Dirichlet Problem For Elliptic Systems Research Articles

The p-ellipticity condition for second order elliptic systems and applications to the Lamé and homogenization problems

The notion of p-ellipticity has recently played a significant role in improving our understanding of issues of solvability of boundary value problems for scalar complex valued elliptic PDEs. In particular, the presence of p-ellipticity ensures higher regularity of solutions of such equations.In this work we extend the notion of p-ellipticity to second order elliptic systems. Recall that for systems, there is no single notion of ellipticity, rather a more complicated picture emerges with ellipticity conditions of varying strength such as the Legendre, Legendre-Hadamard and integral conditions. A similar picture emerges when p-ellipticity is considered. In this paper, we define three new notions of p-ellipticity, establish relationships between them and show that each of them does play an important role in solving boundary value problems.These important roles are demonstrated by establishing extrapolation results for solvability of the Lp Dirichlet problem for elliptic systems, followed by applications of this result in two different scenarios: one for the Lamé system of linear elasticity and another in the theory of homogenization.

Influence of lower-order derivatives on solvability Dirichlet problem for elliptic systems

The article considers the Dirichlet problem for a multidimensional elliptic system containing lower-order derivatives. At the present time elliptic systems of second-order partial differential equations with two independent are well known, what cannot be said about multidimensional elliptic systems. The problem of homotopy classification of such systems has not yet been solved. For elliptic systems phenomena are observed in the nature of the solvability of classical boundary systems, which do not occur in the case of one equation. Among such phenomena, the influence of the loss of smoothness and the influence of lower-order derivatives on the solvability of boundary value problems should be noted. The paper proposes a solution to the Dirichlet problem for a multidimensional elliptic system with lower-order terms of a special form in a half-space. Applying the Fourier transform, the solvability problem is reduced to study of a second-order partial differential equation containing lower-order derivatives. Violations of the violation of the Noetherian property of the Dirichlet problem are considered. The result is formulated as a theorem.

Generalizations of Stampacchia Lemma and applications to quasilinear elliptic systems

We present two generalizations of the classical Stampacchia Lemma. As applications, we consider Dirichlet problem for elliptic systems of the form −∑i=1n∂∂xi∑β=1N∑j=1nai,jα,β(x,u(x))∂uβ(x)∂xj=fα,in Ω,u(x)=0,on ∂Ω.Under ellipticity and degenerate ellipticity conditions of the diagonal coefficients and proportional conditions of the off-diagonal coefficients, we derive some global regularity results.

The Dirichlet problem for elliptic systems with data in Köthe function spaces

We show that the boundedness of the Hardy-Littlewood maximal operator on a K\"othe function space ${\mathbb{X}}$ and on its K\"othe dual ${\mathbb{X}}'$ is equivalent to the well-posedness of the $\mathbb{X}$-Dirichlet and $\mathbb{X}'$-Dirichlet problems in $\mathbb{R}^{n}_{+}$ in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We also discuss a version of the aforementioned result which contains, as a particular case, the Dirichlet problem for elliptic systems with data in the classical Hardy space $H^1$, and the Beurling-Hardy space ${\rm HA}^p$ for $p\in(1,\infty)$. Based on the well-posedness of the $L^p$-Dirichlet problem we then prove the uniqueness of the Poisson kernel associated with such systems, as well as the fact that they generate a strongly continuous semigroup in natural settings. Finally, we establish a general Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems.

Behavior of Solutions to the Dirichlet Problem for Elliptic Systems in Convex Domains

We consider the Dirichlet problem for strongly elliptic systems of order 2m in convex domains. Under a positivity assumption on the Poisson kernel it is proved that the weak solution has bounded derivatives up to order m provided the outward unit normal has no big jumps on the boundary. In the case of second order symmetric systems in plane convex domains the boundedness of the first derivatives is proved without the assumption on the normal.

On Dirichlet problem for elliptic systems on the plane†

The Dirichlet problem for the elliptic system of second order with constant and only leading coefficients is considered in a piecewise smooth domain on the plane. The Fredholm criterion and index formula for this problem in Holder spaces are given. In particular, the examples of the elliptic systems are found such that the index of Dirichlet problem is not equal to 0. Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

The Dirichlet problem for elliptic systems in multiconnected rough regions

We present a new integral formulation for the L 2 Dirichlet boundary value problem associated with second order elliptic systems in multiconnected regions with rough boundaries. This study has been motivated by displacement prescribed problems arising in solid mechanics and our results are relevant to their numerical treatment.

Best Constants in the Miranda-Agmon Inequalities for Solutions of Elliptic Systems and the Classical Maximum Modulus Principle for Fluid and Elastic Half-spaces

The Dirichlet problem for elliptic systems of the second order with constant real and complex coefficients in the half-space 𝔑 k + = {x = (x 1,…,xk ): xk > 0} is considered. It is assumed that the boundary values of a solution u = (u 1,…,u m) have the form ψ 1 ξ 1 + · · · + ψ n ξ n, 1 ≤ n ≤ m, where ξ 1,· · ·,ξ n is an orthogonal system of m-component normed vectors and ψ 1,· · ·,ψ n are continuous and bounded functions on ∂𝔑 k +. We study the mappings [C(∂𝔑 k +)] n ∋ (ψ 1,…,ψ n) → u(x) ∋ 𝔑 m and [C(∂𝔑 k +)] n ∋ (ψ 1,…,ψ n) → u(x) ∋ 𝔑 m generated by real and complex vector valued double layer potentials. We obtain representations for the sharp constants in inequalities between |u(x)| or |(z, u(x))| and ∥u| xk =0∥, where z is a fixed unit m-component vector, | · | is the length of a vector in a finite-dimensional unitary space or in Euclidean space, and (·,·) is the inner product in the same space. Explicit representations of these sharp constants for the Stokes and Lamé systems are given. We show, in particular, that if the velocity vector (the elastic displacement vector) is parallel to a constant vector at the boundary of a half-space and if the modulus of the boundary data does not exceed 1, then the velocity vector (the elastic displacement vector) is majorised by 1 at an arbitrary point of the half-space. An analogous classical maximum modulus principle is obtained for two components of the stress tensor of the planar deformed state as well as for the gradient of a biharmonic function in a half-plane.

Solutions of the dirichlet problem for elliptic systems in a disk

We study 2×2 second-order elliptic systems, which can be written as a single equation with complex coefficients. In an arbitrary bounded region with smooth boundary, we obtain necessary and sufficient conditions on the trace relation of a solution, which we apply in the case of a disk. We prove existence and uniqueness theorems for a solution in a Sobolevskii space of an equation which is not properly elliptic. In particular, we prove that the properties of the problem determine the angle between the bicharacteristics. If it is π-rational, then there is no uniqueness, but if it is π-irrational, then the smoothness of the solution of the Dirichlet problem depends on the order of its approximation by π-rational numbers; but if it is nonreal, then the problem has the usual properties for the elliptic case.

Breakdown of uniqueness of solutions of the Dirichlet problem for elliptic systems in a disc

Breakdown of uniqueness of solutions of the Dirichlet problem for elliptic systems in a disc

On the dirichlet problem for elliptic systems

The system δu + u(a-bu-cv) =0 , δv + v(d-eu-fv) =0 represents stationary solutions of a competing species model. Conditions on the positive constants a,... ,f for existence and uniqueness of positive solutions are given. The results are based on theorems combining degree theory and super- and subfunction techniques for elliptic systems Au+f(x,u) with Dirichlet boundary data.

Finite difference approximations to the Dirichlet problem for elliptic systems

The Dirichlet problem foru=(u 1,...,u n ) $$\Delta u + f(x,u) = 0in\Omega ,u = 0on\Gamma = \partial \Omega $$ wheref=(f 1,...,f n ), is discretized in the usual way (h mesh size): $$\Delta ^h u + f(x,u) = 0in\Omega _h ,u = 0on\Gamma _h $$ We consider variousmonotone, convergent iterative schemes. Among others, they can be used, together with estimation theorems for upper and lower solutions, to show uniqueness for solutions of (2). Numerical results are given for the system $$\Delta u + u(a - bu - c\upsilon ) = 0,\Delta \upsilon + \upsilon (d - eu - f\upsilon ) = 0$$ from mathematical biology (two competing species). It is shown that there is a unique positive solution for certain values of the positive parametersa,..., f. This result is crucial for the asymptotic behavior of solutions of the corresponding parabolic system ast→∞.

Dirichlet problem for elliptic systems of second order differential equations not satisfying Ya. B. Lopatinskii's condition

Dirichlet problem for elliptic systems of second order differential equations not satisfying Ya. B. Lopatinskii's condition