On the existence of solutions of the Beltrami equations with conditions on inverse dilatations

We have found one of possible conditions under which the degenerate Beltrami equation has a continuous solution of the Sobolev class. This solution is H\"{o}lder continuous in the ''weak'' (logarithmic) sense with the exponent power $\alpha=1/2.$ Moreover, it belongs to the class $W^{1, 2}_{\rm loc}.$ Under certain additional requirements, it can also be chosen as a homeomorphic solution. We give an appropriate example of the equation that satisfies all the conditions of the main result of the article, but does not have a homeomorphic Sobolev solution.

Regularity results for a class of obstacle problems with p, q−growth conditions

In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type min{ ∫ΩF(x, Dz) : z ∈ 𝛫ψ(Ω)}. Here 𝛫ψ(Ω) is the set of admissible functions z ∈ u0 + W1,p(Ω) for a given u0 ∈ W1,p(Ω) such that z ≥ ψ a.e. in Ω, ψ being the obstacle and Ω being an open bounded set of ℝn, n ≥ 2. The main novelty here is that we are assuming that the integrand F(x, Dz) satisfies (p, q)-growth conditions and as a function of the x-variable belongs to a suitable Sobolev class. We remark that the Lipschitz continuity result is obtained under a sharp closeness condition between the growth and the ellipticity exponents. Moreover, we impose less restrictive assumptions on the obstacle with respect to the previous regularity results. Furthermore, assuming the obstacle ψ is locally bounded, we prove the local boundedness of the solutions to a quite large class of variational inequalities whose principal part satisfies non standard growth conditions.

Sobolev mappings and moduli inequalities on Carnot groups

We study the mappings that satisfy moduli inequalities on Carnot groups. We prove that the homeomorphisms satisfying the moduli inequalities ($Q$-homeomor\-phisms) with a locally integrable function $Q$ are Sobolev mappings. On this base in the frameworks of the weak inverse mapping theorem, we prove that, on the Carnot groups $\mathbb G,$ the mappings inverse to Sobolev homeomorphisms of finite distortion of the class $W^1_{\nu,\loc}(\Omega;\Omega')$ belong to the Sobolev class $W^1_{1,\loc}(\Omega';\Omega)$.

Inclusion of Hajłasz – Sobolev class Mpα(X) into the space of continuous functions in the critical case

Let (X, d, µ) be a doubling metric measure space with doubling dimension γ, i. e. for any balls B(x, R) and B(x, r), r < R, following inequality holds µ(B(x, R)) ≤ aµ (R/r)γµ(B(x, r)) for some positive constants γ and aµ. Hajłasz – Sobolev space Mpα(X) can be defined upon such general structure. In the Euclidean case Hajłasz – Sobolev space coincides with classical Sobolev space when p > 1, α = 1. In this article we discuss inclusion of functions from Hajłasz – Sobolev space Mpα(X) into the space of continuous functions for p ≤ 1 in the critical case γ = α p. More precisely, it is shown that any function from Hajłasz – Sobolev class Mpα(X), 0 < p ≤ 1, α > 0, has a continuous representative in case of uniformly perfect space (X, d, µ).

Harmonic maps between two concentric annuli in 𝐑3

Given two annuli {\mathbb{A}(r,R)} and {\mathbb{A}(r_{\ast},R_{\ast})} , in {\mathbf{R}^{3}} equipped with the Euclidean metric and the weighted metric {\lvert y\rvert^{-2}} , respectively, we minimize the Dirichlet integral, i.e., the functional \mathscr{F}[f]=\int_{\mathbb{A}(r,R)}\frac{\lVert Df\rVert^{2}}{\lvert f\rvert% ^{2}}, where f is a homeomorphism between {\mathbb{A}(r,R)} and {\mathbb{A}(r_{\ast},R_{\ast})} , which belongs to the Sobolev class {\mathscr{W}^{1,2}} . The minimizer is a certain generalized radial mapping, i.e., a mapping of the form {f(\lvert x\rvert\eta)=\rho(\lvert x\rvert)T(\eta)} , where T is a conformal mapping of the unit sphere onto itself and {\rho(t)={R_{\ast}}\bigl{(}\frac{r_{\ast}}{R_{\ast}}\bigr{)}^{{\frac{R(r-t)}{(% R-r)t}}}} . It should be noticed that, in this case, no Nitsche phenomenon occurs.

On Hölder continuity of solutions of the Beltrami equations on the boundary

In the present paper, it is found conditions on the complex coefficient of the Beltrami equations with the degeneration of the uniform ellipticity in the unit disk under which their generalized homeomorphic solutions are continuous by Hölder on the boundary. These results can be applied to the investigations of various boundary value problems for the Beltrami equations. In a series of recent papers, under the study of the boundary value problems of Dirichlet, Hilbert, Neumann, Poincare and Riemann with arbitrary measurable boundary data for the Beltrami equations as well as for the generalizations of the Laplace equation in anisotropic and inhomogeneous media, it was applied the logarithmic capacity, see e.g. Gutlyanskii V., Ryazanov V., Yefimushkin A. On the boundary value problems for quasiconformal functions in the plane // Ukr. Mat. Visn. - 2015. - 12, no. 3. - P. 363-389; transl. in J. Math. Sci. (N.Y.) - 2016. - 214, no. 2. - P. 200-219; Gutlyanskii V., Ryazanov V., Yefimushkin A. On a new approach to the study of plane boundary-value problems // Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki. - 2017. - No. 4. - P. 12-18; Yefimushkin A. On Neumann and Poincare Problems in A-harmonic Analysis // Advances in Analysis. - 2016. - 1, no. 2. - P. 114-120; Efimushkin A., Ryazanov V. On the Riemann-Hilbert problem for the Beltrami equations in quasidisks // Ukr. Mat. Visn. - 2015. - 12, no. 2. - P. 190–209; transl. in J. Math. Sci. (N.Y.) - 2015. - 211, no. 5. - P. 646–659; Yefimushkin A., Ryazanov V. On the Riemann–Hilbert Problem for the Beltrami Equations // Contemp. Math. - 2016. - 667. - P. 299-316; Gutlyanskii V., Ryazanov V., Yakubov E., Yefimushkin A. On Hilbert problem for Beltrami equation in quasihyperbolic domains // ArXiv.org: 1807.09578v3 [math.CV] 1 Nov 2018, 28 pp. As well known, the logarithmic capacity of a set coincides with the so-called transfinite diameter of the set. This geometric characteristic implies that sets of logarithmic capacity zero and, as a consequence, measurable functions with respect to logarithmic capacity are invariant under mappings that are continuous by Hölder. That circumstance is a motivation of our research. Let \(D\) be a domain in the complex plane \(\mathbb C\) and let \(\mu: D\to\mathbb C\) be a measurable function with \( |\mu(z)| \lt 1 \) a.e. The equation of the form \(f_{\bar{z}}\ =\ \mu(z) f_z \) where \( f_{\bar z}={\bar\partial}f=(f_x+if_y)/2 \), \(f_{z}=\partial f=(f_x-if_y)/2\), \(z=x+iy\), \( f_x \) and \( f_y \) are partial derivatives of the function \(f\) in \(x\) and \(y\), respectively, is said to be a Beltrami equation. The function \(\mu\) is called its complex coefficient, and \( K_{\mu}(z)=\frac{1+|\mu(z)|}{1-|\mu(z)|}\) is called its dilatation quotient. The Beltrami equation is said to be degenerate if \({\rm ess}\,{\rm sup}\,K_{\mu}(z)=\infty\). The existence of homeomorphic solutions in the Sobolev class \(W^{1,1}_{\rm loc}\) has been recently established for many degenerate Beltrami equations under the corresponding conditions on the dilatation quotient \(K_{\mu}\), see e.g. the monograph Gutlyanskii V., Ryazanov V., Srebro U., Yakubov E. The Beltrami equation. A geometric approach. Developments in Mathematics, 26. Springer, New York, 2012 and the further references therein. The main theorem of the paper, Theorem 1, states that a homeomorphic solution \( f:\mathbb D\to\mathbb D \) in the Sobolev class \( W^{1,1}_{\rm loc} \) of the Beltrami equation in the unit disk \(\mathbb D\) has a homeomorphic extension to the boundary that is Hölder continuous if \(K_{\mu}\in L^1(\Bbb D)\) and, for some \(\varepsilon_0\in(0,1)\) and \(C\in[1,\infty)\), $$ \sup\limits_{\varepsilon\in(0,\varepsilon_0)} \int_{\mathbb D\cap D(\zeta,\varepsilon)}K_{\mu}(z) dm(z) \lt C \qquad \forall \zeta \in \partial \mathbb{D} $$ where \(D(\zeta,\varepsilon)=\left\{z\in{\Bbb C}: |z-\zeta| \lt \varepsilon\right\}\).

Anisotropic Sobolev Capacity with Fractional Order

In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure μ that naturally induces an embedding of the anisotropic fractional Sobolev class into the μ-based-Lebesgue-space with 0 < β ≤ n. Also, we investigate the anisotropic fractional α-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as α →0+, will be provided.