相关论文: Energy estimates and cavity interaction for a crit…

Given a family of critical points $u_{\epsilon}:M^n\to\mathbb{C}$ for the complex Ginzburg--Landau energies \begin{align*} &E_\epsilon(u)=\int_{M}\left(\frac{|du|^2}{2}+\frac{(1-|u|^2)^2}{4\epsilon^2}\right), \end{align*} on a manifold $M$,…

In this paper we give an explicit sufficient condition for the affine map $u_\lambda(x):=\lambda x$ to be the global energy minimizer of a general class of elastic stored-energy functionals $I(u)=\int_{\Omega} W(\nabla u)\,dx$ in three…

We study the phenomenon of cavitation for the displacement boundary value problem of radial, isotropic compressible elasticity for a class of stored energy functions of the form $W(F) + h(\det F)$, where $W$ grows like $||F||^n$, and $n$ is…

Let $\Omega\in\mathbb{R}^n$ be the region occupied by a body and let $\mathbf{x}_0$ be a flaw point in $\Omega$. Let $E(\cdot)$ be an energy functional (defined on some appropriate admissible set of deformations of $\Omega$). For $V>0$…

In a convex domain $\O\subset\R^3$, we consider the minimization of a 3D-Ginzburg-Landau type energy $E_\v(u)=1/2\int_\O|\n u|^2+\frac{1}{2\v^2}(a^2-|u|^2)^2$ with a discontinuous pinning term $a$ among $H^1(\O,\C)$-maps subject to a…

In this paper we establish $W^{1,p}$ estimates for solutions $u_\varepsilon$ to Laplace's equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, $C^1$ domain $\Omega_{\varepsilon, \eta}$ in…

We study a variational Ginzburg-Landau type model depending on a small parameter $\varepsilon>0$ for (tangent) vector fields on a $2$-dimensional Riemannian manifold $S$. As $\varepsilon\to 0$, these vector fields tend to have unit length…

Let $\Omega \subset {R}^n,$ $n \geq 3,$ be a bounded open set, $x=(x_1,x_2,\ldots,x_n)$ a generic point which belongs to $\Omega,$ $u \colon \Omega \to {R}^N ,$ $N>1,$ and $ Du=(D_\alpha u^i)$, $D_\alpha = \partial/\partial x_\alpha, $…

We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the {\it pinning domains}. These…

Let $\mathbb{D}(u)$ be the Dirichlet energy of a map $u$ belonging to the Sobolev space $H^1_{u_0}(\Omega;\mathbb{R}^2)$ and let $A$ be a subclass of $H^1_{u_0}(\Omega;\mathbb{R}^2)$ whose members are subject to the constraint $\det \nabla…

We consider a boundary value problem for the $p$-Laplacian, posed in the exterior of small cavities that all have the same $p$-capacity and are anchored to the unit sphere in $\mathbb{R}^d$, where $1<p<d.$ We assume that the distance…

For $ p \in (1,N)$ and a domain $\Omega$ in $\mathbb{R}^N$, we study the following quasi-linear problem involving the critical growth: \begin{eqnarray*} -\Delta_p u - \mu g|u|^{p-2}u = |u|^{p^{*}-2}u \ \mbox{ in } \mathcal{D}_p(\Omega),…

We study a variational model in nonlinear elasticity allowing for cavitation which penalizes both the volume and the perimeter of the cavities. Specifically, we investigate the approximation (in the sense of {\Gamma}-convergence) of the…

We establish a new estimate for the Ginzburg-Landau energies $E_{\epsilon}(u)=\int_M\frac{1}{2}|du|^2+\frac{1}{4\epsilon^2}(1-|u|^2)^2$ of complex-valued maps $u$ on a compact, oriented manifold $M$ with $b_1(M)\neq 0$, obtained by…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^2$. For $\epsilon>0$ small, we construct non-constant solutions to the Ginzburg-Landau equations $-\Delta u=\frac{1}{\epsilon^2}(1-|u|^2)u$ in $\Omega$ such that on $\partial \Omega$ u…

The problem of the sudden growth and coalescence of voids in elastic media is considered. The Dirichlet energy is minimized among incompressible and invertible Sobolev deformations of a two-dimensional domain having $n$ microvoids of radius…

We study the nodal solutions of the Lane Emden Dirichlet problem $-\Delta u = |u|^{p-1}u with DBC on a smooth bounded domain $\Omega$ in $\IR^2$ and where $p>1$. We consider solutions $u_p$ satisfying $p \int_{\Omega}\abs{\nabla u_p}^2\to…

For dimensions $N \geq 4$, we consider the Br\'ezis-Nirenberg variational problem of finding \[ S(\epsilon V) := \inf_{0\not\equiv u\in H^1_0(\Omega)} \frac{\int_\Omega |\nabla u|^2 \, dx +\epsilon \int_\Omega V\, |u|^2 \,…

This paper studies the Dirichlet problem for Laplace's equation in a domain $\Omega_{\varepsilon, \eta}$ perforated with small holes, where $\varepsilon$ represents the scale of the minimal distances between holes and $\eta$ the ratio…

This paper is concerned with the incompressible Euler equations. In Onsager's critical classes we provide explicit formulas for the Duchon-Robert measure in terms of the regularization kernel and a family of vector-valued measures…