Adaptive boundary element methods with convergence rates

This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasi-optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods. In particular, no saturation-type assumption is used. The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.Comment: 48 pages. A journal version. The previous version (v3) is a bit lengthie

Axisymmetric critical points of a nonlocal isoperimetric problem on the two-sphere

On the two dimensional sphere, we consider axisymmetric critical points of an isoperimetric problem perturbed by a long-range interaction term. When the parameter controlling the nonlocal term is sufficiently large, we prove the existence of a local minimizer with arbitrary many interfaces in the axisymmetric class of admissible functions. These local minimizers in this restricted class are shown to be critical points in the broader sense (i.e., with respect to all perturbations). We then explore the rigidity, due to curvature effects, in the criticality condition via several quantitative results regarding the axisymmetric critical points.Comment: 26 pages, 6 figures. This version is to appear in ESAIM: Control, Optimisation and Calculus of Variation

A prescribed scalar and boundary mean curvature problem and the Yamabe classification on asymptotically Euclidean manifolds with inner boundary

We consider the problem of finding a metric in a given conformal class with prescribed non-positive scalar curvature and non-positive boundary mean curvature on an asymptotically Euclidean manifold with inner boundary. We obtain a necessary and sufficient condition in terms of a conformal invariant of the zero sets of the target curvatures for the existence of solutions to the problem and use this result to establish the Yamabe classification of metrics in those manifolds with respect to the solvability of the prescribed curvature problem.Comment: 25 page

On uniqueness of weak solutions to the second boundary value problem for generated prescribed Jacobian equations

We prove that two Aleksandrov solutions of a generated prescribed Jacobian equation have the same gradients at points where they are both differentiable. For the optimal transportation case where two solutions can be translated to agree at a point without changing the $g$-subdifferential at that point, we recover the uniqueness up to a constant of solutions. For the general case, our result is a new proof with less regularity assumptions of a key theorem recently used to prove the uniqueness of solutions

A Scaling Approach to Elliptic Theory for Geometrically-Natural Differential Operators with Sobolev-Type Coefficients

We develop local elliptic regularity for operators having coefficients in a range of Sobolev-type function spaces (Bessel potential, Sobolev-Slobodeckij, Triebel-Lizorkin, Besov) where the coefficients have a regularity structure typical of operators in geometric analysis. The proofs rely on a nonstandard technique using rescaling estimates and apply to operators having coefficients with low regularity. For each class of function space for an operator's coefficients, we exhibit a natural associated range of function spaces of the same type for the domain of the operator and we provide regularity inference along with interior estimates. Additionally, we present a unified set of multiplication results for the function spaces we consider.Comment: 73 pages, 3 figure