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Uniform bounds for quotients of Green functions on C 1 , 1 -domains

H. Hueber, M. Sieveking (1982)

Annales de l'institut Fourier

Let Δ u = Σ i 2 x i 2 , L u = Σ i , j a i j 2 x i x j u + Σ i b i x i u + c u be elliptic operators with Hölder continuous coefficients on a bounded domain Ω R n of class C 1 , 1 . There is a constant c > 0 depending only on the Hölder norms of the coefficients of L and its constant of ellipticity such that c - 1 G Δ Ω G L Ω c G Δ Ω on Ω × Ω , where γ Δ Ω (resp. G L Ω ) are the Green functions of Δ (resp. L ) on Ω .

Unique continuation for |Δu| ≤ V |∇u| and related problems.

Thomas H. Wolff (1990)

Revista Matemática Iberoamericana

Much of this paper will be concerned with the proof of the followingTheorem 1. Suppose d ≥ 3, r = max {d, (3d - 4)/2}. If V ∈ Llocr(Rd), then the differential inequality |Δu| ≤ V |∇u| has the strong unique continuation property in the following sense: If u belongs to the Sobolev space Wloc2,p and if |Δu| ≤ V |∇u| andlimR→0 R-N ∫|x| < R |∇u|p' = 0for all N then u is constant.

Unique continuation with weak type lower order terms: The variable coefficient case.

Guozhen Lu (1995)

Publicacions Matemàtiques

This paper deals with the unique continuation problems for variable coefficient elliptic differential equations of second order. We will prove that the unique continuation property holds when the variable coefficients of the leading term are Lipschitz continuous and the coefficients of the lower order terms have small weak type Lorentz norms. This will improve an earlier result of T. Wolff in this direction.

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