Fractional Sobolev-Poincaré inequalities in irregular domains

This paper is devoted to the study of fractional (q, p)-Sobolev-Poincaré in- equalities in irregular domains. In particular, the author establishes (essentially) sharp fractional (q, p)-Sobolev-Poincaré inequalities in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional (q, p)-Sobolev-Poincaré inequalities together with a separation property are s-diam John domains for certain s, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P., Sobolev-Poincaré implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out

A decomposition theorem for BV functions

The Jordan decomposition states that a function f: R \u2192 R is of bounded variation if and only if it can be written as the dierence of two monotone increasing functions. In this paper we generalize this property to real valued BV functions of many variables, extending naturally the concept of monotone function. Our result is an extension of a result obtained by Alberti, Bianchini and Crippa. A counterexample is given which prevents further extensions

Basic properties of nonsmooth Hormander's vector fields and Poincare's inequality

We consider a family of vector fields defined in some bounded domain of R^p, and we assume that they satisfy Hormander's rank condition of some step r, and that their coefficients have r-1 continuous derivatives. We extend to this nonsmooth context some results which are well-known for smooth Hormander's vector fields, namely: some basic properties of the distance induced by the vector fields, the doubling condition, Chow's connectivity theorem, and, under the stronger assumption that the coefficients belong to C^{r-1,1}, Poincare's inequality. By known results, these facts also imply a Sobolev embedding. All these tools allow to draw some consequences about second order differential operators modeled on these nonsmooth Hormander's vector fields.Comment: 60 pages, LaTeX; Section 6 added and Section 7 (6 in the previous version) changed. Some references adde

Hardy's inequality for functions vanishing on a part of the boundary

We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.Comment: 26 pages, 2 figures, includes several improvements in Sections 6-8 allowing to relax the assumptions in the main results. Final version published at http://link.springer.com/article/10.1007%2Fs11118-015-9463-

Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity

We define a class of deformations in W^1,p(\u3a9,R^n), p>n 121, with positive Jacobian that do not exhibit cavitation. We characterize that class in terms of the non-negativity of the topological degree and the equality between the distributional determinant and the pointwise determinant of the gradient. Maps in this class are shown to satisfy a property of weak monotonicity, and, as a consequence, they enjoy an extra degree of regularity. We also prove that these deformations are locally invertible; moreover, the neighbourhood of invertibility is stable along a weak convergent sequence in W^1,p, and the sequence of local inverses converges to the local inverse. We use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps. We apply those results to prove existence of minimizers in some models for nematic elastomers and magnetoelasticity

A bridge between Dubovitskiĭ-Federer theorems and the coarea formula

The Morse-Sard theorem requires that a mapping v : ℝn → ℝm is of class Ck, k &gt; max(n - m, 0). In 1957 Dubovitskiĭ generalized this result by proving that almost all level sets for a Ck mapping has Hs-negligible intersection with its critical set, where s = max(n - m - k + 1, 0). Here the critical set, or m-critical set is defined as Zv,m = {x ∈ ℝn : rank ∇v(x) &lt; m}. Another generalization was obtained independently by Dubovitskiĭ and Federer in 1966, namely for Ck mappings v : ℝn → ℝd and integers m ≤ d they proved that the set of m-critical values v(Zv,m) is q◦-negligible for q◦ = m ≤ 1 + n m+1/k. They also established the sharpness of these results within the Ck category. Here we prove that Dubovitskiĭ's theorem can be generalized to the case of continuous mappings of the Sobolev-Lorentz class Wkp,1(ℝn, ℝd), p = n/k (this is the minimal integrability assumption that guarantees the continuity of mappings). In this situation the mappings need not to be everywhere differentiable and in order to handle the set of nondifferentiability points, we establish for such mappings an analog of the Luzin N-property with respect to lower dimensional Hausdorff content. Finally, we formulate and prove a bridge theorem that includes all the above results as particular cases. As a limiting case in this bridge theorem we also establish a new coarea type formula: if E ⊂ {x ∈ ℝn : rank ∇v(x) ≤ mg, then ∫Jmv(x) dx = ∫^n-m(E∩v^-1(y))d^m(y). E________ℝd The mapping v is ℝd-valued, with arbitrary d, and the formula is obtained without any restrictions on the image v(ℝn)(such as m-rectifiability or σ-finiteness with respect to the m-Haussdorf measure). These last results are new also for smooth mappings, but are presented here in the general Sobolev context. The proofs of the results are based on our previous joint papers with J. Bourgain (2013, 2015).</p

A bridge between Dubovitskiĭ-Federer theorems and the coarea formula

The Morse-Sard theorem requires that a mapping v : ℝn → ℝm is of class Ck, k > max(n - m, 0). In 1957 Dubovitskiĭ generalized this result by proving that almost all level sets for a Ck mapping has Hs-negligible intersection with its critical set, where s = max(n - m - k + 1, 0). Here the critical set, or m-critical set is defined as Zv,m = {x ∈ ℝn : rank ∇v(x) Here we prove that Dubovitskiĭ's theorem can be generalized to the case of continuous mappings of the Sobolev-Lorentz class Wkp,1(ℝn, ℝd), p = n/k (this is the minimal integrability assumption that guarantees the continuity of mappings). In this situation the mappings need not to be everywhere differentiable and in order to handle the set of nondifferentiability points, we establish for such mappings an analog of the Luzin N-property with respect to lower dimensional Hausdorff content. Finally, we formulate and prove a bridge theorem that includes all the above results as particular cases. As a limiting case in this bridge theorem we also establish a new coarea type formula: if E ⊂ {x ∈ ℝn : rank ∇v(x) ≤ mg, then ∫Jmv(x) dx = ∫^n-m(E∩v^-1(y))d^m(y). E________ℝd The mapping v is ℝd-valued, with arbitrary d, and the formula is obtained without any restrictions on the image v(ℝn)(such as m-rectifiability or σ-finiteness with respect to the m-Haussdorf measure). These last results are new also for smooth mappings, but are presented here in the general Sobolev context. The proofs of the results are based on our previous joint papers with J. Bourgain (2013, 2015).</p