A direct uniqueness proof for equations involving the p-Laplace operator

By M. Belloni, B. Kawohl. 2002

151 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/11242adbcae0eb200206dfc82b4624e983acc111

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Simplicity of the principal eigenvalue for indefinite quasilinear problems (2007, background): “1, [25]] and [5] the main idea is to show that η is again a minimizer of J .”
DIRICHLET PROBLEMS FOR THE 1-LAPLACE OPERATOR, INCLUDING THE EIGENVALUE PROBLEM (2007, background): “The first eigenfunction of the p-Laplacian operator with p ∈ (1,∞) is known to be nonzero in Ω and unique (modulo sign change and scaling), for a variational proof see [8].”
The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p → ∞ (2004, background): “4) and [9], see also [1, 3, 12, 16, 40] for related results on the p-Laplace operator.”
Convexity properties of Dirichlet integrals and Picone-type inequalities (2014, background): “See also [3] and [27] for some generalizations in the case q = p.”
Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators (2003, methodology): “To prove its uniqueness we follow the idea from [ 4 ] and assume that there are two positive minimizers u and U of (3.1).”
Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant (2003, background): “A simple proof of this long-known fact was recently given in [3].”
ON THE HIGHER EIGENVALUES FOR THE ∞-EIGENVALUE PROBLEM (2004, background): “For a new and direct proof of the simplicity we refer to [6].”
Uniqueness (2021, unknown): “…in [37], where the path γ ( t ) = ((1 − t ) u p + tv p ) 1 /p is used to prove the integral monotonicity (4.6) Alternatively, it is simple to show that D = { v > 0; v 1 /p ∈ W 1 ,p 0 (Ω) } is a convex cone and as proved in [12, p. 230] (see also [50]), the functional is strictly convex on D .”
A NONLINEAR EIGENVALUE PROBLEM (2004, background): “1A short comment on the uniqueness proof in [11] has been added later.”
Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems (2021, background): “The best constant (1.2) λq(Ω) := inf 06=u∈W 1,p0 (Ω)\{0} ∫ Ω |∇u| p dx (∫ Ω |u| q dx ) p q of the inequality (1.1) is achieved at a solution uq ∈W 1,p 0 (Ω) of the following Dirichlet boundary value problem (see Ôtani [62], Franzina-Lamberti [43]) (1.3) −∆pu := −div(|∇u| p−2∇u) = λq(Ω)‖u‖ p−q Lq(Ω)”
Positive eigenfunctions for the p-Laplace operator revisited (2006, methodology): “In [5] they established the uniqueness of minimizers (which are positive by nature).”