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Hardy’s inequality in a limiting case on general bounded domains

Jaeyoung Byeon

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

E-mail Address: byeon@kaist.ac.kr

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and

Futoshi Takahashi

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

E-mail Address: futoshi@sci.osaka-cu.ac.jp

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https://doi.org/10.1142/S0219199718500700Cited by:1 (Source: Crossref)

Abstract

In this paper, we study Hardy’s inequality in a limiting case:

∫Ω|∇u|Ndx≥CN(Ω)∫Ω|u(x)|N|x|N(logR|x|)Ndx∫Ω|∇u|Ndx≥CN(Ω)∫Ω∣∣u(x)∣∣N|x|N(logR|x|)Ndx<math display="block" altimg="eq-00001.gif"><mrow><msub><mrow><mo class="MathClass-op">∫</mo></mrow><mrow><mi mathvariant="normal">Ω</mi></mrow></msub><mo class="MathClass-open" stretchy="false">|</mo><mo class="MathClass-op">∇</mo><mi>u</mi><msup><mrow><mo class="MathClass-close" stretchy="false">|</mo></mrow><mrow><mi>N</mi></mrow></msup><mi>d</mi><mi>x</mi><mo class="MathClass-rel">≥</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>N</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo class="MathClass-close" stretchy="false">)</mo><msub><mrow><mo class="MathClass-op">∫</mo></mrow><mrow><mi mathvariant="normal">Ω</mi></mrow></msub><mfrac><mrow><mo class="MathClass-rel">|</mo><mi>u</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mi>N</mi></mrow></msup></mrow><mrow><mo class="MathClass-open" stretchy="false">|</mo><mi>x</mi><msup><mrow><mo class="MathClass-close" stretchy="false">|</mo></mrow><mrow><mi>N</mi></mrow></msup><msup><mrow><mo class="MathClass-open" stretchy="false">(</mo><mo class="qopname">log</mo><mfrac><mrow><mi>R</mi></mrow><mrow><mo class="MathClass-rel">|</mo><mi>x</mi><mo class="MathClass-rel">|</mo></mrow></mfrac><mo class="MathClass-close" stretchy="false">)</mo></mrow><mrow><mi>N</mi></mrow></msup></mrow></mfrac><mi>d</mi><mi>x</mi></mrow></math>

for functions

u \in W_{0}^{1, N} (Ω)

$u \in W_{0}^{1, N} (Ω)$ , where

Ω

$Ω$ is a bounded domain in

$ℝ^{N}$ with

$R = {sup}_{x \in Ω} | x |$ . We study the attainability of the best constant

$C_{N} (Ω)$ in several cases. We provide sufficient conditions that assure

$C_{N} (Ω) > C_{N} (B_{R})$ and

$C_{N} (Ω)$ is attained, here

$B_{R}$ is the

$N$ -dimensional ball with center the origin and radius

$R$ . Also, we provide an example of

$Ω \subset ℝ^{2}$ such that

$C_{2} (Ω) > C_{2} (B_{R}) = 1 / 4$ and

$C_{2} (Ω)$ is not attained.

Keywords:

AMSC: 35A23, 26D10

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