Class Of Second-order Elliptic Equations Research Articles

Well posedness for one class of elliptic equations with drift

We studied one class of second-order elliptic equations with intermediate coefficient and proved that the semi-periodic problem on a strip is unique solvable in Hilbert space. We assume that the intermediate coefficient of the equation is continuously differentiable and grows rapidly near infinity, for example, it grows faster than (|x|+1 ) ln (|x|+3). However, we do not impose bounds on its derivatives. We believe that the lower-order coefficient is continuous, can be unlimited and change sign.

Sobolev-type regularity and Pohozaev-type identities for some degenerate and singular problems

Sobolev-type regularity results are proved for solutions to a class of second-order elliptic equations with a singular or degenerate weight, under non-homogeneous Neumann conditions. As an application a Pohozaev-type identity for weak solutions is derived.

Maximal L1 regularity for solutions to inhomogeneous incompressible Navier-Stokes equations

This paper is devoted to the maximal L1 regularity and asymptotic behavior for solutions to the inhomogeneous incompressible Navier-Stokes equations under a scaling-invariant smallness assumption on the initial velocity. We obtain a new global L1-in-time estimate for the Lipschitz seminorm of the velocity field without any smallness assumption on the initial density fluctuation. In the derivation of this estimate, we study the maximal L1 regularity for a linear Stokes system with variable coefficients. The analysis tools are a use of the semigroup generated by a generalized Stokes operator to characterize some Besov norms and a new gradient estimate for a class of second-order elliptic equations of divergence form. Our method can be used to study some other issues arising from incompressible or compressible viscous fluids.

Stability of stationary subsonic compressible Euler flows with mass-additions in two-dimensional straight ducts

We show existence, uniqueness and stability for a family of stationary subsonic compressible Euler flows with mass-additions in two-dimensional rectilinear ducts, subjected to suitable time-independent multi-dimensional boundary conditions at the entrances and exits. The stationary subsonic Euler equations consist a quasi-linear system of elliptic-hyperbolic composite-mixed type, while addition-of-mass destructs the usual methods based upon conservation of mass and Lagrangian coordinates to separate the elliptical and hyperbolic modes of the system. We establish a new decomposition and nonlinear iteration scheme to overcome this major difficulty. It reveals that mass-additions introduce very strong interactions in the elliptic and hyperbolic modes, and lead to a class of second-order elliptic equations with multiple integral nonlocal terms. The linearized problem is solved by studying algebraic and analytical properties of infinite weakly coupled boundary-value problems of ordinary differential equations, each with multiple nonlocal terms, after applications of Fourier analysis methods.

Renormalized solutions of elliptic equations with variable exponents and general measure data

A class of second-order elliptic equations with variable nonlinearity exponents and the right-hand side in the form of the general Radon measure with finite total variation is considered. The existence of a renormalized solution of the Dirichlet problem is proved as a consequence of stability with respect to the convergence of the right-hand side of the equation. Bibliography: 37 titles.

Optimal Estimates for the Conductivity Problem by Green’s Function Method

We study a class of second-order elliptic equations of divergence form, with discontinuous coefficients and data, which models the conductivity problem in composite materials. We establish optimal gradient estimates by showing the explicit dependence of the elliptic coefficients and the distance between interfacial boundaries of inclusions. These extend the known results in the literature and answer open problem (b) proposed by Li and Vogelius (2000) for the isotropic conductivity problem. We also obtain more interesting higher-order derivative estimates, which answers open problem (c) of Li and Vogelius (2000). It is worth pointing out that the equations under consideration in this paper are non-homogeneous.

Error analysis on bi-parameter finite elements

A bi-parameter finite element is introduced in this paper, in which a new local grid refinement technique is presented for a class of second-order elliptic equations. The new approach is based on a bi-parameter finite element space defined, respectively, on the whole domain with size H and on a subdomain with size h(h ⪡ H). The stability and convergence analysis of the bi-parameter finite element approximation is studied. The behavior of approximation with respect to the parameters H and h is presented. In particular, the error estimates for the finite element Green's function expressed by H and h are derived.

Positive eigenfunctions for a class of second-order elliptic equations with strong nonlinearity

(1977). Positive eigenfunctions for a class of second-order elliptic equations with strong nonlinearity. Communications in Partial Differential Equations: Vol. 2, No. 6, pp. 615-625.