Mixed problems in a Lipschitz domain for strongly elliptic second-order systems

We consider mixed problems for strongly elliptic second-order systems in a bounded domain with Lipschitz boundary in the space ℝ ⁿ . For such problems, equivalent equations on the boundary in the simplest L ₂-spaces H ^s of Sobolev type are derived, which permits one to represent the solutions via surface potentials. We prove a result on the regularity of solutions in the slightly more general spaces H _p ^s of Bessel potentials and Besov spaces B _p ^s . Problems with spectral parameter in the system or in the condition on a part of the boundary are considered, and the spectral properties of the corresponding operators, including the eigenvalue asymptotics, are discussed.     

Positive solutions of second-order differential systems with nonhomogeneous boundary conditions

This paper is to investigate the positive solutions of the systems of second-order ordinary differential equations with nonhomogeneous multi-point boundary conditions. By the lower and upper solutions method, Schauder fixed point theorem and fixed point index theory, under certain conditions, it is proved that there exist appropriate regions of parameters in which the problem has at least two, at least one or no positive solution.     

Regularity results for quasilinear elliptic systems with nonlinear boundary conditions

It is proved that a solution of the boundary-value problem for a second-order quasilinear system with controlled order of nonlinearity is partially smooth all the way to the boundary of a domain. The boundary condition is imposed by means of a second-order nonlinear operator which can be regarded as a generalization of the “directional derivative” to the case of quasilinear systems. Bibliography: 6 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 23–50.     

Potential type operators and transmission problems for strongly elliptic second-order systems in Lipschitz domains

We consider a strongly elliptic second-order system in a bounded n-dimensional domain Ω⁺ with Lipschitz boundary Γ, n ≥ 2. The smoothness assumptions on the coefficients are minimized. For convenience, we assume that the domain is contained in the standard torus $ \mathbb{T}^n $ \mathbb{T}^n . In previous papers, we obtained results on the unique solvability of the Dirichlet and Neumann problems in the spaces H _p ^σ and B _p ^σ without use of surface potentials. In the present paper, using the approach proposed by Costabel and McLean, we define surface potentials and discuss their properties assuming that the Dirichlet and Neumann problems in Ω⁺ and the complementing domain Ω⁻ are uniquely solvable. In particular, we prove the invertibility of the integral single layer operator and the hypersingular operator in Besov spaces on Γ. We describe some of their spectral properties as well as those of the corresponding transmission problems.     

Artificial boundary conditions for elliptic systems on polyhedral truncation surfaces

For a sufficiently broad class of formally selfadjoint boundary value problems in the domains with conical outlets to infinity, including exterior boundary value problems, we suggest an algorithm for constructing some artificial boundary conditions on polyhedral truncation surfaces that guarantees higher precision of approximation as the infinite domain is replaced with a large but bounded domain. The errors are estimated. Anisotropic 3-dimensional elasticity problems are discussed as an example.     

Green's functions for elliptic and parabolic systems with Robin-type boundary conditions

The aim of this paper is to investigate Green's function for parabolic and elliptic systems satisfying a possibly nonlocal Robin-type boundary condition. We construct Green's function for parabolic systems with time-dependent coefficients satisfying a possibly nonlocal Robin-type boundary condition assuming that weak solutions of the system are locally Hölder continuous in the interior of the domain, and as a corollary we construct Green's function for elliptic system with a Robin-type condition. Also, we obtain Gaussian bound for Robin Green's function under an additional assumption that weak solutions of Robin problem are locally bounded up to the boundary. We provide some examples satisfying such a local boundedness property, and thus have Gaussian bounds for their Green's functions.     

Bifurcation from trivial solution for elliptic systems with nonlinear boundary conditions

In this paper, we investigate semilinear elliptic systems having a parameter with nonlinear Neumann boundary conditions over a smooth bounded domain. The objective of our study is to analyse bifurcation component of positive solutions from trivial solution and their stability. The results are obtained via classical bifurcation theorem from a simple eigenvalue, by studying the eigenvalue problem of elliptic systems.     

Solutions for systems of nonlinear elliptic equations with nonlinear boundary conditions

We look for solutions of systems of nonlinear elliptic equations with nonlinear boundary conditions and values in some compact convex set M. If the nonlinear terms satisfy a sign condition on the boundary of M and the inhomogeneous terms assume their values in this set existence of solutions is proved. The proof is based on the homotopy invariance of the Leray-Schauder degree and Weinberger's strong maximum principle.