Volume 232, Nº 4 (2018)

We establish the existence of C^1+α-regular viscosity solutions of fully nonlinear secondorder parabolic equations like ∂_tu+H(v,Dv,D²v, t, x) = 0 allowing the main part of H to have small discontinuities with respect to x and be just measurable in t.

We study an Ibragimov–Mamontov type equation with singular Bessel operators playing the role of second order derivatives. We obtain a representation of the solution by using a special Bessel transform.

It is shown that bounded sequences satisfying nonlinear differential constraints, strongly precompact under an exact condition of nondegeneration of these conditions. The proof is based on new localization principles for ultraparabolic H-measures with continuous indices.

We consider a first order strictly hyperbolic system of n equations with constant coefficients in a bounded domain. It is assumed that the domain is strictly convex relative to characteristics, so that the projection along each characteristic is an involution having two fixed singular points. The natural statement of boundary value problems for such systems requires that singular points go to singular points under such transformations. We present a necessary and sufficient condition for the existence of such domains, called characteristically closed.

We construct and justify a homogenized model of the variational inequality with the Laplace operator and a nonlinear boundary constraint on the flow on arbitrary shaped cavities generating perforation of the domain with critical values of parameters.

We consider the Dirichlet and Neumann problems for the Laplace operator in periodic waveguides. Integro-differential connections between the solution and its normal derivative, interpreted as a finite-dimensional version of the Steklov–Poincaré operator, are imposed on the artificial face of the truncated waveguide. These connections are obtained from the orthogonality and normalization conditions for the Floquet waves which are oscillating incoming/outgoing, as well as exponentially decaying/growing in the periodic waveguide. Under certain conditions, we establish the unique solvability of the problem and obtain error estimates for the solution itself, as well as for scattering coefficients in the solution. We give examples of trapped waves in periodic waveguides.