Ngô Quốc Anh

July 4, 2010

Nash-Moser’s iteration technique

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 23:05

We now continue this topic. Today we discuss a very powerful technique known as the Nash-Moser iteration technique. Let us consider a sub-solution u of the following variational inequality

\displaystyle\int_\Omega {{A^{\alpha \beta }}(x){\nabla _\alpha }u{\nabla _\beta }\varphi dx} \leqslant 0, \quad \varphi \in H_0^1(\Omega )

where \varphi \geqslant 0. We assume A is of class L^\infty(\Omega) and do satisfy an ellipticity condition. For the sake of simplicity we assume u\geqslant 0. Then by using the test function

\varphi=u^p\eta^2, \quad p>1

with \eta a cut-off function defined as in this topic we arrive at

\displaystyle\int_{B_R} {{A^{\alpha \beta }}(x){\nabla _\alpha }u{\nabla _\beta }({u^p}{\eta ^2})dx} \leqslant 0

which yields

\displaystyle p\int_{B_R} {{A^{\alpha \beta }}(x){u^{p - 1}}{\eta ^2}{\nabla _\alpha }u{\nabla _\beta }udx} + 2\int_{B_R} {{A^{\alpha \beta }}(x){u^p}\eta {\nabla _\alpha }u{\nabla _\beta }\eta dx} \leqslant 0.

Thus by the ellipticity condition

\displaystyle\int_{{B_R}} {{u^{p - 1}}{\eta ^2}{{\left| {\nabla u} \right|}^2}dx} \leqslant \frac{c}{p}\int_{{B_R}} {\left| {\nabla u} \right|{u^{\frac{{p - 1}}{2}}}{u^{\frac{{p + 1}}{2}}}\left| {\nabla \eta } \right|\eta dx}

and thus

\displaystyle\int_{{B_R}} {{u^{p - 1}}{\eta ^2}{{\left| {\nabla u} \right|}^2}dx} \leqslant \frac{c}{{{p^2}}}\int_{{B_R}} {{u^{p + 1}}{{\left| {\nabla \eta } \right|}^2}dx} .

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