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On the L-maximization of the solution of Poisson's equation: Brezis–Gallouet–Wainger type inequalities and applications

  • Davit Harutyunyan (a1) and Hayk Mikayelyan (a2)

Abstract

For the solution of the Poisson problem with an L right hand side

\begin{cases} -\Delta u(x) = f (x) & {\rm in}\ D, \\ u=0 & {\rm on}\ \partial D \end{cases}
we derive an optimal estimate of the form
\|u\|_\infty\leq \|f\|_\infty \sigma_D(\|f\|_1/\|f\|_\infty),
where σD is a modulus of continuity defined in the interval [0, |D|] and depends only on the domain D. The inequality is optimal for any domain D and for any values of $\|f\|_1$ and $\|f\|_\infty .$ We also show that
\sigma_D(t)\leq\sigma_B(t),\text{ for }t\in[0,|D|],
where B is a ball and |B| = |D|. Using this optimality property of σD, we derive Brezis–Galloute–Wainger type inequalities on the L norm of u in terms of the L1 and L norms of f. As an application we derive L − L1 estimates on the k-th Laplace eigenfunction of the domain D.

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On the L-maximization of the solution of Poisson's equation: Brezis–Gallouet–Wainger type inequalities and applications

  • Davit Harutyunyan (a1) and Hayk Mikayelyan (a2)

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