Higher-order Sobolev embeddings and isoperimetric inequalities

We study higher-order compact Sobolev embeddings on a domain ∩ ⊆Rn endowed with a probability measure ∨ and satisfying certain isoperimetric inequality. Given m ∈ N, we present a condition on a pair of rearrangement-invariant spaces X( ∩,∨) and Y ( ∩,∨) which suffices to guarantee a compact embedding of the Sobolev space V m X ( ∩,∨) into Y (∩,∨). The condition is given in terms of compactness of certain one-dimensional operator depending on the isoperimetric function of ( ∩,∨). We then apply this result to the characterization of higher-order compact Sobolev embeddings on concrete measure spaces, including John domains, Maz'ya classes of Euclidean domains and product probability spaces, whose standard example is the Gauss space.

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.

Sobolev and isoperimetric inequalities with monomial weights

In this paper, we establish higher-order Sobolev and Rellich-type inequalities on non-compact Riemannian manifolds supporting an isoperimetric inequality. We highlight two notable settings: manifolds with non-negative Ricci curvature and having Euclidean volume growth (supporting Brendle's isoperimetric inequality), and manifolds with non-positive sectional curvature (satisfying the Cartan–Hadamard conjecture or supporting Croke's isoperimetric inequality). Our proofs rely on various symmetrization techniques, and the key ingredient is an iterated Talenti's comparison principle. The non-iterated version is analogous to the main result of Chen and Li [J. Geom. Anal., 2023].

By using optimal mass transport theory we prove a sharp isoperimetric inequality in $${\textsf {CD}} (0,N)$$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. In the case of n-dimensional Riemannian manifolds with nonnegative Ricci curvature, we outline an alternative proof of the rigidity result of Brendle (Comm Pure Appl Math 2021:13717, 2021). As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate symmetrization techniques and optimal volume non-collapsing properties. The equality cases in the latter inequalities are also characterized by stating that sufficiently smooth, nonzero extremal functions exist if and only if the Riemannian manifold is isometric to the Euclidean space.

Schneider introduced an inter-dimensional difference body operator on convex bodies, and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in Rn from those in Rn, were replaced by inter-dimensional simplicial operators, which generate convex bodies in Rnm from those in Rn (or vice versa). In this work, we treat the Lp extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary m-dimensional convex bodies containing the origin. We establish mth-order Lp isoperimetric inequalities, including the mth-order versions of the Lp Petty projection inequality, Lp Busemann-Petty centroid inequality, Lp Santaló inequalities, and Lp affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals (Rn,‖⋅‖E)→(Rm,‖⋅‖F).

It is shown that the minimum value for c is $4/3 \sqrt 3 \pi ^2 \cong .0780$ for $\phi $ of function class $C^0 $ piecewise $C^2 $ in real Euclidean 3-space.

Affine vs. Euclidean isoperimetric inequalities

On manifolds with non-negative Ricei curvature and Sobolev inequalities

The sharp Sobolev and isoperimetric inequalities split twice

Sobolev inequalities in arbitrary domains

Let ( M , g ) (M,g) be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature K ≤ − 1 K\le -1 . If f f is a compactly supported function of bounded variation on M M , then f f satisfies the Sobolev inequality \[ 4 π ∫ M f 2 d A + ( ∫ M | f | d A ) 2 ≤ ( ∫ M ‖ ∇ f ‖ d A ) 2 . 4\pi \int _M f^2\,dA+ \left (\int _M |f|\,dA \right )^2\le \left (\int _M\|\nabla f\|\,dA \right )^2. \] Conversely, letting f f be the characteristic function of a domain D ⊂ M D\subset M recovers the sharp form 4 π A ( D ) + A ( D ) 2 ≤ L ( ∂ D ) 2 4\pi A(D)+A(D)^2\le L(\partial D)^2 of the isoperimetric inequality for simply connected surfaces with K ≤ − 1 K\le -1 . Therefore this is the Sobolev inequality “equivalent” to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces. Under the same assumptions on ( M , g ) (M,g) , if c : [ a , b ] → M c\colon [a,b]\to M is a closed curve and w c ( x ) w_c(x) is the winding number of c c about x x , then the Sobolev inequality implies \[ 4 π ∫ M w c 2 d A + ( ∫ M | w c | d A ) 2 ≤ L ( c ) 2 , 4\pi \int _M w_c^2\,dA+ \left (\int _M|w_c|\,dA \right )^2\le L(c)^2, \] which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature ≤ − 1 \le -1 .

We first review improvements of (first-order) Sobolev and Hardy inequalities by the addition of suitable lower-order terms (these improved inequalities have been pioneered by Brezis and Nirenberg (1983) and Brezis and Vazquez (1997)). Recently, corresponding results concerning first-order Hardy–Sobolev and higher-order Sobolev and Hardy inequalities have been proved.

Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities

Optimal Sobolev embeddings for the Ornstein-Uhlenbeck operator