We prove that the inverse of the Stokes operator in two dimensions, maps the Hardy space $\mathcal {H}^1$ into $L^\infty$.

In this paper, we examine the question of optimality conditions in terms of Young's measures for a relaxed problem associated to a one-dimensional nonconvex problem of the calculus of variations of the type $\inf \int_a^b f(x,u(x),u' (x))dx.$ Once these conditions are established, we set sufficient conditions for the existence of solutions of the nonconvex problem.

Solutions of a class of fifth-order model evolution equations corresponding to initial data in relatively weak function spaces are shown to exhibit a smoothing effect of the type of Kato. These models include the next hierarchy of the Korteweg-de Vries equation. It is interesting to observe that conditions that guarantee smoother solutions in some of these weaker function spaces are exactly the ones that allow for the equation to admit solitary-wave solutions of the characteristic $\text{sech}^{2}$ profile of the K-dV equation.

Let $B$ be the unit ball of $\Bbb R^N$, $N \ge 3$. We consider the problem $\Delta u = f(|x|)u^{p-\varepsilon}$ in $B$, $u > 0$ in $B$, $u = 0$ on $\partial B$, where $f\in C^\infty(\Bbb R, \Bbb R)$, $p = (N+2)/(N-2)$, $\varepsilon\ge 0$. First, we study the behavior of the minimizing radially symmetric solutions of the problem when $\varepsilon\to 0^+$. According to the (local or global) monotony of $f$, they converge to a solution of the critical problem or they blow up. The two cases are described. As a consequence, for large $N$ and all $\varepsilon > 0$, the critical problems with $f(r) = 1+\varepsilon r^k$ have minimizing radially symmetric solutions. They necessarily blow up when $\varepsilon\to 0^+$. Here again, we describe the blow up.

We give the general limit form of a sequence of obstacle problems relative to a family of equivalent regular Dirichlet forms of diffusion type.

An existence theorem is established for the initial-boundary-value problem for the system $\frac{\partial}{\partial t} \alpha (u) - \Delta u \ni \sigma (u)|\nabla\phi|^2,$ div$(\sigma (u) \nabla \phi) =0$ in the case where $\alpha$ is a maximal monotone graph in $\Bbb R$ and $\sigma$ is a continuous, nonnegative function on $\Bbb R$ such that $ \sigma(s)=0$ if and only if $s\ge a$ for some $a>0.$ A solution is constructed as the limit of a sequence of classical weak solutions of the regularized problems. Due to the possible degeneracy of the system, oscillations in the approximating solution sequence can persist. We prove that the failure of strong convergence is concentrated in a set which does not really matter in our passage to the limit.

In this paper we consider the existence, uniqueness and the asymptotic behavior of the solutions of Schrödinger equations in bounded domains with nonlinear dissipative boundary conditions; the existence and uniqueness are proved by means of nonlinear semigroup theory and the asymptotic behavior is obtained by establishing decay rates for the energy.

We study entire solutions (i.e., solutions defined in all of $\mathcal {R}^{n+1}$) of second-order nonlinear parabolic equations with linear principal part. Under appropriate hypotheses, we establish existence and uniqueness of an entire solution vanishing at infinity. More generally, we discuss existence and uniqueness of an entire solution approaching a given heat polynomial at infinity. When specialized to the linear homogeneous parabolic equation, containing no zero-order term, our results yield a Liouville theorem stating that an entire and bounded solution must be constant; certain asymptotic behavior of the coefficients at infinity however is required. Our methods involve the establishment of a priori bounds on entire solutions, first for the nonhomogeneous heat equation and then for a more general linear parabolic equation; we use these bounds with a Schauder continuation technique to study the nonlinear equation.

Consider a system of functional differential equations with finite delay $$ x'(t)=F(t,x_t),\quad x\in\Bbb R^n. \tag1 $$ We extend a result of Hering and show that if there exists a continuous functional $V:\Bbb R^+\times C_H\to\Bbb R^+$ and a constant $\gamma>0$ such that $$ \begin{align} &W_1(\phi(0)|)\le V(t,\phi)\le W_2(|\phi|_h)+W_3(\|\phi\|),\\ &V'_{(1)}(t,\phi)\le-W_4(|\phi|_h),\quad\text{and}\quad W_1(r)-W_3(r)>0\quad\text{for }r\in(0,\gamma), \end{align} $$ then the zero solution of (1) is uniformly asymptotically stable. Here $|\cdot|_h$ is a seminorm on the space $C([-h,0],\Bbb R^n)$. This result generalizes some very important theorems in the literature including Burton and Krasovskii's theorems. We also show that Krasovskii's second theorem is a corollary of his first theorem.

We consider the operator $ H=\varepsilon h+\cos(\alpha\cdot j+\vartheta) $ for $j\in\mathbb{Z}^\nu, $ where $h$ is self-adjoint, translation invariant and finite range. The vector $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_\nu)$ is assumed to have the diophantine property $ |j\cdot\alpha\,\text{mod}\,2\pi|\ge C/|j|^2, $ where $j\ne0$ and $C$ is some constant. For $\varepsilon$ sufficiently small we prove that $H$ has pure point spectrum for almost every $\vartheta$. Moreover, every polynomially bounded eigenfunction of $H$ decays exponentially fast. Finally, we will show how this operator comes up in the study of electrons in a transverse magnetic field subject to a two dimensional periodic potential.