Weighted spaces of Sobolev type and degenerate elliptic equations

Weighted Sobolev Spaces and Degenerate Elliptic Equations

In the case ω = 1, this space is denoted W (Ω). Sobolev spaces without weights occur as spaces of solutions for elliptic and parabolic partial differential equations. In various applications, we can meet boundary value problems for elliptic equations whose ellipticity is “disturbed” in the sense that some degeneration or singularity appears. This “bad” behaviour can be caused by the coefficient...

A variational weak weighted derivative: Sobolev spaces and degenerate elliptic equations∗

A new class of weak weighted derivatives and its associated Sobolev spaces is introduced and studied. The proposed notion uses a variational formulation in its definition which generalizes the usual weighting of the classical weak derivative. Such a construction naturally leads to Sobolev spaces containing classes of discontinuous functions. Weak closedness with respect to both varying function...

Existence of solutions in weighted Sobolev spaces for some degenerate semilinear elliptic equations

We prove an existence result for the Dirichlet problem associated to some degenerate quasilinear elliptic equations in a bounded open set Ω in R in the setting of weighted Sobolev spaces W 1,p 0 (Ω, ω). 2000 Mathematics Subject Classification: 35J70, 35J60.

Sobolev Spaces and Elliptic Equations

Lipschitz domains. Our presentations here will almost exclusively be for bounded Lipschitz domains. Roughly speaking, a domain (a connected open set) Ω ⊂ R is called a Lipschitz domain if its boundary ∂Ω can be locally represented by Lipschitz continuous function; namely for any x ∈ ∂Ω, there exists a neighborhood of x, G ⊂ R, such that G ∩ ∂Ω is the graph of a Lipschitz continuous function und...

Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains