On equivalent conditions for the validity of Poincaré inequality on weighted Sobolev space with applications to the solvability of degenerated PDEs involving p-Laplacian

Caccioppoli-type estimates and Hardy-type inequalities derived from weighted p-harmonic problems

“…This is commented below in detail. In addition, the classical (unweighted) Poincaré's inequality on an arbitrary bounded domain can be concluded from [55] and it is confirmed to hold with best constant in [26,Remark 7.6]. The inequality with weights of the form x α exp βx γ provided in [55,Theorem 5.5] can also be retrieved by the methods from [40] with the same constant, while the inequality with weights of the form The optimal constants in inequalities of Hardy type are an object of a certain discussion already, which we briefly summarize in Table 2.…”

Existence and uniqueness of the solution of linear degenerate PDEs in weighted Sobolev space with nonhomogeneous boundary condition

“…It appears that when H (see ) admits Poincaré inequality, then function g 0 can be omitted in the definition of function g representing functionals on that subspace. This follows from the following statement proven in Dhara and Kałamajska . Here we state it in the abbreviated version, dealing only with Sobolev spaces related to P =2.…”

“…Remark (Dhara and Kałamajska) 1) Under the assumptions in Proposition and Theorem we have . Hence τ ∈ B 2 (Ω), and so according to Proposition we have .…”

“…The following definition was introduced in . We adopt it here for the space denoted in Dhara and Kałamajska by in the special case when Ψ( λ )=Φ( λ )=| λ | 2 . Because of the special structure chosen, we propose the more classical name for this particular case.…”

Caccioppoli--type estimates and Hardy--type inequalities derived from degenerated $p$--harmonic problems

“…which perhaps could be applied to study further properties of solutions. For example, the Hardy-Poincaré inequalities like above, where a(•) = b(•), are often equivalent to the solvability of degenerated PDEs of the type div a(x)|∇u(x)| p−2 ∇u(x) = x * , where x * is an arbitrary functional on weighted Sobolev space W 1,p ̺,0 (Ω) defined as the completion of C ∞ 0 (Ω) in the norm of Sobolev space W 1,p ̺ (Ω), see Theorem 7.12 in [19]. We hope that by the investigation of the qualitative properties of supersolutions to degenerated PDEs and by constructions of Hardy-type inequalities, we can get deeper insight into the theory of degenerated elliptic PDEs.…”