ngô Quốc Anh @ 6:26

This is a classical question: how could we understand the coercivity of the operator -\Delta + h? Given a smooth compact Riemannian manifold of dimension n\geqslant 3, the original definition for the coercivity of such an operator is just

\displaystyle\int_M {({{\left| {\nabla u} \right|}^2} + h{u^2})d{v_g}} \geqslant C\underbrace {\int_M {({{\left| {\nabla u} \right|}^2} + {u^2})d{v_g}} }_{\left\| u \right\|_{{H^1}}^2}

for any u \in H^1(M) for some constant C>0. If h is a function, by taking a positive constant we immediately see that it is necessary

\displaystyle \int_M h dv_g >0.

Consequently, \sup_M h>0 and in particular if h is constant, h must be positive. If h is not constant, then it is not clear to classify h. In that case, we interpret this terminology in a different manner as follows

\displaystyle\mu=\mathop {\inf }\limits_{u \in {H^1}(M)} \frac{{\displaystyle\int_M {({{\left| {\nabla u} \right|}^2} + h{u^2})d{v_g}} }}{{\displaystyle\int_M {{u^2}d{v_g}} }} > 0.

It is simple to show that the coercivity of -\Delta + h is equivalent to \mu>0. The fact that \mu>0 implies that  the coercivity of -\Delta + h is simple just by definition. For the reverse case, it involves a bit of calculus of variation. For interested reader, we refer to a book by O. Druet, E. Hebey, and F. Robert [here].

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