Second-order Elliptic Systems Research Articles

A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space

In this article we consider parabolic systems and $L_p$ regularityof the solutions. With zero boundary condition the solutionsexperience bad regularity near the boundary. This article addressesa possible way of describing the regularity nature. Our space domainis a half space and we adapt an appropriate weight into our functionspaces. In this weighted Sobolev space setting we develop aFefferman-Stein theorem, a Hardy-Littlewood theorem and sharpfunction estimations. Using these, we prove uniqueness and existenceresults for second-order elliptic and parabolic partial differentialsystems in weighed Sobolev spaces.

A priori estimates and existence of positive solutions for higher-order elliptic equations

In this paper, we present a new sufficient condition to get a priori L∞-estimates for positive solutions of higher-order elliptic equations in a smooth bounded convex domain of RN with Navier boundary conditions or for radially symmetric solutions in the ball with Dirichlet boundary conditions. A priori L∞-estimates for positive solutions of the second-order elliptic system in a smooth bounded convex domain of RN with Dirichlet boundary conditions are also established. As usual, these a priori bounds allow us to obtain existence results. Also, by truncation technique combined with minimax method, we obtain existence of positive solution for higher-order elliptic equations of the form (1.1) below when we only assume that the nonlinearity is a nondecreasing positive function satisfying: liminfs→+∞f(s)s>Λ1, limsups→0f(s)s<Λ1, where Λ1 is the first eigenvalue of (−Δ)m with Navier boundary conditions and the weak subcritical growth condition: lims→+∞⁡f(s)sσ=0, where σ=N+2mN−2m.

Boundary Gradient Estimates for Parabolic and Elliptic Systems from Linear Laminates

We study boundary gradient estimates for second-order divergence type parabolic and elliptic systems in $C^{1,\alpha}$ domains. The coefficients and data are assumed to be H\older in the time variable and all but one spatial variables. This type of systems arises from the problems of linearly elastic laminates and composite materials.

Multiplicity of Positive Solutions for a Second-Order Elliptic System of Kirchhoff Type

We study elliptic problems of Kirchhoff type inΩ⊂RN (N≥3). Using variational tools, we establish the existence of at least two nontrivial and nonnegative solutions.

On [formula omitted] estimates in homogenization of elliptic equations of Maxwellʼs type

For a family of second-order elliptic systems of Maxwellʼs type with rapidly oscillating periodic coefficients in a C1,α domain Ω, we establish uniform estimates of solutions uε and ∇×uε in Lp(Ω) for 1<p⩽∞. The proof relies on the uniform W1,p and Lipschitz estimates for solutions of scalar elliptic equations with periodic coefficients.

Fundamental solutions of a class of first-order linear elliptic systems

The Cauchy–Riemann type systems in ℝ n+1 are systems of first-order partial differential equations whose solutions satisfy the Laplace equation. The systems can be written in the form , where E 0 = I m is the identity matrix, E i (i ≥ 1) are m × m constant matrices satisfying the condition E i E j + E j E i = − 2δ ij I m . We introduce a more general condition for E i , they are variable coefficients and satisfy the condition E i (x) · E j (x) + E j (x) · E i (x) = − 2a ij (x)I m , where [a ij ] is a symmetric and positive matrix. Each 𝒞2-solution of the system now becomes a solution of a second-order elliptic system with the leading part . In this article, we introduce the Levi functions and construct fundamental solutions of the systems ‘in the large’.

Dirichlet problem for weakly coupled elliptic systems on the plane

We consider second-order elliptic systems on the plane with constant (and only leading) matrix coefficients. We show that for these systems the notion of being weakly coupled (in the sense of A.V. Bitsadze) is equivalent to the well-known complementarity condition for the Dirichlet problem. In the framework of the function theoretic approach, we introduce analogs of double-layer potentials for solutions of weakly coupled systems. By using these potentials, we obtain a complete description of solutions of weakly elliptic systems in Holder classes as well as in the Hardy classes hp(D) and \(C\left( {\bar D} \right)\).

Homogenization of elliptic systems with Neumann boundary conditions

For a family of second-order elliptic systems with rapidly oscillating periodic coefficients in a $C^{1,\alpha }$ domain, we establish uniform $W^{1,p}$ estimates, Lipschitz estimates, and nontangential maximal function estimates on solutions with Neumann boundary conditions.

Uniqueness of positive radially symmetric solutions of the Dirichlet problem for a nonlinear elliptic system of second order

We prove the existence and uniqueness of positive radially symmetric solutions of the Dirichlet problem for a nonlinear elliptic system of second order. A numerical method for constructing such solutions is also given.

Optimal partial regularity results for nonlinear elliptic systems in Carnot groups

In this paper, we consider partial regularityfor weak solutions of second-order nonlinear elliptic systems inCarnot groups. By the method of A-harmonic approximation, weestablish optimal interior partial regularity of weak solutions tosystems under controllable growth conditions with sub-quadraticgrowth in Carnot groups.

Regularity Result for Quasilinear Elliptic Systems with Super Quadratic Natural Growth Condition

We consider boundary regularity for weak solutions of second-order quasilinear elliptic systems under natural growth condition with super quadratic growth and obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. Combined with existing results on interior partial regularity, this result yields an upper bound on the Hausdorff dimension of the singular set at the boundary.

Positive solutions to a nonlinear fourth-order partial differential equation

A class of fourth-order partial differential equations with Dirichlet boundary conditions which has important applications in phase transformation theory and in the description of the motion of a very thin layer of viscous incompressible fluid is studied. By transforming the higher order problem into a second-order elliptic system and using a fixed point method, the existence and uniqueness of steady state solutions are obtained. Moreover, by using a semi-discrete method, the existence of strong solutions of the evolution equation is obtained if the initial function is in the neighborhood of a positive steady state solution. Finally, the solution of the evolution equation converges to its steady state solution as the time t→+∞.

Boundary regularity result for quasilinear elliptic systems

We consider boundary regularity for weak solutions of second-order quasilinear elliptic systems under controllable growth condition, and obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. Combined with existing results on interior partial regularity, this result yields an upper bound on the Hausdorff dimension of the singular set at the boundary.

Gradient Estimates for Parabolic and Elliptic Systems from Linear Laminates

We establish several gradient estimates for second-order divergence type parabolic and elliptic systems. The coefficients and data are assumed to be Hölder or Dini continuous in the time variable and all but one spatial variable. This type of system arises from the problems of linearly elastic laminates and composite materials. For the proof, we use Campanato’s approach in a novel way. Non-divergence type equations under similar conditions are also discussed.

A [formula omitted]-theory of elliptic and parabolic partial differential systems in [formula omitted] domains

In this paper we consider the second-order parabolic partial differential systems and elliptic systems with C1 space domains. We prove the existence and uniqueness results in the Sobolev spaces with weights. In this solution spaces the derivatives of solutions are allowed to blow up near the boundary and also the coefficients of the systems may oscillate to a great extent or blow up near the boundary.

Large solutions of elliptic systems of second order and applications to the biharmonic equation

In this work we study the nonnegative solutions of the elliptic system \[ \Delta u=|x|^{a}v^{\delta},\qquad\Delta v=|x|^{b}u^{\mu}% \] in the superlinear case $\mu\delta>1,$ which blow up near the boundary of a domain of $\mathbb{R}^{N},$ or at one isolated point. In the radial case we give the precise behavior of the large solutions near the boundary in any dimension $N$. We also show the existence of infinitely many solutions blowing up at $0.$ Furthermore, we show that there exists a global positive solution in $\mathbb{R}^{N}\backslash\left\{ 0\right\} ,$ large at $0,$ and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation \[ \Delta^{2}u=\left\vert x\right\vert ^{b}\left\vert u\right\vert ^{\mu}. \] Our results are based on a new dynamical approach of the radial system by means of a quadratic system of order 4, introduced in [4], combined with the nonradial upper estimates of [5].

Symmetry of solutions for quasimonotone second-order elliptic systems in ordered Banach spaces

We consider symmetry properties of solutions to nonlinear elliptic boundary value problems defined on bounded symmetric domains of \({\mathbb R^n}\) . The solutions take values in ordered Banach spaces E, e.g. \({E=\mathbb R^N}\) ordered by a suitable cone. The nonlinearity is supposed to be quasimonotone increasing. By considering cones that are different from the standard cone of componentwise nonnegative elements we can prove symmetry of solutions to nonlinear elliptic systems which are not covered by previous results. We use the method of moving planes suitably adapted to cover the case of solutions of nonlinear elliptic problems with values in ordered Banach spaces.

Boundary regularity for elliptic systems under a natural growth condition

We consider weak solutions $${u \in u_0 + W^{1,2}_0(\Omega,\mathbb{R}^N) \cap L^{\infty}(\Omega,\mathbb{R}^N)}$$ of second-order nonlinear elliptic systems of the type $$- {\rm div} \,a (\, \cdot \,, u, Du ) = b(\, \cdot \,,u,Du)\qquad \text{ in }\Omega$$ with an inhomogeneity satisfying a natural growth condition. In dimensions $${n \in \{2,3,4\}}$$ , we show that $${\mathcal{H}^{n-1}}$$ -almost every boundary point is a regular point for Du, provided that the boundary data and the coefficients are sufficiently smooth.

Exponential decay and Fredholm properties in second-order quasilinear elliptic systems

We consider second-order quasilinear elliptic systems on unbounded domains in the setting of Sobolev spaces. We complete our earlier work on the Fredholm and properness properties of the associated differential operators by giving verifiable conditions for the linearization to be Fredholm of index zero. This opens the way to using the degree for C 1 -Fredholm maps of index zero as a tool in the study of such quasilinear systems. Our work also enables us to check the Fredholm assumption which plays an important role in Rabier's approach to proving exponential decay to zero at infinity of solutions.

Initial-boundary value problems for a class of homogeneous second-order elliptic systems on a plane with a singular point

We solve initial-boundary value problems for certain model second-order elliptic systems on a plane with a singular point.