Prod'homme, Maxime

This thesis deals with the optimal transport problem, in particular with regularity properties shared by optimal transport maps. The first part of this manuscript provides a new proof of the Caffarelli contraction theorem, stating that the optimal transport map from the gaussian measure to a measure with a uniformly log-concave density with respect...

Colombo, Maria Haffter, Silja
Published in
Annals of PDE

We consider the SQG equation with dissipation given by a fractional Laplacian of order α

Pegon, Marc

This thesis is dedicated to the study of two separate geometric variational problems involving nonlocal energies: firstly, the geometry and singularities of fractional harmonic maps,and secondly, an iso perimetric problem with a repulsive integrable potential inspired by Gamow’s liquid drop model for the atomic nucleus. On the first topic, we impro...

Chamorro, Diego Lemarié-Rieusset, Pierre-Gilles Mayoufi, Kawther

We study the role of the pressure in the partial regularity theory for weak solutions of the Navier–Stokes equations. By introducing the notion of dissipative solutions, due to Duchon & Robert, we will provide a generalization of the Caffarelli, Kohn and Nirenberg theory. Our approach gives a new enlightenment of the role of the pressure in this th...

Passarelli di Napoli, Antonia
Published in
Nonlinear Differential Equations and Applications NoDEA

We establish the C1,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C^{1,\alpha}}$$\end{document} partial regularity of vectorial minimizers of non autonomous convex ...

Mácha, Václav
Published in
Open Mathematics

In the presented work, we study the regularity of solutions to the generalized Navier-Stokes problem up to a C 2 boundary in dimensions two and three. The point of our generalization is an assumption that a deviatoric part of a stress tensor depends on a shear rate and on a pressure. We focus on estimates of the Hausdorff measure of a singular set ...

Kokarev, Gerasim
Published in
Advances in Mathematics

We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. The principal results include the general regularity properties of ...

Cowan, Craig Ghoussoub, Nassif
Published in
Calculus of Variations and Partial Differential Equations

We examine the fourth order problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta ^2 u = \lambda f(u) $$\end{document} in \documentclass[12pt]{minimal} \usepac...

Duzaar, Frank Habermann, Jens
Published in
Journal of Evolution Equations

We study non-linear parabolic systems with non-standard p(z)-growth conditions and establish that the gradient of weak solutions is locally Hölder continuous with Hölder exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{up...

Bögelein, V. Foss, M. Mingione, G.
Published in
Mathematische Zeitschrift

Under the only assumption of continuous coefficients, we prove a partial Hölder continuity result for solutions to parabolic systems with polynomial growth. A key component throughout the argument is the use of DiBenedetto’s intrinsic geometry (Degenerate Parabolic Equations. Universitext. Springer, New York, 1993) to accommodate the inhomogeneity ...