Solvability in the small of mmm-th order elliptic equations in weighted grand Sobolev spaces (original) (raw)
Related papers
Existence and regularity theory in weighted Sobolev spaces and applications
2016
In the thesis we discuss several questions related to the study of degenerate, possibly nonlinear PDEs of elliptic type. At first we discuss the equivalent conditions between the validity of weighted Poincaré inequalities, structure of the functionals on weighted Sobolev spaces, isoperimetric inequalities and the existence and uniqueness of solutions to the degenerate nonlinear elliptic PDEs with nonhomogeneous boundary condition, having the form: { div (ρ(x)|∇u|p−2∇u) = x∗, u− w ∈ W 1,p ρ,0 (Ω), (0.0.1) involving any given x∗ ∈ (W 1,p ρ,0 (Ω))∗ and w ∈ W 1,p ρ (Ω), where u ∈ W 1,p ρ (Ω) and W 1,p ρ (Ω) denotes certain weighted Sobolev space, W 1,p ρ,0 (Ω) is the completion of C∞ 0 (Ω). As a next step, we undertake a natural question how to interpret the nonhomogenous boundary conditions in weighted Sobolev spaces, when the natural analytical tools, like trace embedding theorems, are missing. Our further goal is to contribute to solvability and uniqueness for degenerate elliptic PDE...
Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains
International Journal of Mathematics and Mathematical Sciences, 2008
We study in this paper a class of second-order linear elliptic equations in weighted Sobolev spaces on unbounded domains of R n , n ≥ 3. We obtain an a priori bound, and a regularity result from which we deduce a uniqueness theorem.
Weighted Sobolev Spaces and Degenerate Elliptic Equations
Boletim da Sociedade Paranaense de Matemática, 2008
In this paper, we survey a number of recent results obtained in the study of weighted Sobolev spaces (with power-type weights, Ap -weights, padmissible weights, regular weights and the conjecture of De Giorgi) and the existence of entropy solutions for degenerate quasilinear elliptic equations.
2021
In this paper, we study the existence and uniqueness of weak solution to a Dirichlet boundary value problems for the following nonlinear degenerate elliptic problems −div [ ω1A(x,∇u) + ν2B(x, u,∇u) ] + ν1C(x, u) + ω2|u|u = f − divF, where 1 < p < ∞, ω1, ν2, ν1 and ω2 are Ap-weight functions, and A : Ω × R −→ R, B : Ω×R×R −→ R, C : Ω×R −→ R are Caratéodory functions that satisfy some conditions and the right-hand side term f − divF belongs to Lp(Ω, ω ′ 2 ) + n ∏ j=1 L ′ (Ω, ω ′ 1 ). We will use the BrowderMinty Theorem and the weighted Sobolev spaces theory to prove the existence and uniqueness of weak solution in the weighted Sobolev space W 1,p 0 (Ω, ω1, ω2).
arXiv (Cornell University), 2022
We consider m-th order linear, uniformly elliptic equations Lu = f with non-smooth coefficients in Banach-Sobolev spaces W m Xw (Ω) generated by weighted general Banach Function Spaces (BFS) Xw(Ω) on a bounded domain Ω ⊂ R n. Supposing boundedness of the Hardy-Littlewood Maximal and Calderón-Zygmund singular operators in Xw(Ω) we obtain solvability in the small in W m Xw (Ω) and establish interior Schauder type a priori estimates for the corresponding elliptic operator. These results will be used in order to obtain Fredholmness of the operator under consideration in Xw(Ω) with suitable weight. In addition, we analyze some examples of weighted BFS that verify our assumptions and in which the corresponding Schauder type estimates and Fredholmness of the operator hold true. This approach and the obtained results are new even for well studied spaces as the Morrey spaces, grand Lebesgue spaces, and Lebesgue spaces with variable exponents.
Sum of weighted Lebesgue spaces and nonlinear elliptic equations
Nodea-nonlinear Differential Equations and Applications, 2011
We study the sum of weighted Lebesgue spaces, by considering an abstract measure space (Omega,mathcalA,mu){(\Omega ,\mathcal{A},\mu)}(Omega,mathcalA,mu) and investigating the main properties of both the Banach space L\left( \Omega \right) =\left\{u_{1}+u_{2}:u_{1} \in L^{q_{1}} \left(\Omega \right),u_{2} \in L^{q_{2}} \left( \Omega \right) \right\}, L^{q_{i}} \left( \Omega \right) :=L^{q_{i}} \left( \Omega ,d\mu \right),andtheNemytskiı˘operatordefinedonit.Thenweapplyourgeneralresultstoproveexistenceandmultiplicityofsolutionstoaclassofnonlinearp−Laplacianequationsoftheformand the Nemytskiĭ operator defined on it. Then we apply our general results to prove existence and multiplicity of solutions to a class of nonlinear p-Laplacian equations of the formandtheNemytskiı˘operatordefinedonit.Thenweapplyourgeneralresultstoproveexistenceandmultiplicityofsolutionstoaclassofnonlinearp−Laplacianequationsoftheform-\triangle _{p}u+V\left( \left| x\right| \right) \left| u\right| ^{p-2}u=f\left( \left| x\right| ,u\right) \quad {\rm in} \mathbb{R}^{N}$$ where V is a nonnegative measurable potential, possibly singular and vanishing at infinity, and f is a Carathéodory function satisfying a double-power growth condition in u.
Weighted Sobolev theorem in Lebesgue spaces with variable exponent
Journal of Mathematical Analysis and Applications, 2007
For the Riesz potential operator I α there are proved weighted estimates I α f L q(•) (Ω,w q p) C f L p(•) (Ω,w) , Ω ⊆ R n , 1 q(x) ≡ 1 p(x) − α n within the framework of weighted Lebesgue spaces L p(•) (Ω, w) with variable exponent. In case Ω is a bounded domain, the order α = α(x) is allowed to be variable as well. The weight functions are radial type functions "fixed" to a finite point and/or to infinity and have a typical feature of Muckenhoupt-Wheeden weights: they may oscillate between two power functions. Conditions on weights are given in terms of their Boyd-type indices. An analogue of such a weighted estimate is also obtained for spherical potential operators on the unit sphere S n ⊂ R n .