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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)


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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1Brezis, H. and Gallouet, T.. Nonlinear Schrödinger evolution equations. Nonlinear Anal. Theory Methods Appl. 4 (1980), 677681.
2Brezis, H. and Wainger, S.. A note on limiting cases of Sobolev embedding and convolution inequalities. Commun. Partial Differ. Equ. 5 (1987), 773789.
3Elliott, L. H. and Loss, M.. Analysis. Graduate Studies in Mathematics, vol. 14 (Providence, RI: American Mathematical Society, second edition, 2001).
4Engler, H.. An alternative proof of the Brezis—Wainger inequality. Commun. Partial Differ. Equ. 14 (1989), 541544.
5Foias, C., Manley, O., Rosa, R. and Temam, R.. Navier-Stokes Equations and Turbulence (Cambridge: Cambridge University Press, 2001). ISBN 0-521-36032-3.
6Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order (Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, 1983). XIII, 513 S. DM 128. ISBN 3-540-13025-X (Grundlehren der mathematischen Wissenschaften 224).
7Giga, M. H., Giga, Y. and Saal, J.. Nonlinear partial differential equations. Asymptotic behaviour of solutions and self-similar solutions. Progress in Nonlinear Differential Equations and Their Applications, vol. 79, ISBN: 978-0-8176-4173-3 (Boston, MA: Birkhäuser, Boston, Ltd., 2010).
8Górka, P.. Brezis–Waigner inequality in Riemannian manifolds. J. Inequal. Appl. 11 (2008), 16. Article ID 715961.
9Kato, T. and Ponce, G.. Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41 (1988), 891907.
10Kresin, G. and Maz'ya, V.. Maximum principles and sharp constants for solutions of elliptic and parabolic systems. Mathematical Surveys and Monographs, Vol. 183, pp. viii+317, ISBN: 978-0-8218-8981-7 (Providence, RI: American Mathematical Society, 2012).
11Maz'ya, V. and Shaposhnikova, T.. Brézis–Gallouet–Wainger type inequality for irregular domains. Complex Variables and Elliptic Equations 56 (2011) 10–11, 991–1002. A tribute to Victor I. Burenkov; Guest Editors: Robert P. Gilbert, Massimo Lanza de Cristoforis and Alexander Pankov.
12Morii, K., Sato, T. and Wadade, H.. Brézis–Gallouet–Wainger type inequality with a double logarithmic term in the Hölder space: Its sharp constants and extremal functions. Nonlinear Anal.: Theory Methods Appl. 73 (2010), 17471766.
13Morii, K., Sato, T., Sawano, Y. and Wadade, H.. Sharp constants of Brézis–Gallout–Wainger type inequalities with a double logarithmic term on bounded domains in Besov and Triebel-Lizorkin spaces. Boundary Value Prob. (2010), 138. ID:584521.
14Sawano, Y.. Brezis–Gallouet–Wainger type inequality for Besov-Morrey spaces. Studia Math. 196 (2010), 91101.
15Talenti, G.. Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1976), 697718.
16Vishik, M.. Incompressible flows of an ideal liquid with unbounded vorticity. Commun. Math. Phys. 213 (2000), 697731.
17Weinberger, H. F.. Symmetrization in uniformly elliptic problems. In 1962 Studies in Mathematical Analysis and Related Topics, pp. 424428 (Stanford, California: Stanford Univ. Press, 1962).


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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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