Bi-Sobolev mappings and elliptic equations in the plane

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...

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...

Elliptic Equations with Limiting Sobolev Exponents

where a ( x ) is a given function on M . The original interest in such questions grew out of Yamabe's problem (see [40], [39], [2], [27], [15]) which corresponds to the special case where a ( x ) = ( ( N 2)/4(N l ) ) R ( x ) and R ( x ) is the scalar curvature of M . It turns out that, despite its simple form, equation (1) (or ( 2 ) ) has a very rich structure and provides an amazing source of ...

Multiple solutions for Kirchhoff elliptic equations in Orlicz-Sobolev spaces

Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains