Optimal location of resources and Steiner symmetry in a population dynamics model in heterogeneous environments

The subject of this paper is inspired by Cantrell and Cosner (1989) and Cosner, Cuccu and Porru (2013). Cantrell and Cosner (1989) investigate the dynamics of a population in heterogeneous environments by means of diffusive logistic equations. An important part of their study consists in finding sufficient conditions which guarantee the survival of the species. Mathematically, this task leads to the weighted eigenvalue problem \(-\Delta u =\lambda m u \) in a bounded smooth domain \(\Omega\subset \mathbb{R}^N\), \(N\geq 1\), under homogeneous Dirichlet boundary conditions, where \(\lambda \in \mathbb{R}\) and \(m\in L^\infty(\Omega)\). The domain \(\Omega\) represents the environment and \(m(x)\), called the local growth rate, says where the favourable and unfavourable habitats are located. Then, Cantrell and Cosner (1989) consider a class of weights \(m(x)\) corresponding to environments where the total sizes of favourable and unfavourable habitats are fixed, but their spatial arrangement is allowed to change; they determine the best choice among them for the population to survive. In our work we consider a sort of refinement of the result above. We write the weight \(m(x)\) as sum of two (or more) terms, i.e. \(m(x)=f_1(x)+f_2(x)\), where \(f_1(x)\) and \(f_2(x)\) represent the spatial densities of the two resources which contribute to form the local growth rate \(m(x)\). Then, we fix the total size of each resource allowing its spatial location to vary. As our first main result, we show that there exists an optimal choice of \(f_1(x)\) and \(f_2(x)\) and find the form of the optimizers. Our proof relies on some results in Cosner, Cuccu and Porru (2013) and on a new property (to our knowledge) about the classes of rearrangements of functions. Moreover, we show that if \(\Omega\) is Steiner symmetric, then the best arrangement of the resources inherits the same kind of symmetry. (Actually, this is proved in the more general context of the classes of rearrangements of measurable functions.

Weak quasicircles have Lipschitz dimension 1

We prove that the Lipschitz dimension of any bounded turning Jordan circle or arc is equal to 1. Equivalently, the Lipschitz dimension of any weak quasicircle or arc is equal to 1.

Two examples related to conical energies

In a recent article (2021) we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and Mörters (2000), the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.

On logarithmic Hölder continuity of mappings on the boundary

We study mappings satisfying the so-called inverse Poletsky inequality. Under integrability of the corresponding majorant, it is proved that these mappings are logarithmic Hölder continuous in the neighborhood of the boundary points. In particular, the indicated properties hold for homeomorphisms whose inverse satisfy the weighted Poletsky inequality.

Completely monotone sequences and harmonic mappings

In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.

Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function

Let \(\Omega\) be a subdomain in \(\mathbb{R}^n\) and \(M_\Omega\) be the local Hardy-Littlewood maximal function. In this paper, we show that both the commutator and the maximal commutator of \(M_\Omega\) are bounded and continuous from the first order Sobolev spaces \(W^{1,p_1}(\Omega)\) to \(W^{1,p}(\Omega)\) provided that \(b\in W^{1,p_2}(\Omega)\), \(1<p_1,p_2,p<\infty\) and \(1/p=1/p_1+1/p_2\). These are done by establishing several new pointwise estimates for the weak derivatives of the above commutators. As applications, the bounds of these operators on the Sobolev space with zero boundary values are obtained.

Baernstein’s star-function, maximum modulus points and a problem of Erdős

The paper is devoted to the development of Baernstein's method of \(T^{*}\)-function. We consider the relationship between the number of separated maximum modulus points of a meromorphic function and the \(T^{*}\)-function. The results of Bergweiler, Bock, Edrei, Goldberg, Heins, Ostrovskii, Petrenko, Wiman are generalized. We also give examples showing that the obtained estimates are sharp.

Uniformization of metric surfaces using isothermal coordinates

We establish a uniformization result for metric surfaces – metric spaces that are topological surfaces with locally finite Hausdorff 2-measure. Using the geometric definition of quasiconformality, we show that a metric surface that can be covered by quasiconformal images of Euclidean domains is quasiconformally equivalent to a Riemannian surface. To prove this, we construct an atlas of suitable isothermal coordinates.

Note on an elementary inequality and its application to the regularity of p-harmonic functions

We study the Sobolev regularity of \(p\)-harmonic functions. We show that \(|Du|^{\frac{p-2+s}{2}}Du\) belongs to the Sobolev space \(W^{1,2}_{\operatorname{loc}}\), \(s>-1-\frac{p-1}{n-1}\), for any \(p\)-harmonic function \(u\). The proof is based on an elementary inequality.

Improved critical Hardy inequality and Leray-Trudinger type inequalities in Carnot groups

In this paper, we prove an improvement of the critical Hardy inequality in Carnot groups. We show that this improvement is sharp and can not be improved. We apply this improved critical Hardy inequality together with the Moser-Trudinger inequality due to Balogh, Manfredi and Tyson (2003) to establish the Leray-Trudinger type inequalities which extend the inequalities of Psaradakis and Spector (2015) and Mallick and Tintarev (2018) to the setting of Carnot groups.