Volume 14, Issue 1

In the present paper we consider the Carnot–Carathéodory distance δE{\delta_{E}} to a closed set E in the sub-Riemannian Heisenberg groups ℍn{{\mathbb{H}}^{n}}, n⩾1{n\geqslant 1}. The ℍ{{\mathbb{H}}}-regularity of δE{\delta_{E}} is proved under mild conditions involving a general notion of singular points. In case E is a Euclidean Ck{C^{k}} submanifold, k⩾2{k\geqslant 2}, we prove that δE{\delta_{E}} is Ck{C^{k}} out of the singular set. Explicit expressions for the volume of the tubular neighborhood when the boundary of E is of class C2{C^{2}} are obtained, out of the singular set, in terms of the horizontal principal curvatures of ∂⁡E{\partial E} and of the function 〈N,T〉/|Nh|{\langle N,T\rangle/|N_{h}|} and its tangent derivatives.

We consider second-order divergence form elliptic operators with W1,1{W^{1,1}} coefficients, in a uniform domain Ω with Ahlfors regular boundary. We show that the A∞{A_{\infty}} property of the elliptic measure associated to any such operator implies that Ω is a set of locally finite perimeter whose boundary, ∂⁡Ω{\partial\Omega}, is rectifiable. As a corollary we show that for this type of operators, absolute continuity of the surface measure with respect to the elliptic measure is enough to guarantee rectifiability of the boundary. In the case that the coefficients are continuous we obtain additional information about Ω.

We investigate the asymptotic behavior of solutions of Hamilton–Jacobi equations with large Hamiltonian drift terms in an open subset of the two-dimensional Euclidean space. The drift is given by ε-1⁢(Hx2,-Hx1){{\varepsilon}^{-1}(H_{x_{2}},-H_{x_{1}})} of a Hamiltonian H , with ε>0{{\varepsilon}>0}. We establish the convergence, as ε→0+{{\varepsilon}\to 0+}, of solutions of the Hamilton–Jacobi equations and identify the limit of the solutions as the solution of systems of ordinary differential equations on a graph. This result generalizes the previous one obtained by the author to the case where the Hamiltonian H admits a degenerate critical point and, as a consequence, the graph may have more than four segments at a node.

In this work we prove convergence results of sequences of Riemannian 4-manifolds with almost vanishing L2{L^{2}}-norm of a curvature tensor and a non-collapsing bound on the volume of small balls. In Theorem 1.1 we consider a sequence of closed Riemannian 4-manifolds, whose L2{L^{2}}-norm of the Riemannian curvature tensor tends to zero. Under the assumption of a uniform non-collapsing bound and a uniform diameter bound, we prove that there exists a subsequence that converges with respect to the Gromov–Hausdorff topology to a flat manifold. In Theorem 1.2 we consider a sequence of closed Riemannian 4-manifolds, whose L2{L^{2}}-norm of the Riemannian curvature tensor is uniformly bounded from above, and whose L2{L^{2}}-norm of the traceless Ricci-tensor tends to zero. Here, under the assumption of a uniform non-collapsing bound, which is very close to the Euclidean situation, and a uniform diameter bound, we show that there exists a subsequence which converges in the Gromov–Hausdorff sense to an Einstein manifold. In order to prove Theorem 1.1 and Theorem 1.2, we use a smoothing technique, which is called L2{L^{2}}-curvature flow. This method was introduced by Jeffrey Streets. In particular, we use his “tubular averaging technique” in order to prove distance estimates of the L2{L^{2}}-curvature flow, which only depend on significant geometric bounds. This is the content of Theorem 1.3.

The notion of weakly monotone functions extends the classical definition of monotone function, that can be traced back to Lebesgue. It was introduced, in the framework of Sobolev spaces, by Manfredi, in connection with the analysis of the regularity of maps of finite distortion appearing in the theory of nonlinear elasticity. Diverse authors, including Iwaniecz, Kauhanen, Koskela, Maly, Onninen, Zhong, thoroughly investigated continuity properties of monotone functions in the more general setting of Orlicz–Sobolev spaces, in view of the analysis of continuity, openness and discreteness properties of maps under minimal integrability assumptions on their distortion. The present paper complements and augments the available Orlicz–Sobolev theory of weakly monotone functions. In particular, a variant is proposed in a customary condition ensuring the continuity of functions from this class, which avoids a technical additional assumption, and applies in certain situations when the latter is not fulfilled. The continuity outside sets of zero Orlicz capacity, and outside sets of (generalized) zero Hausdorff measure are also established when everywhere continuity fails.

We consider different fractional Neumann Laplacians of order s∈(0,1){s\in(0,1)} on domains Ω⊂ℝn{\Omega\subset\mathbb{R}^{n}}, namely, the restricted Neumann Laplacian (-ΔΩN)Rs{{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{R}}}}, the semirestricted Neumann Laplacian (-ΔΩN)Srs{{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{Sr}}}} and the spectral Neumann Laplacian (-ΔΩN)Sps{{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{Sp}}}}. In particular, we are interested in the attainability of Sobolev constants for these operators when Ω is a half-space.

For any fixed integer D>1{D>1} we show that there exists M∈[e-1,1]{M\in[e^{-1},1]} such that for any open, bounded, convex domain Ω⊂ℝD{\Omega\subset{\mathbb{R}}^{D}} with smooth boundary for which the maximum of the distance function to the boundary of Ω is less than or equal to M , the principal frequency of the p -Laplacian on Ω is an increasing function of p on (1,∞){(1,\infty)}. Moreover, for any real number s>M{s>M} there exists an open, bounded, convex domain Ω⊂ℝD{\Omega\subset{\mathbb{R}}^{D}} with smooth boundary which has the maximum of the distance function to the boundary of Ω equal to s such that the principal frequency of the p -Laplacian is not a monotone function of p∈(1,∞){p\in(1,\infty)}.