### A Cauchy problem for ${u}_{t}-\Delta u={u}^{p}\phantom{\rule{4pt}{0ex}}\text{with}\phantom{\rule{4pt}{0ex}}0\<p\<1$. Asymptotic behaviour of solutions

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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 ${m}^{\text{'}}$ (the magnetization) of two variables ${x}^{\text{'}}$: ${E}_{\epsilon}\left({m}^{\text{'}}\right)=\epsilon \int {|{\nabla}^{\text{'}}\xb7{m}^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla}^{\text{'}}{|}^{-1/2}{\nabla}^{\text{'}}\xb7{m}^{\text{'}}\right|}^{2}d{x}^{\text{'}}$. We are interested in the behavior of minimizers as $\epsilon \to 0$. They are expected to be ${S}^{1}$-valued maps ${m}^{\text{'}}$ of vanishing distributional divergence ${\nabla}^{\text{'}}\xb7{m}^{\text{'}}=0$, so that appropriate boundary conditions enforce line discontinuities. For finite $\epsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel walls. Néel...

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

The celebrated criterion of Petrowsky for the regularity of the latest boundary point, originally formulated for the heat equation, is extended to the so-called p-parabolic equation. A barrier is constructed by the aid of the Barenblatt solution.

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

Let $\Omega $ be a bounded open subset of ${\mathbb{R}}^{n}$, $n>2$. In $\Omega $ we deduce the global differentiability result $$u\in {H}^{2}(\Omega ,{\mathbb{R}}^{N})$$ for the solutions $u\in {H}^{1}(\Omega ,{\mathbb{R}}^{n})$ of the Dirichlet problem $$u-g\in {H}_{0}^{1}(\Omega ,{\mathbb{R}}^{N}),-\sum _{i}{D}_{i}{a}^{i}(x,u,Du)={B}_{0}(x,u,Du)$$ 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.

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.

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...

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 $$div\left({u}^{-}{v}^{\to}\right)\in {H}^{1}\left({R}^{d}\right)$$ which include as a particular case, the result of [3].