We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain the Adams criterion for the solubility of the single obstacle problem and establish connections with fine potential theory. We also study when the minimal p-weak upper gradient of a function remains minimal when restricted to a nonopen subset. Many of the results are new even for open E (apart from those which are trivial in this case) and also on R-n.

We study the Dirichlet problem for p-harmonic functions on metric spaces with respect to arbitrary compactifications. A particular focus is on the Perron method, and as a new approach to the invariance problem we introduce Sobolev-Perron solutions. We obtain various resolutivity and invariance results, and also show that most functions that have earlier been proved to be resolutive are in fact Sobolev-resolutive. We also introduce (Sobolev)-Wiener solutions and harmonizability in this nonlinear context, and study their connections to (Sobolev)-Perron solutions, partly using Q-compactifications.

Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.

Jiménez, Leslie

Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. Univ Chile, Chile.

Given a compact Riemann surface X with an action of a finite group G, the group algebra Q[G] provides an isogenous decomposition of its Jacobian variety JX, known as the group algebra decomposition of JX. We consider the set of equisymmetric Riemann surfaces M(2n -1, D-2n, theta) for all n amp;gt;= 2. We study the group algebra decomposition of the Jacobian JX of every curve X is an element of M (2n - 1, D-2n, theta) for all admissible actions, and we provide affine models for them. We use the topological equivalence of actions on the curves to obtain facts regarding its Jacobians. We describe some of the factors of JX as Jacobian (or Prym) varieties of intermediate coverings. Finally, we compute the dimension of the corresponding Shimura domains.

Newtonian spaces generalize first-order Sobolev spaces to abstract metric measure spaces. In this paper, we study regularity of Newtonian functions based on quasi-Banach function lattices. Their (weak) quasi-continuity is established, assuming density of continuous functions. The corresponding Sobolev capacity is shown to be an outer capacity. Assuming sufficiently high integrability of upper gradients, Newtonian functions are shown to be (essentially) bounded and (Hölder) continuous. Particular focus is put on the borderline case when the degree of integrability equals the “dimension of the measure”. If Lipschitz functions are dense in a Newtonian space on a proper metric space, then locally Lipschitz functions are proven dense in the corresponding Newtonian space on open subsets, where no hypotheses (besides being open) are put on these sets.