Noncoercive convection–diffusion elliptic problems with Neumann boundary conditions

Abstract

We study the existence and uniqueness of solutions of the convective-diffusive elliptic equationUnder the assumption that V ∈ L p ( ) N with p = N if N ≥ 3 (resp. p > 2 if N = 2), we prove that the problem has a solution u ∈ H 1 ( ) if f dx = 0, and also that the kernel is generated by a function u ∈ H 1 ( ), unique up to a multiplicative constant, which satisfies u > 0 a.e. on . We also prove that the equation −div(D∇u) + div(V u) + ν u = f has a unique solution for all ν > 0 and the map f → u is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equationThe dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems. Mathematics Subject Classification (2000)35D05 · 35B30 · 35J25 · 35K20 · 47B44

A Finite Element Method for a Noncoercive Elliptic Problem with Neumann Boundary Conditions

“…Moreover, the detailed analysis on the existence and uniqueness of solutions of (2.3) in [8] shows that the linear operator L γ : V → V * is an isomorphism for any γ ∈ R + . The restriction of L to V is an isomorphism from V to V * := {f ∈ V * | ⟨f, 1⟩ = 0}.…”

“…Furthermore, the analysis in [8] shows in particular that the eigenvalues of L form a sequence of increasing nonnegative numbers tending to +∞. The first eigenvalue of L is 0, is of multiplicity 1 and is associated with an a.e.…”

“…This type of equations appears for example in studying the steady case of the diffusion of magnetic particles in a magnetic liquid including convection [4,12], or if α is the gradient of a potential, α = ∇Ψ, as the steady version of the Fokker-Planck equation [8].…”

“…Then, the results of Riesz-Schauder theory tell us that there is at least one solution of (1.1) if and only if f is perpendicular to the kernel of the adjoint operator L * : V → V * . The detailed analysis in [8] shows that the kernels of L and L * are one-dimensional, ker L = spanû witĥ u > 0 a.e. in Ω and ker L * = span 1.…”

A finite element method for a noncoercive elliptic problem with Neumann boundary conditions

“…If div α ≥ 0 in Ω and α · n ≤ 0 on Γ the bilinear form a(·, ·) is coercive on V , consequently existence and uniqueness follow from Lax-Milgram theorem. For general α the solvability of problem (3) was studied in [2]. Let α ∈ L p (Ω), p > 2, then the problem (3) has a solutionû unique up to a multiplicative constant, andû > 0.…”