Equivalence of critical and subcritical sharp Trudinger-Moser-Adams inequalities
By Nguyen Huong Lam, G. Lu, Lu Zhang. 2015
89 citations recorded by the source corpus.
Sharp Trudinger-Moser inequalities on the first order Sobolev spaces and their analogous Adams inequalities on high order Sobolev spaces play an important role in geometric analysis, partial differential equations and other branches of modern mathematics. Such geometric inequalities have been studied extensively by many authors in recent years and there is a vast literature. There are two types of such optimal inequalities: critical and subcritical sharp inequalities, both are with best constants. Critical sharp inequalities are under the restriction of the full Sobolev norms for the functions under consideration, while the subcritical inequalities are under the restriction of the partial Sobolev norms for the functions under consideration. There are subtle differences between these two type of inequalities. Surprisingly, we prove in this paper that these critical and subcritical Trudinger-Moser and Adams inequalities are actually equivalent. Moreover, we also establish the asymptotic behavior of the supremum for the subcritical Trudinger-Moser and Adams inequalities on the entire Euclidean spaces (Theorem 1.1 and Theorem 1.3) and provide a precise relationship between the supremums for the critical and subcritical Trudinger-Moser and Adams inequalities (Theorem 1.2 and Theorem 1.4).
Subfields: Applied Mathematics, Geometry and Topology, Mechanics of Materials
View Paper: https://www.semanticscholar.org/paper/74f37dc6964be5b757b5a7f56b65e8cbfe453e29
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In Applied Mathematics
In Geometry and Topology
In Mechanics of Materials
Citations in the corpus
- Trudinger–Moser Inequalities in Fractional Sobolev–Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation (2018, background): “Furthermore, Lam, Lu and Zhang showed the asymptotic behaviors of the supremums of the subcritical and critical Trudinger–Moser inequalities and established their equivalence of such inequalities in [36].”