Vol 242, No 6 (2019)

Let (X, d, μ) and (Y, d′, μ′) be metric spaces α-regular by Ahlfors with α > 0 and locally finite Borel measures μ and μ′, respectively. We consider the class ACS_E of absolutely continuous functions on a.a. compact subsets E ⊂ X and establish the membership of mappings f: X → Y to a given class.

For parabolic equations with nonstandard growth conditions, we prove local boundedness of weak solutions in terms of nonlinear parabolic potentials of the right-hand side of the equation.

We find the necessary and sufficient conditions under which an unbounded metric space X has, at infinity, a unique pretangent space \( {\Omega}_{\infty, \tilde{r}}^X \) for every scaling sequence \( \tilde{r} \). In particular, it is proved that \( {\Omega}_{\infty, \tilde{r}}^X \) is unique and isometric to the closure of X for every logarithmic spiral X and every \( \tilde{r} \). It is also shown that the uniqueness of pretangent spaces to subsets of a real line is closely related to the “asymptotic asymmetry” of these subsets.

In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z), whereas its reaction term f(u) is a continuous non-linear function. Assuming that f(t)/t → 0 as t → ∞, we establish a theorem on existence of weak \( C\left(\overline{D}\right)\cap {W}_{\mathrm{loc}}^{1,2}(D) \) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components. As consequences, we give applications to some concrete model semilinear equations of mathematical physics, arising from modeling processes in anisotropic and inhomogeneous media. With a view to the further development of the theory of boundary-value problems for the semilinear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity.