Abstract
We study here Dirichlet boundary value problem for a quasi-linear elliptic equation with anisotropic p-Laplace operator in its principle part and L^1-control in coefficient of the low-order term. As characteristic feature of such problem is a specification of the matrix of anisotropy A=A^{sym}+A^{skew} in BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space W^{1,p}_0(\Omega), we specify a suitable functional class in which we look for solutions and prove existence of weak solutions in the sense of Minty using a non standard approximation procedure and compactness arguments in variable spaces.
Highlights
In this paper we deal with the following boundary value problem (−∆p (A, y) + |y|p−2 yu = − div f in Ω, y = 0 on ∂Ω, u ∈ L1 (Ω), u(x) ≥ 0 a.e. in Ω, where−∆p (A, y) = − div |(∇y, A∇y)| p−2 A∇y (1.1) (1.2)is the anisotropic p-Laplacian, 2 ≤ p < +∞, A is the matrix of anisotropy, yd ∈ L2 (Ω) and f ∈ L∞ (Ω; RN ) are given distributions.The interest to elliptic equations whose principal part is an anisotropic pLaplace operator arises from various applied contexts related to composite materials such as nonlinear dielectric composites, whose nonlinear behavior is modeled by the so-called power-low
The existence, uniqueness, and variational properties of the weak solution to the above boundary value problem (BVP) usually are drastically dierent from the corresponding properties of solutions to the elliptic equations with coercive L∞ -matrices of anisotropy
(3.1)(3.2) is the fact that the skew-symmetric part D of the matrix A is merely measurable and its sub-multiplicative norm belongs to the BM O-space. This circumstance can entail a number of pathologies with respect to the standard properties of BVPs for elliptic equations with anisotropic p-Laplacian even with 'a good' symmetric part A and a smooth right-hand side f
Summary
The existence, uniqueness, and variational properties of the weak solution to the above BVP usually are drastically dierent from the corresponding properties of solutions to the elliptic equations with coercive L∞ -matrices of anisotropy (we refer to [6, 2628, 31] for the details and other results in this eld) Another distinguishing feature of the boundary value problem (3.1)(3.2) is the fact that the skew-symmetric part D of the matrix A is merely measurable and its sub-multiplicative norm belongs to the BM O-space (rather than the space L∞ Ω ). This circumstance can entail a number of pathologies with respect to the standard properties of BVPs for elliptic equations with anisotropic p-Laplacian even with 'a good' symmetric part A and a smooth right-hand side f. 1,p smooth compactly supported functions in W0,B (Ω)
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