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A compactness result in thin-film micromagnetics and the optimality of the Néel wall

Radu Ignat, Felix Otto (2008)

Journal of the European Mathematical Society

In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for S 1 -valued maps m ' (the magnetization) of two variables x ' : E ε ( m ' ) = ε | ' · m ' | 2 d x ' + 1 2 | ' | - 1 / 2 ' · m ' 2 d x ' . We are interested in the behavior of minimizers as ε 0 . They are expected to be S 1 -valued maps m ' of vanishing distributional divergence ' · m ' = 0 , so that appropriate boundary conditions enforce line discontinuities. For finite ε > 0 , these line discontinuities are approximated by smooth transition layers, the so-called Néel walls. Néel...

A counterexample to Schauder estimates for elliptic operators with unbounded coefficients

Enrico Priola (2001)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider a homogeneous elliptic Dirichlet problem involving an Ornstein-Uhlenbeck operator in a half space R + 2 of R 2 . We show that for a particular initial datum, which is Lipschitz continuous and bounded on R + 2 , the second derivative of the classical solution is not uniformly continuous on R + 2 . In particular this implies that the well known maximal Hölder-regularity results fail in general for Dirichlet problems in unbounded domains involving unbounded coefficients.

A geometric improvement of the velocity-pressure local regularity criterion for a suitable weak solution to the Navier-Stokes equations

Jiří Neustupa (2014)

Mathematica Bohemica

We deal with a suitable weak solution ( 𝐯 , p ) to the Navier-Stokes equations in a domain Ω 3 . We refine the criterion for the local regularity of this solution at the point ( 𝐟 x 0 , t 0 ) , which uses the L 3 -norm of 𝐯 and the L 3 / 2 -norm of p in a shrinking backward parabolic neighbourhood of ( 𝐱 0 , t 0 ) . The refinement consists in the fact that only the values of 𝐯 , respectively p , in the exterior of a space-time paraboloid with vertex at ( 𝐱 0 , t 0 ) , respectively in a ”small” subset of this exterior, are considered. The consequence is that...

A global differentiability result for solutions of nonlinear elliptic problems with controlled growths

Luisa Fattorusso (2008)

Czechoslovak Mathematical Journal

Let Ω be a bounded open subset of n , n > 2 . In Ω we deduce the global differentiability result u H 2 ( Ω , N ) for the solutions u H 1 ( Ω , n ) of the Dirichlet problem u - g H 0 1 ( Ω , N ) , - i D i a i ( x , u , D u ) = B 0 ( x , u , D u ) with controlled growth and nonlinearity q = 2 . The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.

A gradient estimate for solutions of the heat equation. II

Charles S. Kahane (2001)

Czechoslovak Mathematical Journal

The author obtains an estimate for the spatial gradient of solutions of the heat equation, subject to a homogeneous Neumann boundary condition, in terms of the gradient of the initial data. The proof is accomplished via the maximum principle; the main assumption is that the sufficiently smooth boundary be convex.

A Harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries are C1,α.

Luis A. Caffarelli (1987)

Revista Matemática Iberoamericana

This is the first in a series of papers where we intend to show, in several steps, the existence of classical (or as classical as possible) solutions to a general two-phase free-boundary system. We plan to do so by:(a) constructing rather weak generalized solutions of the free-boundary problems,(b) showing that the free boundary of such solutions have nice measure theoretical properties (i.e., finite (n-1)-dimensional Hausdorff measure and the associated differentiability properties),(c) showing...

A Note on Div-Curl Lemma

Gala, Sadek (2007)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 42B30, 46E35, 35B65.We prove two results concerning the div-curl lemma without assuming any sort of exact cancellation, namely the divergence and curl need not be zero, and d i v ( u v ) H 1 ( R d ) which include as a particular case, the result of [3].

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