Boundary value problems for singular p- and p(x)- Laplacian equations in a domain with conical point on the boundary

This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p- and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

On the W 2p Estimate for Oblique Derivative Problem in Lipschitz Domains

The aim of this paper is to establish $W^2_p$ estimate for non-divergence form 2nd-order elliptic equations with the oblique derivative boundary condition in domains with small Lipschitz constants. Our result generalizes those in [ 16] and [ 17], which work for $C^{1,\alpha }$ domains with $\alpha> 1-1/p$. As an application, we also obtain a solvability result. An extension to fully nonlinear elliptic equations with the oblique derivative boundary condition is also discussed.

Oblique Derivative Problem Solution for the Lavrentyev-Bitsadze Equation in a Half-Plane

The paper solves the boundary value problem of an oblique derivative for the Lavrent'ev – Bitsadze equation in a half-plane. The Lavrent'ev – Bitsadze equation is an equation of mixed (elliptic-hyperbolic) type. Mixed-type equations arise when solving many applied problems (for example, when simulating transonic flows of a compressible medium).In the paper, the domain of ellipticity is a half-plane, and that of hyperbolicity is its adjacent strip. On one of the straight lines bounding the strip, an oblique derivative is specified (in the direction that forms an acute angle with this straight line), and on the other straight line, which is the interface between the strip and the half-plane, the solutions are matched by boundary conditions of the fourth kind. In the hyperbolicity strip, the solution is represented by the d'Alembert formula, and in the half-plane, where the equation is elliptic, the bounded solution is represented by the Poisson integral with unknown density. For this unknown density of the Poisson integral, a singular integral equation is obtained, which is reduced to the Riemann boundary value problem with a shift for holomorphic functions. The solution of the Riemann problem is reduced to the solution of two functional equations. Solutions of these functional equations and the Sokhotsky formula for an integral of Cauchy type allowed us to find the unknown density of the Poisson integral. This allowed us to find a solution to the oblique derivative problem as the sum of a functional series (up to an arbitrary constant term).