Ngô Quốc Anh

August 16, 2010

The Moser-Trudinger inequality for domains with holes

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 2:42

In this entry, we are interested in the following result

Theorem (Moser-Trudinger’s inequality for domains with holes). Let \Omega be a bounded smooth domain in \mathbb R^2. Let S_1 and S_2 be two subsets of \overline \Omega satisfying

{\rm dist}(S_1,S_2) \geqslant \delta_0>0

and let \gamma_0 be a number satisfying \gamma_0 \in \left(0,\frac{1}{2}\right). Then for any \varepsilon>0, there exists a constant c=c(\varepsilon, \delta_0, \gamma_0)>0 such that

\displaystyle\int_\Omega {{e^u}} \leqslant C\exp \left[ {\frac{1}{{32\pi - \varepsilon }}\int_\Omega {{{\left| {\nabla u} \right|}^2}} + C} \right]

holds for all u \in H_0^1(\Omega) satisfying

\displaystyle\frac{{\int_{{S_1}} {{e^u}} }}{{\int_\Omega {{e^u}} }} \geqslant {\gamma _0}, \quad \frac{{\int_{{S_2}} {{e^u}} }}{{\int_\Omega {{e^u}} }} \geqslant {\gamma _0}.


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