Sharp Singular Trudinger–Moser Inequalities in Lorentz–Sobolev Spaces

By G. Lu, Hanli Tang. 2016

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Abstract In this paper, we first establish a singular (0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any bounded domain in ℝn${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.1). Next, we prove the critical singular (0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any unbounded domain in ℝn${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.2). Then, we set up a subcritical singular (0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any unbounded domain in ℝn${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.3). Finally, we establish the subcritical nonsingular (β=0${(\beta=0}$) Trudinger–Moser inequality on any unbounded domain in ℝn${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.5). The constants in all these inequalities are sharp. In [9], for the proof of Theorem 1.2 in the nonsingular case β=0${\beta=0}$, the following inequality was used (see [17]): u ∗ ⁢ ( r ) - u ∗ ⁢ ( r 0 ) ≤ 1 n ⁢ w n 1 / n ⁢ ∫ r r 0 U ⁢ ( s ) ⁢ s 1 / n ⁢ d ⁢ s s , $u^{\ast}(r)-u^{\ast}(r_{0})\leq\frac{1}{nw_{n}^{{1/n}}}\int_{r}^{r_{0}}U(s)s^{% {1/n}}\frac{ds}{s},$ where U⁢(x)${U(x)}$ is the radial function built from |∇⁡u|${|\nabla u|}$ on the level set of u, i.e., ∫ | u | > t | ∇ u | d x = ∫ 0 | { | u | > t } | U ( s ) d s . $\int_{|u|>t}\lvert\nabla u|\,dx=\int_{0}^{|\{|u|>t\}|}U(s)\,ds.$ The construction of such U uses the deep Fleming–Rishel co-area formula and the isoperimetric inequality and is highly nontrivial. Moreover, this argument will not work in the singular case 0<β<n${0<\beta<n}$. One of the main novelties of this paper is that we can avoid the use of this deep construction of such a radial function U (see remarks at the end of the introduction). Moreover, our approach adapts the symmetrization-free argument developed in [19, 21, 23], where we derive the global inequalities on unbounded domains from the local inequalities on bounded domains using the level sets of the functions under consideration.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/397dccd103a2904d02ab4a9187b3e1499c09e022

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In Applied Mathematics

  1. Nguyen Huong Lam
  2. G. Lu
  3. Caifeng Zhang
  4. Hanli Tang
  5. Y. Yang
  6. Federica Sani
  7. Jiuyi Zhu
  8. Lu Zhang

In Computational Theory and Mathematics

  1. Nguyen Huong Lam
  2. G. Lu
  3. Caifeng Zhang
  4. Hanli Tang
  5. Y. Yang
  6. Federica Sani
  7. Jiuyi Zhu
  8. Lu Zhang

In Mechanics of Materials

  1. G. Lu
  2. Nguyen Huong Lam

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