Sobolev and isoperimetric inequalities with monomial weights

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.

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.

We introduce some methods to study the isoperimetric problem in 3-dimensional Riemannian manifolds and we show that in the positive curva- ture case we can control the topology of the isoperimetric regions. We consider the case of the projective space, which was first solved by Ritorea nd Ros, and we apply it to the Willmore problem. The isoperimetric problem is a classical topic in geometry but at the same time many basic questions about it remain unsolved. In this paper we first introduce some of the methods used in the study of that problem, although we will not try to be exhaustive at all. We will explain some relatively flexible ideas, like symmetrization or stability, which can be adapted to a certain number of situations. As an example we will study the problem for radial metrics on the 3-sphere. Then we will show that in 3-manifolds with positive Ricci curvature the topol- ogy of the isoperimetric regions can be controled. In particular we will prove that, when the volume of the ambient space is large, any isoperimetric surface must be either an sphere or a torus. As consequence, we will solve the isoperimetric problem in the real projective space or, equivalently, the isoperimetric problem for antipodal invariant regions in the 3-sphere. That result was first obtained by Ritorea nd Ros (35), but here we will give a somewhat different proof. Finally, as application of the above results, we will solve the Willmore conjec- ture for tori in euclidean space which are symmetric with respect to a point. 2. The isoperimetric problem In this paper we will only consider the three dimensional case. Let M be a Riemaniann 3-dimensional manifold with or without boundary and volume V(M ) ∈ )0, ∞). Given a positive number v< V(M ), we want to study the compact surfaces Σ ⊂ M such that

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).

Higher-order Sobolev embeddings and isoperimetric inequalities

Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities

Affine vs. Euclidean isoperimetric inequalities

We consider a class of monomial weights 𝑥A = |𝑥1|𝑎1…|𝑥N|𝑎N in ℝN, where ai is a nonnegative real number for each i ∈ {1,…,N}, and we establish the ε — ε property and the boundedness of isoperimetric sets with different monomial weights for the perimeter and volume. Moreover, we present cases of nonexistence of the isoperimetric inequality when it is not possible to associate the corresponding Sobolev inequality. Finally, for N = 2, we developed an original type of symmetrization, which we call star-shaped Steiner symmetrization, and we apply it to a class of isoperimetric problems with different monomial weights.

The sharp Sobolev and isoperimetric inequalities split twice

On manifolds with non-negative Ricei curvature and Sobolev inequalities

The classical isoperimetric inequality in Euclidean space. Three different approaches.- The curve shortening flow and isoperimetric inequalities on surfaces.- $H^k$-flows and isoperimetric inequalities.- Estimates on the Willmore functional and isoperimetric inequalities.- Singularities in the volume-preserving mean curvature flow.- Bounds on the Heegaard genus of a hyperbolic manifold.- The isoperimetric profile for small volumes.- Local existence of flows driven by the second fundamental form and formation of singularities.- Invariance properties.- Singular behaviour of convex surfaces.- Convexity estimates.- Rescaling near a singularity.- Cylindrical and gradient estimates.- Mean curvature flow with surgeries.

We discuss extensions of some results of V.G.Maz'ya to Riemannian manifolds. His proofs of the relationships between capacities, isoperimetric inequalities and Sobolev inequalities did not use specific properties of the Euclidean space. His method, transplanted to manifolds, gives a unified approach to such results as parabolicity criteria, eigenvalues estimates, heat kernel estimates, etc.

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.

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 .

After Hormander's fundamental paper on hypoellipticity [54], the study of partial differential equations arising from families of noncommuting vector fields has developed significantly. In this paper we study some basic functional and geometric properties of general families of vector fields that include the Hormander type as a special case. Similar to their classical counterparts, such properties play an important role in the analysis of the relevant differential operators (both linear and nonlinear). To motivate our results, we recall some classical inequalities. Let E C R be a Caccioppoli set (a measurable set having a locally finite perimeter); then one has the isoperimetric inequality