Abstract
In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is −div((1 + |∇u| 2) (p−2)/2 ∇u) − div(c(x)|u| p−2 u) = f in Ω, (1 + |∇u| 2) (p−2)/2 ∇u + c(x)|u| p−2 u · n = 0 on ∂Ω,
Highlights
In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is
We assume that the principal part −div(a(x, Du)) is not degenerate when p > 2, i.e. in the model case −div(a(x, ∇u)) = −div((1 + |∇u|2)(p−2)/2∇u). Such an assumption is not required when p ≤ 2, that is for such values of p we prove uniqueness results for operators whose prototype is the so-called p-Laplace operator, −∆pu = −div(|∇u|p−2∇u)
We prove two uniqueness results depending on the values of p: Theorem 3.1
Summary
In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is (1.1). When p is close to 1, i.e. p ≤ 2 − 1/N, the distributional solution to problem (1.1) does not belong to a Sobolev space and in general is not a summable function; this implies that its mean value has not meaning and any existence result for distributional solution with null mean value cannot hold. This difficulty is overcome in [18]. In contrast when we consider Neumann boundary conditions and two solutions u, v ∈ H1(Ω) having null median, by using Tk(u − v) we can prove equality (1.3), but Poincare-Wirtinger inequality does not allow to get.
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