Abstract
Firstly, we define an order for differential forms. Secondly, we also define the supersolution and subsolution of the -harmonic equation and the obstacle problems for differential forms which satisfy the -harmonic equation, and we obtain the relations between the solutions to -harmonic equation and the solution to the obstacle problem of the -harmonic equation. Finally, as an application of the obstacle problem, we prove the existence and uniqueness of the solution to the -harmonic equation on a bounded domain with a smooth boundary , where the -harmonic equation satisfies where is any given differential form which belongs to .
Highlights
A large amount of work about the A-harmonic equation for the differential forms has been done
We define the supersolution and subsolution of the A-harmonic equation and the obstacle problems for differential forms which satisfy the A-harmonic equation, and we obtain the relations between the solutions to A-harmonic equation and the solution to the obstacle problem of the A-harmonic equation
As an application of the obstacle problem, we prove the existence and uniqueness of the solution to the A-harmonic equation on a bounded domain Ω with a smooth boundary ∂Ω, where the Aharmonic equation satisfies d A x, du 0, x ∈ Ω; u ρ, x ∈ ∂Ω, where ρ is any given differential form which belongs to W1,p Ω, Λl−1
Summary
A large amount of work about the A-harmonic equation for the differential forms has been done. As an application of the obstacle problem, we prove the existence and uniqueness of the solution to the A-harmonic equation on a bounded domain Ω with a smooth boundary ∂Ω, where the Aharmonic equation satisfies d A x, du 0, x ∈ Ω; u ρ, x ∈ ∂Ω, where ρ is any given differential form which belongs to W1,p Ω, Λl−1 . We just started our research on the obstacle problem for differential forms satisfying the A-harmonic equation and we hope that our work will stimulate further research in this direction.
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