In this paper, we study the existence of homoclinic solutions for a class of fourth order differential equations. By using variational methods, the existence and the non-existence of nontrivial homoclinic solutions are obtained, depending on a parameter.

We study differentiability properties of a potential of the type K⋆μ, where μ is a finite Radon measure in RN and the kernel K satisfies |∇jK(x)|≤C|x|−(N−1+j),j=0,1,2. We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vallée Poussin sense associated with the kernel |x|−(N−1). We require that the first order remainder at a point is small when measured by means of a normalized weak capacity “norm” in balls of small radii centered at the point. This implies weak LN/(N−1) differentiability and thus Lp differentiability in the Calderón–Zygmund sense for 1≤p<N/(N−1). We show that K⋆μ is a.e. differentiable in the capacity sense, thus strengthening a recent result by Ambrosio, Ponce and Rodiac. We also present an alternative proof of a quantitative theorem of the authors just mentioned, giving pointwise Lipschitz estimates for K⋆μ. As an application, we study level sets of newtonian potentials of finite Radon measures.

We obtain improved fractional Poincaré inequalities in John domains of a metric space (X,d) endowed with a doubling measure μ under some mild regularity conditions on the measure μ. We also give sufficient conditions on a bounded domain to support fractional Poincaré type inequalities in this setting.

We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence {(dn/dzn)(R(z)expT(z))}∞n=1. Here, R(z) is a rational function with at least two poles, all of which are distinct, and T(z) is a polynomial. This is an extension of a recent measure-theoretic refinement of Pólya’s Shire theorem for rational functions.

We prove a structure theorem for any n-rectifiable set E⊂Rn+1,n≥1, satisfying a weak version of the lower ADR condition, and having locally finite Hn (n-dimensional Hausdorff) measure. Namely, that Hn-almost all of E can be covered by a countable union of boundaries of bounded Lipschitz domains contained in Rn+1∖E. As a consequence, for harmonic measure in the complement of such a set E, we establish a non-degeneracy condition which amounts to saying that Hn|E is “absolutely continuous” with respect to harmonic measure in the sense that any Borel subset of E with strictly positive Hn measure has strictly positive harmonic measure in some connected component of Rn+1∖E. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set E as above is the boundary of a connected domain Ω⊂Rn+1 which satisfies an infinitesimal interior thickness condition, then Hn|∂Ω is absolutely continuous (in the usual sense) with respect to harmonic measure for Ω. Local versions of these results are also proved: if just some piece of the boundary is n-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results in “Rectifiability of harmonic measure” [Geom. Funct. Anal. 26 (2016), 703–728], we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is n-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely n-unrectifiable piece having vanishing harmonic measure.