On integral conditions in the mapping theory

It is established interconnections between various integral conditions that play an important role in the theory of space mappings and in the theory of degenerate Beltrami equations in the plane.Comment: 15 pages, changes related to Corollary 3.2, see (3.28

Dirichlet problem for Poisson equations in Jordan domains

We study the Dirichlet problem for the Poisson equations △u(z) = g(z) with g ∈ Lp, p > 1, and continuous boundary data φ : ∂D → ℝ in arbitrary Jordan domains D in ℂ and prove the existence of continuous solutions u of the problem.Мы изучаем задачу Дирихле для уравнений Пуассона △u(z) = g(z) с g ∈ Lp, p > 1, и непрерывными граничными данными φ : ∂D → ℝ в произвольных жордановых областях D ⊂ ℂ и доказываем существование непрерывных решений u этой задачи.Ми вивчаємо задачу Дiрихле для рiвнянь Пуасона △u(z) = g(z) с g ∈ Lp, p > 1, та неперервними граничними даними φ : ∂D → ℝ в довiльних жорданових областях D ⊂ ℂ та доводимо iснування неперервних рiшень u цiєї задач

To the theory of semi-linear Beltrami equations

The present paper is devoted to the study of semi-linear Beltrami equations which are closely relevant to the corresponding semi-linear Poisson type equations of mathematical physics on the plane in anisotropic and inhomogeneous media. In its first part, applying completely continuous ope\-ra\-tors by Ahlfors-Bers and Leray--Schauder approach, we prove existence of regular solutions of the semi-linear Beltrami equations with no boundary conditions. Moreover, here we derive their representation through solutions of the Vekua type equations and generalized analytic functions with sources. As consequences, it is given a series of applications of these results to semi-linear Poisson type equations and to the corresponding equations of mathematical physics describing such phenomena as diffusion with physical and chemical absorption, plasma states and stationary burning in anisotropic and inhomogeneous media. The second part of the paper contains existence, representation and regularity results for nonclassical solutions to the Hilbert (Dirichlet) boundary value problem for semi-linear Beltrami equations and to the Poincare (Neumann) boundary value problem for semi-linear Poisson type equations with arbitrary boundary data that are measurable with respect to logarithmic capacity.Comment: 28 pages. arXiv admin note: text overlap with arXiv:2107.1066

On Dirichlet problem for degenerate Beltrami equations with sources

The present paper is devoted to the study of the Dirichlet problem ${\rm{Re}}\,\omega(z)\to\varphi(\zeta)$ as $z\to\zeta,$ $z\in D,\zeta\in \partial D,$ with continuous boundary data $\varphi :\partial D\to\mathbb R$ for Beltrami equations $\omega_{\bar{z}}=\mu(z) \omega_z+\sigma (z)$, $|\mu(z)|<1$ a.e., with sources $\sigma :D\to\mathbb C$ in the case of locally uniform ellipticity. In this case, we establish a series of effective integral criteria of the type of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz on singularities of the equations at the boundary for existence, representation and regularity of solutions in arbitrary bounded domains $D$ of the complex plane $\mathbb C$ with no boun\-da\-ry component degenerated to a single point for sources $\sigma$ in $L_p(D)$, $p>2$, with compact support in $D$. Moreover, we prove in such domains existence, representation and regularity of weak solutions of the Dirichlet problem for the Poisson type equation ${\rm div} [A(z)\nabla\,u(z)] = g(z)$ whose source $g\in L_p(D)$, $p>1$, has compact support in $D$ and whose mat\-rix valued coefficient $A(z)$ guarantees its locally uniform ellipticity.Comment: 31 pages. arXiv admin note: substantial text overlap with arXiv:2111.1037

Stray-fields-based magnetoresistance mechanism in Ni80Fe20-Nb-Ni80Fe20 trilayers

We report on the transport and magnetic properties of hybrid trilayers and bilayers that consist of low spin-polarized Ni80Fe20 exhibiting in-plane but no uniaxial anisotropy and low-Tc Nb. We reveal a magnetoresistance effect that is pronounced. In our trilayers the magnetoresistance exhibits an increase of two orders of magnitude when the superconducting state is reached: from the conventional normal-state values 0.6 % it goes up to 1000 % for temperatures below Tc. In contrast, in the bilayers the effect is only minor since from 3% in the normal state increases only to 70 % for temperatures below Tc. Magnetization data of both the longitudinal and transverse magnetic components are presented. Most importantly, we present data not only for the normal state of Nb but also in its superconducting state. Strikingly, these data show that below its Tc SC the Nb interlayer under the influence of the outer Ni80Fe20 layers attains a magnetization component transverse to the external field. By comparing the transport and magnetization data we propose a candidate mechanism that could motivate the pronounced magnetoresistance effect observed in the trilayers. Adequate magnetostatic coupling of the outer Ni80Fe20 layers is motivated by stray fields that emerge naturally in their whole surface due to the multidomain magnetic structure that they attain near coercivity. Atomic force microscopy is employed in order to examine the possibility that such magnetostatic coupling could be promoted by interface roughness. Referring to the bilayers, although out-of-plane rotation of the magnetization of the single Ni80Fe20 layer is still observed, in these structures magnetostatic coupling does not occur due to the absence of a second Ni80Fe20 one so that the observed magnetoresistance peaks are only modest.Comment: 9 pages, 7 figure

Transient receptor potential melastatin 7 cation channel kinase new player in angiotensin II–induced hypertension

Transient receptor potential melastatin 7 (TRPM7) is a bifunctional protein comprising a magnesium (Mg2+)/cation channel and a kinase domain. We previously demonstrated that vasoactive agents regulate vascular TRPM7. Whether TRPM7 plays a role in the pathophysiology of hypertension and associated cardiovascular dysfunction is unknown. We studied TRPM7 kinase–deficient mice (TRPM7Δkinase; heterozygous for TRPM7 kinase) and wild-type (WT) mice infused with angiotensin II (Ang II; 400 ng/kg per minute, 4 weeks). TRPM7 kinase expression was lower in heart and aorta from TRPM7Δkinase versus WT mice, effects that were further reduced by Ang II infusion. Plasma Mg2+ was lower in TRPM7Δkinase versus WT mice in basal and stimulated conditions. Ang II increased blood pressure in both strains with exaggerated responses in TRPM7Δkinase versus WT groups (P&lt;0.05). Acetylcholine-induced vasorelaxation was reduced in Ang II–infused TRPM7Δkinase mice, an effect associated with Akt and endothelial nitric oxide synthase downregulation. Vascular cell adhesion molecule–1 expression was increased in Ang II–infused TRPM7 kinase–deficient mice. TRPM7 kinase targets, calpain, and annexin-1, were activated by Ang II in WT but not in TRPM7Δkinase mice. Echocardiographic and histopathologic analysis demonstrated cardiac hypertrophy and left ventricular dysfunction in Ang II–treated groups. In TRPM7 kinase–deficient mice, Ang II–induced cardiac functional and structural effects were amplified compared with WT counterparts. Our data demonstrate that in TRPM7Δkinase mice, Ang II–induced hypertension is exaggerated, cardiac remodeling and left ventricular dysfunction are amplified, and endothelial function is impaired. These processes are associated with hypomagnesemia, blunted TRPM7 kinase expression/signaling, endothelial nitric oxide synthase downregulation, and proinflammatory vascular responses. Our findings identify TRPM7 kinase as a novel player in Ang II–induced hypertension and associated vascular and target organ damage

Double proximity effect in hybrid planar Superconductor-(Normal metal/Ferromagnet)-Superconductor structures

We have investigated the differential resistance of hybrid planar Al-(Cu/Fe)-Al submicron bridges at low temperatures and in weak magnetic fields. The structure consists of Cu/Fe-bilayer forming a bridge between two superconducting Al-electrodes. In superconducting state of Al-electrodes, we have observed a double-peak peculiarity in differential resistance of the S-(N/F)-S structures at a bias voltage corresponding to the minigap. We claim that this effect (the doubling of the minigap) is due to an electron spin polarization in the normal metal which is induced by the ferromagnet. We have demonstrated that the double-peak peculiarity is converted to a single peak at a coercive applied field corresponding to zero magnetization of the Fe-layer