Sobolev and isoperimetric inequalities with monomial weights

By X. Cabré, Xavier Ros-Oton. 2012

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We consider the monomial weight $|x_1|^{A_1}...|x_n|^{A_n}$ in $\mathbb R^n$, where $A_i\geq0$ is a real number for each $i=1,...,n$, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure $dx$ replaced by $|x_1|^{A_1}...|x_n|^{A_n}dx$, and they contain the best or critical exponent (which depends on $A_1$, ..., $A_n$). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions. When $A_i$ are nonnegative \textit{integers}, these inequalities are exactly the classical ones in the Euclidean space $\mathbb R^D$ (with no weight) when written for axially symmetric functions and domains in $\mathbb R^D=\mathbb R^{A_1+1}\times...\times\mathbb R^{A_n+1}$.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6452956cd4812661ea855722bbecc2304aff46d5

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Citations in the corpus

Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions (2019, background): “Sobolev inequalities for weighted Sobolev spaces have been deeply studied in different contexts and by many authors (see for example [4, 25, 26]) as they play a key role in the regularity theory for elliptic PDEs.”
Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities on half‐spaces via mass transport and consequences (2013, methodology): “As before, the result for the particular case of the weighted Sobolev inequalities is already contained in work of Cabré and Ros-Oton [9], with the very different proof mentioned above.”
Fractional elliptic problems with critical growth in the whole of $\R^n$ (2015, background): “With this respect, for further comments that compare monomial and Muckenhoupt weights, see the end of Section 1 in [15].”
Sharp isoperimetric inequalities via the ABP method (2013, unknown): “n, are important examples for which (1.13) holds. The isoperimetric inequality —and the corresponding Sobolev inequality (1.3)— with the above monomial weights were studied by the ﬁrst two authors in [11, 12]. These inequalities arose in [11] while studying reactiondiﬀusion problems with symmetry of”
Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature (2020, methodology): “In particular, Theorem 1.1 implies the main result of Cabré, Ros-Oton and Serra [10, Theorem 1.3], i.e., for every open set ⊂ Rn with enough smooth boundary and∫ ∩ w < ∞ one has Pw,H ( ; )(∫ ∩ w ) n+α−1 n+α ≥ Pw,H (W ; )(∫ W∩ w ) n+α−1 n+α , (2.7) where Pw,H denotes the anisotropic weighted perimete”
Regularity of Stable Solutions up to Dimension 7 in Domains of Double Revolution (2012, methodology): “ined inequality is precisely the one given in Proposition 1.7. Thus, the previous proposition extends the classical Sobolev inequality to the case of non-integer exponents aand b. In another article, [5], we prove inequality (1.11) with (R +)2 replaced by (R +)d and with ˙a˝b replaced by the monomia”