Tero Kilpeläinen

Sobolev Spaces with Zero Boundary Values on Metric Spaces

We generalize the definition of the first order Sobolev spaces with zero boundary values to an arbitrary metric space endowed with a Borel regular measure. We show that many classical results extend to the metric setting. These include completeness, lattice properties and removable sets.

Sobolev Inequalities on Sets with Irregular Boundaries

We derive (weighted) Sobolev-Poincaré inequalities for s-John domains and s-cusp domains, both with optimal exponents. These results are obtained as consequences of a more comprehensive criterion.

Removable sets for continuous solutions of quasilinear elliptic equations

We show that sets of n i p + fi(p i 1) Hausdor measure zero are re- movable for fi-Holder continuous solutions to quasilinear elliptic equations similar to the p-Laplacian. The result is optimal. We also treat larger sets in terms of a growth condition. In particular, our results apply to quasiregular mappings.

Hölder continuity of solutions to quasilinear elliptic equations involving measures

AbstractWe show that the solutionu of the equation

Generalized dirichlet problem in nonlinear potential theory

- div(|\nabla u|^{p - 2} \nabla u) = \mu

On the fusion problem for degenerate elliptic equations

is locally β-Hölder continuous provided that the measure μ satisfies the condition μ(B(x,r))⩽Mrn − p + α(p − 1) for some α>β. A corresponding result for more general operators is also proven.

Weighted Sobolev spaces and capacity.

AbstractWe construct singular solutions to equations