Linear Elliptic Systems Research Articles

Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems

For a Hamiltonian $H \in C^2(R^{N \times n})$ and a map $u:\Omega \subseteq R^n \to R^N$, we consider the supremal functional \begin{eqnarray} E_\infty (u,\Omega) := \|H(Du)\|_{L^\infty(\Omega)} . \end{eqnarray} The ``Euler-Lagrange PDE associated to (1) is the quasilinear system \begin{eqnarray} A_\infty u := (H_P \otimes H_P + H[H_P]^\bot H_{PP})(Du):D^2 u = 0. \end{eqnarray} (1) and (2) are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author [28]. Herein we show that the Dirichlet problem for (2) admits for all $n = N \geq 2$ infinitely-many smooth solutions on the punctured ball, in the case of $H(P)=|P|^2$ for the $\infty$-Laplacian and of $H(P)= {|P|^2}{\det(P^\top P)^{-1/n}}$ for optimised Quasiconformal maps. Nonuniqueness for the linear degenerate elliptic system $A(x):D^2u =0$ follows as a corollary. Hence, the celebrated $L^\infty$ scalar uniqueness theory of Jensen [24] has no counterpart when $N \geq 2$. The key idea in the proofs is to recast (2) as a first order differential inclusion $Du(x) \in \mathcal{K} \subseteq R^{n\times n}$, $x\in \Omega$.

Borderline gradient continuity for nonlinear parabolic systems

We consider the evolutionary \(p\)-Laplacean system $$\begin{aligned} \partial _t u-\triangle _p u=F,\qquad p > \frac{2n}{n+2} \end{aligned}$$ in cylindrical domains of \( \mathbb R^{n}\times \mathbb R\), and prove the continuity of the spatial gradient \(Du\) under the Lorentz space assumption \(F\in L(n+2,1)\). When \(F\) is time independent the condition improves in \(F \in L(n,1)\). This is the limiting case of a result of DiBenedetto claiming that \(Du\) is Holder continuous when \(F \in L^{q}\) for \(q>n+2\). At the same time, this is the natural nonlinear parabolic analog of a linear result of Stein, claiming the gradient continuity of solutions to the linear elliptic system \(\triangle u \in L(n,1)\) is continuous. New potential estimates are derived and moreover suitable nonlinear potentials are used to describe fine properties of solutions.

Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications

We establish a quantization result for the angular part of the energy of solutions to elliptic linear systems of Schrodinger type with antisymmetric potentials in two dimensions. This quantization is a consequence of uniform Lorentz‐Wente type estimates in degenerating annuli. Moreover this result is optimal in the sense that we exhibit a sequence of functions satisfying our hypothesis whose radial part of the energy is not quantized. We derive from this angular quantization the full energy quantization for general critical points to functionals which are conformally invariant or also for pseudoholomorphic curves on degenerating Riemann surfaces.

Local inversions in ultrasound-modulated optical tomography

Ultrasound-modulated optical tomography is a hybrid imaging modality that aims to combine the high contrast of optical waves with the high resolution of ultrasound. We follow the model of the influence of ultrasound modulation on the light intensity measurements developed in Bal and Schotland (2010 Phys. Rev. Lett. 104 043902). We present sufficient conditions ensuring that the absorption and diffusion coefficients modeling light propagation can locally be uniquely and stably reconstructed from the corresponding available information. We present an iterative procedure to solve such a problem based on the analysis of linear elliptic systems of redundant partial differential equations.

On positive solution for a class of quasilinear elliptic systems with sign-changing weights

In this paper, we consider the problem for the existence of positive solutions of quasi- linear elliptic system ⎧ ⎪ ⎪ −Δpu = λa(x)u α v γ , x ∈ Ω, −Δqv = λb(x)u η v β , x ∈ Ω, u = v = 0, x ∈ ∂ Ω, where the λ > 0i s ap arameter,Ω is a bounded domain in RN(N > 1) with smooth bound- ary ∂ Ω ,a nd theΔpz = div(|∇z| p−2 ∇z) is the p-Laplacian operator. Here a(x) and b(x) are C 1 sign-changing functions that maybe are negative near the boundary. Using the method of sub-super solutions and comparison principle, which studied the existence of positive solutions for quasilinear elliptic system. The main results of the present paper are new and extend the previously known results.

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’.

A Free Boundary Problem Arising from Segregation of Populations with High Competition

In this work, we show how to obtain a free boundary problem as the limit of a fully non linear elliptic system of equations that models population segregation (Gause-Lotka-Volterra type). We study the regularity of the solutions. In particular, we prove Lipschitz regularity across the free boundary. The problem is motivated by the work done by Caffarelli, Karakhanyan and Fang-Hua Lin for the linear case.

Counterexamples in the theory of coerciveness for linear elliptic systems related to generalizations of Korn's second inequality

AbstractWe show that the following generalized version of Korn's second inequality with nonconstant measurable matrix valued coefficients P: Ω ⊂ ℝ3 → ℝ3× 3 is in general false, even if P ∈ SO(3), while the Legendre‐Hadamard condition and ellipticity on ℂn for the quadratic form |Du P+(DuP)T|2 is satisfied. Thus Gårding's inequality may be violated for formally positive quadratic forms.

Estimates for solutions to the model Venttsel problem in Campanato spaces

We consider a linear elliptic system of second order equations with constant coefficients in a neighborhood of the plane part of the boundary. The boundary condition contains the conormal derivative and the second order operator containing the tangent derivatives (a Venttsel type problem). For a weak solution we obtain the fundamental Campanato estimates. We clarify the dependence of the smoothness of a weak solution to the corresponding inhomogeneous problem in Campanato spaces on the smoothness of the right-hand sides of the system and boundary conditions. Bibliography: 8 titles.

Weighted analytic regularity in polyhedra

We explain a simple strategy to establish analytic regularity for solutions of second order linear elliptic boundary value problems. The abstract framework presented here helps to understand the proof of analytic regularity in polyhedral domains given in the authors’ paper in [M. Costabel, M. Dauge, S. Nicaise, Analytic regularity for linear elliptic systems in polygons and polyhedra, Math. Models Methods Appl. Sci. 22 (8) (2012)]. We illustrate this strategy by considering problems set in smooth domains, in corner domains and in polyhedra.

Asymptotically or super linear cooperative elliptic systems in the whole space

Under the assumption that F is asymptotically or super linear as | U |→∞ with U =( u, v ) ∈ R 2 , we obtain the existence of ground state solutions of a class of cooperative elliptic systems in R N by using a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou. To the best of our knowledge, there is no result published concerning the systems in the whole space R N .

Energy quantization for biharmonic maps

In the present work we establish an energy quantization (or energy identity) result for solutions to scaling invariant variational problems in dimension 4 which includes biharmonic maps (extrinsic and intrinsic). To that aim we first establish an angular energy quantization for solutions to critical linear 4th order elliptic systems with antisymmetric potentials. The method is inspired by the one introduced by the authors previously in [LaR] for 2nd order problems.

О решениях одной системы уравнений с частными производными с двумя независимыми переменными

In the paper we consider first order linear elliptic and hyperbolic systems with constant coefficients and two independent variables. For such systems we study the variety of all the solutions and that of the solutions growing at infinity at most as a power function.

Cotton-Type and Joint Invariants for Linear Elliptic Systems

Cotton-type invariants for a subclass of a system of two linear elliptic equations, obtainable from a complex base linear elliptic equation, are derived both by spliting of the corresponding complex Cotton invariants of the base complex equation and from the Laplace-type invariants of the system of linear hyperbolic equations equivalent to the system of linear elliptic equations via linear complex transformations of the independent variables. It is shown that Cotton-type invariants derived from these two approaches are identical. Furthermore, Cotton-type and joint invariants for a general system of two linear elliptic equations are also obtained from the Laplace-type and joint invariants for a system of two linear hyperbolic equations equivalent to the system of linear elliptic equations by complex changes of the independent variables. Examples are presented to illustrate the results.

Sharp Regularity for Elliptic Systems Associated with Transmission Problems

The paper concerns regularity theory for linear elliptic systems with divergence form arising from transmission problems. Estimates in BMO, Dini and Holder spaces are derived simultaneously and the gaps among of them are filled by using Campanato–John–Nirenberg spaces. Results obtained in the paper are parallel to the classical regularity theory for elliptic systems.

Existence and multiplicity of solutions for asymptotically linear nonperiodic Hamiltonian elliptic system

This paper is concerned with the following nonperiodic Hamiltonian elliptic system {−Δu+V(x)u=Hv(x,u,v)−Δv+V(x)v=Hu(x,u,v)x∈RN,u(x),v(x)→0as|x|→∞, where z=(u,v):RN→R×R, N≥3. Assuming the potential V is nonperiodic and sign-changing, H(x,z) is nonperiodic in x and asymptotically quadratic in z=(u,v), existence and multiplicity of solutions are obtained via a variational approach.

Multiplicity result for asymptotically linear noncooperative elliptic systems

In this paper, we obtain a multiplicity result for a class of asymptotically linear noncooperative elliptic systems by using saddle point reduction technique and two abstract critical point theorems. Copyright © 2012 John Wiley & Sons, Ltd.

Hartogs Phenomenon for Systems of Differential Equations

A property of extension is studied for solutions of linear elliptic systems of differential equations. We show that the dimension of the characteristic variety of the system plays a key role in the problem.

Concentration phenomena for neutronic multigroup diffusion in random environments

We study the asymptotic behavior of the principal eigenvalue of a weakly coupled, cooperative linear elliptic system in a stationary ergodic heterogeneous medium. The system arises as the so-called multigroup diffusion model for neutron flux in nuclear reactor cores, the principal eigenvalue determining the criticality of the reactor in a stationary state. Such systems have been well studied in recent years in the periodic setting, and the purpose of this work is to obtain results in random media. Our approach connects the linear eigenvalue problem to a system of quasilinear viscous Hamilton–Jacobi equations. By homogenizing the latter, we characterize the asymptotic behavior of the eigenvalue of the linear problem and exhibit some concentration behavior of the eigenfunctions.

The weak maximum principle for a class of strongly coupled elliptic differential systems

A classical counterexample due to E. De Giorgi, shows that the weak maximum principle does not remain true for general linear elliptic differential systems. Since then, there were some efforts to establish the weak maximum principle for special elliptic differential systems, but the existing works are addressing only the cases of weakly coupled systems, or almost-diagonal systems, or even some systems coupling in various lower order terms. In this paper, by contrast, we present maximum modulus estimates for weak solutions to some coupled elliptic differential systems with different principal parts, under some mild assumptions. The systems under consideration are strongly coupled in the second order terms and other lower order terms, without restrictions on the size of ratios of the different principal part coefficients, or on the number of equations and space variables.