31 research outputs found
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 -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
Analysis of a General Family of Regularized Navier-Stokes and MHD Models
We consider a general family of regularized Navier-Stokes and
Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian
manifolds with or without boundary, with n greater than or equal to 2. This
family captures most of the specific regularized models that have been proposed
and analyzed in the literature, including the Navier-Stokes equations, the
Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha
model, the Simplified Bardina model, the Navier-Stokes-Voight model, the
Navier-Stokes-alpha-like models, and certain MHD models, in addition to
representing a larger 3-parameter family of models not previously analyzed. We
give a unified analysis of the entire three-parameter family using only
abstract mapping properties of the principle dissipation and smoothing
operators, and then use specific parameterizations to obtain the sharpest
results. We first establish existence and regularity results, and under
appropriate assumptions show uniqueness and stability. We then establish
results for singular perturbations, including the inviscid and alpha limits.
Next we show existence of a global attractor for the general model, and give
estimates for its dimension. We finish by establishing some results on
determining operators for subfamilies of dissipative and non-dissipative
models. In addition to establishing a number of results for all models in this
general family, the framework recovers most of the previous results on
existence, regularity, uniqueness, stability, attractor existence and
dimension, and determining operators for well-known members of this family.Comment: 37 pages; references added, minor typos corrected, minor changes to
revise for publicatio