Since p-Laplacian is ( p − 1 ) -homogeneous, we may assume that φ satisfies ∫ Ω m ( x ) φ p d x = 1, and hence ∥ ∇ φ ∥ p p = μ 1 ( m ) ∫ Ω m ( x ) φ p d x = μ 1 ( m ) holds by taking φ as a test function.
By choosing v=v i as a test function, we obtain ∫ Ω | ∇ u | p − 2 ∇ u ∇ v i dx = λ ∫ Ω | u | p − 2 u v i dx + ∫ Γ | u | p − 2 u v i ds.
When the set consisting of a single point, this reduces to the Dirac distribution, denoted by δ, which associates to a test function f its value at the.
In this paper, in order to remedy the partial matching problem, we detect and exclude the outliers first by an outlier test function, and then apply the proposed PMHD to the remaining feature points.
end{gathered} (2.14) By (2.14), by choosing a suitable test function φ, one may obtain the stability of the entropy solution.
The following modified comparison principle plays a crucial role in our proofs, which can be obtained by establishing a suitable test function and Gronwall's inequality.
The following modified comparison principle plays a crucial role in our proofs, which can be obtained by establishing a suitable test function and using Lemma 1 and Gronwall's inequality.
If the diffusion coefficient of the equation is degenerate on the boundary, no matter we can define the trace of the solution on the boundary or not, by choosing a suitable test function, the stability of the solutions always can be established without a boundary condition.
Furthermore, nonexistence of nontrivial global weak solutions to initial value problems is studied by choosing a special test function.
As we have said before, by choosing a suitable test function, we can prove the stability of the solutions without any boundary value condition only if (alpha>0).
