Search results for "iPSC"

Sobolev and bounded variation functions on metric measure spaces

2014

International audience

Reliable numerical solution of a class of nonlinear elliptic problems generated by the Poisson-Boltzmann equation

2020

We consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson-Boltzmann equation (PBE). We prove mathematical correctness of the problem, study a suitable class of approximations, and deduce guaranteed and fully computable bounds of approximation errors. The latter goal is achieved by means of the approach suggested in [S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp., 69:481-500, 2000] for convex variational problems. Moreover, we establish the error identity, which defines the error measure natural for the considered class of problems and show that it yields computa…

Rademacherin lause

2008

Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces

2020

This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a -hypersurface without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure . As a consequence, we show that any Lipschitz map defined on a subset of a Carnot group of Hausdorff dimension 12, with…

Lipschitz Carnot-Carathéodory Structures and their Limits

2022

AbstractIn this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carathéodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not…

Fractional Maximal Functions in Metric Measure Spaces

2013

Abstract We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.

Numeeriset yhtälönratkaisumenetelmät

2017

Tämän tutkielman tavoitteena on käsitellä numeerisia yhtälönratkaisumenetelmiä matematiikan aineenopettajan näkökulmasta ja toimia lukion numeerisen matematiikan kurssin opettajan taustamateriaalina. Keskeinen sisältö käsittelee Lipschitz-jatkuvuutta, iteraatiota sekä Newton-Raphsonin menetelmää. Yhtälönratkaisu ja kahden lausekkeen yhtäsuuruuksien vertailu palautuu aina matematiikan klassiseen ongelmaan funktion nollakohdan etsimisestä. Keskeiset numeerisen yhtälönratkaisun metodit ovat rekursio ja iteraatio. Rekursio tarkoittaa oleellisesti toistoa. Iteraatiossa edellinen likiratkaisu ohjaa tarkentavasti seuraavan likiratkaisun laskentaa. Tällöin muodostuu tarkentuvien likiratkaisuiden lu…

Existence and classification of critical points for nondifferentiable functions

2004

A general min-max principle established by Ghoussoub is extended to the case of functionals which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function. Some topological properties of the min-max-generated critical points in such a framework are then pointed out.

Solvability of a first order system in three-dimensional non-smooth domains

1985

summary:A system of first order partial differential equations is studied which is defined by the divergence and rotation operators in a bounded nonsmooth domain $\Omega\subset \bold R^3$. On the boundary $\delta\Omega$, the vanishing normal component is prescribed. A variational formulation is given and its solvability is investigated.

Two examples related to conical energies

2022

In a recent article we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and M\"{o}rters, the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.