Ginzburg-Landau equation and motion by mean curvature, I: Convergence

In this paper we study the asymptotic behavior (∈→0) of the Ginzburg-Landau equation: $$u_l^\varepsilon - \Delta u^\varepsilon + \frac{1}{{\varepsilon ^2 }}f(u^\varepsilon ) = 0.$$ . where the unknownu ^∈ is a real-valued function of [0. ∞)× ^Rd , and the given nonlinear functionf(u) = 2u(u ²?1) is the derivative of a potential W(u) = (u ²?l)²/2 with two minima of equal depth. We prove that there are a subsequence ∈_n and two disjoint, open subsetsP, N of (0, ∞) ×R ^d satisfying $$u^{\varepsilon _n } \to 1_\mathcal{P} - 1_\mathcal{N} , as n \to \infty . $$ uniformly inP andN (here 1 _A is the indicator of the setA). Furthermore, the Hausdorff dimension of the interface Γ = complement of (P∪N) ? (0, ∞)×R ^d is equal tod and it is a weak solution of the mean curvature flow as defined in [13,92]. If this weak solution is unique, or equivalently if the level-set solution of the mean curvature flow is “thin,” then the convergence is on the whole sequence. We also show thatu ^∈n has an expansion of the form $$u^{\varepsilon _n } (t,x) = q\left( {\frac{{d(t,x) + O(\varepsilon _n )}}{{\varepsilon _n }}} \right).$$ whereq(r) = tanh(r) is the traveling wave associated to the cubic nonlinearityf, O(∈) → 0 as ∈ → 0, andd(t, x) is the signed distance ofx to thet-section of Γ. We prove these results under fairly general assumptions on the initial data,u ⁰. In particular we donot assume thatu ^∈(0.x) = q(d(0,x)/∈), nor that we assume that the initial energy, ε^∈(u ^∈(0, .)), is uniformly bounded in ∈. Main tools of our analysis are viscosity solutions of parabolic equations, weak viscosity limit of Barles and Perthame, weak solutions of mean curvature flow and their properties obtained in [13] and Ilmanen’s generalization of Huisken’s monotonicity formula.     

Spectral asymptotics and trace formulas for differential operators with unbounded coefficients

In the space L ²[0, π], we consider the operators $$ L = L_0 + V, L_0 = - y'' + (\nu ^2 - 1/4)r^{ - 2} y (\nu \geqslant 1/2) $$ with the Dirichlet boundary conditions. The potential V is the operator of multiplication by a function (in general, complex-valued) in L ²[0, π] satisfying the condition $$ \int\limits_0^\pi {r^\varepsilon } (\pi - r)^\varepsilon |V(r)|dr < \infty , \varepsilon \in [0,1] $$ . We prove the trace formula Σ _n=1 ^∞ [µ _n ? λ _n ? Σ _k=1 ^m α _k ⁽ⁿ⁾ ] = 0.     

H convergence for quasi-linear elliptic equations with quadratic growth

We consider in this paper the limit behavior of the solutionsu ^? of the problem $$\begin{gathered} - div(a^\varepsilon Du^\varepsilon ) + \gamma u^\varepsilon = H^\varepsilon (x, u^\varepsilon , Du^\varepsilon ), \hfill \\ u^\varepsilon \in H_0^1 (\Omega ) \cap L^\infty (\Omega ), \hfill \\ \end{gathered}$$ whereH ^? has quadratic growth inDu ^? anda ^? (x) is a family of matrices satisfying the general assumptions of abstract homogenization. We also consider the problem $$\begin{gathered} - div(a^\varepsilon Du^\varepsilon ) + G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) = f \in H^{ - 1} (\Omega ), \hfill \\ u^\varepsilon \in H_0^1 (\Omega ), G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) \in L^1 (\Omega ), u^\varepsilon G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) \in L^1 (\Omega ) \hfill \\ \end{gathered}$$ whereG ^? has quadratic growth inDu ^? and satisfiesG ^? (x, s, ξ)s ≥ 0. Note that in this last modelu ^? is in general unbounded, which gives extra difficulties for the homogenization process. In both cases we pass to the limit and obtain an homogenized equation having the same structure.     

BOUNDARY VALUE PROBLEMS OF SINGULARLY PERTURBED INTEGRO－DIFFERENTIAL EQUATIONS

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Relative average widths of Sobolev spaces in L 2(ℝ d )

In this paper, in order to consider the problems of relative width on ? ^d , we proposed definitions of relative average width which combine the ideas of the relative width and the average width. We established the smallest number M which make the following equality $$ \overline K _\sigma (U(W_2^\alpha ),M(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) $$ hold, where U(W ₂ ^α ) is the Riesz potential or Bessel potential of the unit ball in L ²(? ^k ) and the notations $ \overline K _\sigma $ (·, ·,L ₂(? ^d )) and $ \overline d _\sigma $ (·, L ₂(? ^d )) denote respectively the relative average width in the sense of Kolmogorov and the average width in the sense of Kolmogorov in their given order. In 2001, Subbotin and Telyakovskii got similar results on the relative width of Kolmogorov type. We also proved that $$ \overline K _\sigma (U(W_2^\alpha ) \cap B(L_2 (\mathbb{R}^d )),U(W_2^\beta ) \cap B(L_2 (\mathbb{R}^d ))L_2 (\mathbb{R}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 (\mathbb{R}^d )), $$ where 0 × β × α.     

Homogenization of the Neumann problem in perforated domains: an alternative approach

The main result of this paper is a compactness theorem for families of functions in the space SBV (Special functions of Bounded Variation) defined on periodically perforated domains. Given an open and bounded set ${\Omega\subseteq\mathbb{R}^n}$ , and an open, connected, and (?1/2, 1/2) ⁿ -periodic set ${P\subseteq\mathbb{R}^n}$ , consider for any ???>?0 the perforated domain ?? _?? :=????????? P. Let ${(u_\varepsilon)\subset SBV^p(\Omega_{\varepsilon})}$ , p?>?1, be such that ${\int_{\Omega_{\varepsilon}}\left|{\nabla{u}_\varepsilon}\right|^pdx+\mathcal{H}^{n-1}(S_{u_\varepsilon}\,\cap\,\Omega_{\varepsilon}) +\left\Vert{u_\varepsilon}\right\Vert_{L^p(\Omega_{\varepsilon})}}$ is bounded. Then, we prove that, up to a subsequence, there exists ${u\in GSBV^p\,\cap\, L^p(\Omega)}$ satisfying ${\lim_\varepsilon\left\Vert{u-u_\varepsilon}\right\Vert_{L^1(\Omega_{\varepsilon})}=0}$ . Our analysis avoids the use of any extension procedure in SBV, weakens the hypotheses on P to the minimal ones and simplifies the proof of the results recently obtained in Focardi et?al. (Math Models Methods Appl Sci 19:2065?C2100, 2009) and Cagnetti and Scardia (J Math Pures Appl (9), to appear). Among the arguments we introduce, we provide a localized version of the Poincaré-Wirtinger inequality in SBV. As a possible application we study the asymptotic behavior of a brittle porous material represented by the perforated domain ?? _?? . Finally, we slightly extend the well-known homogenization theorem for Sobolev energies on perforated domains.     

On the Distance to the Closest Matrix with Triple Zero Eigenvalue

The 2-norm distance from a matrix A to the set ${\mathcal{M}}$ of n × n matrices with a zero eigenvalue of multiplicity ≥3 is estimated. If $$Q(\gamma _1 ,\gamma _2 ,\gamma _3 ) = \left( {\begin{array}{*{20}c} A &amp; {\gamma _1 I_n } &amp; {\gamma _3 I_n } \\ 0 &amp; A &amp; {\gamma _2 I_n } \\ 0 &amp; 0 &amp; A \\ \end{array} } \right), n \geqslant 3,$$ then $$\rho _2 (A,{\mathcal{M}}) \geqslant {\mathop {max}\limits_{\gamma _1 ,\gamma _2 \geqslant 0,\gamma _3 \in {\mathbb{C}}}} \sigma _{3n - 2} (Q(\gamma _1 ,\gamma _2 ,\gamma _3 )),$$ where σⁱ(·)is the ith singular value of the corresponding matrix in the decreasing order of singular values. Moreover, if the maximum on the right-hand side is attained at the point $\gamma ^ * = (\gamma _1^ * ,\gamma _2^ * ,\gamma _3^ * )$ , where $\gamma _1^ * \gamma _2^ * \ne 0$ , then, in fact, one has the exact equality $$\rho _2 (A,{\mathcal{M}}) = \sigma _{3n - 2} (Q(\gamma _1^ * ,\gamma _2^ * ,\gamma _3^ * )).$$ This result can be regarded as an extension of Malyshev's formula, which gives the 2-norm distance from A to the set of matrices with a multiple zero eigenvalue.     

Exponential sums over primes and the prime twin problem

The purpose of this paper is to obtain an effective estimate of the exponential sum $\sum_{n\le x}\Lambda(n)e\left(\left(\frac{a}{q}+\beta\right)n\right)$ (where e(??)=e ^{2?? i ??} , ??,?????, (a,q)=1 and ?? is the von Mangoldt function) in the range ${(\log x)}^{1/2+\varepsilon}\le q\le \frac{x}{{(\log\log\log x)}^{1+\varepsilon}}$ and $|\beta|<\frac{1}{q{(\log\log\log x)}^{1+\varepsilon}}$ . It improves Daboussi??s estimate [2, Theorem 1] in the range q??(log?x) ^D and x(log?x)^?D ??q, D>0 and is valid in a wider range for ??.     

Symplectic mean curvature flow in CP 2

Let Σ be an immersed symplectic surface in CP ² with constant holomorphic sectional curvature k > 0. Suppose Σ evolves along the mean curvature flow in CP ². In this paper, we show that the symplectic mean curvature flow exists for long time and converges to a holomorphic curve if the initial surface satisfies ${|A|^2 \leq \lambda|H|^2 + \frac{2\lambda-1}{\lambda}k}$ and ${\cos\alpha\geq\sqrt{\frac{7\lambda-3}{3\lambda}}\left(\frac{1}{2} < \lambda\leq\frac{2}{3}\right) {\rm or} |A|^2\leq \frac{2}{3}|H|^2+\frac{4}{5}k\cos\alpha\, {\rm and} \cos\alpha\geq 1-\varepsilon}$ , for some ${\varepsilon}$ .     

A weighted Hardy inequality and nonexistence of positive solutions

In this article, we prove that the following weighted Hardy inequality $$\begin{array}{ll}\big(\frac{|{d-p}|}{p}\big)^{p}\, \int\limits_{\Omega}\, \frac{|{u}|^{p}}{|{x}|^{p}}\;d\mu \\ \quad \quad \le \int\limits_{\Omega}\,|{\nabla u}|^{p}\;d\mu+ \big(\frac{|{d-p}|}{p}\big)^{p-1}\,\textrm{sgn}(d-p)\,\int\limits_{\Omega}|{u}|^{p}\,\frac{(x^{t}Ax)^{p/2}}{|{x}|^{p}}\; d\mu \quad \quad \quad (1) \end{array}$$ holds with optimal Hardy constant ${\big(\frac{|d-p|}{p}\big)^{p}}$ for all ${u \in W^{1,p}_{\mu,0}(\Omega)}$ if the dimension d ≥ 2, 1 < p < d, and for all ${u \in W^{1,p}_{\mu,0}(\Omega{\setminus}\{0\})}$ if p > d ≥ 1. Here we assume that Ω is an open subset of ${\mathbb{R}^{d}}$ with ${0 \in \Omega}$ , A is a real d × d-symmetric positive definite matrix, c > 0, and $$ d \mu: = \rho(x) \,dx \qquad \textrm{with} \quad \rho(x) = c \cdot \exp(-\frac{1}{p}(x^{t}Ax)^{p/2}), \quad x \in\Omega.\quad \quad (2) $$ If p > d ≥ 1, then we can deduce from (1) a weighted Poincaré inequality on ${W^{1,p}_{\mu,0}(\Omega \setminus\{0\})}$ . Due to the optimality of the Hardy constant in (1), we can establish the nonexistence (locally in time) of positive weak solutions of a p-Kolmogorov parabolic equation perturbed by a singular potential in dimension d = 1, for 1 < p <  + ∞, and when Ω =  ]0, + ∞[.     

On weak solutions of linear coercive integral equations in spacesL 2(X; ℋ k )

Пусть (X,A, μ) - полное про странство с σ-конечно й мерой, и пусть \(\overline {\mu \times \mu } \) . - замык ание меры μ×μ. Пусть далееg: X×X→C - квадратично интегрируемая функц ия по мере \(\overline {\mu \times \mu } \) . Рассматривается лин ейное интегральное у равнение (слабого) типа (1) (1) $$u(t) + A(\mathop \smallint \limits_x g(t,s)u(s)d\mu ) = f(t)\Pi .B.B\,X,$$ гдеА - максимальное р асширение L _k ^′ (в простр анстве ХëрмандераH ₁=B₂к) соотв ествующего линейного (псевдодиф ференциального) опер атораL: S→S; иS обозначает класс Щварца функций Rⁿ→-C. Уст анавливается сущест вование (слабых) решений (1) при н екотором условии коэрпитивно сти на оператор (2) (2) $$(L\Psi )(t) = \Psi (t) + \int\limits_x {g(t,s)L(\Psi (s))d\mu ,} $$ где Ψ принадлежит про странстувуD(Х, S) всех конечно-значных функ ций изX→S. Далее, изучается обобщенна я обратимость максим ального расширения оператора L. Наконец, пр иводится некоторое алгебраическое усло вие, обеспечивающее коэрцитивность L.     

Two Linear Transformations each Tridiagonal with Respect to an Eigenbasis of the other; Comments on the Parameter Array

Let ${\mathbb K} $ denote a field. Let it d denote a nonnegative integer and consider a sequence p=( $\theta_i, \theta^*_i,i=0...d; \varphi_j, \phi_j,j=1...{\it d})$ consisting of scalars taken from ${\mathbb K} $ . We call p a parameter array whenever: (PA1) $\theta_i \not=\theta_j, \; \theta^*_i\not=\theta^*_j$ if $$i\not=j$, $(0 \leq i, j\leq d)$; (PA2) $ \varphi_i\not=0$, $\phi_i\not=0$ $(1 \leq i \leq d)$; (PA3) $\varphi_i = \phi_1 \sum_{h=0}^{i-1} ({\theta_h-\theta_{d-h}})/({\theta_0-\theta_d}) + (\theta^*_i-\theta^*_0)(\theta_{i-1}-\theta_d)$ $(1 \leq i \leq d)$; (PA4) $\phi_i = \varphi_1 \sum_{h=0}^{i-1} ({\theta_h-\theta_{d-h}})/({\theta_0-\theta_d}) + (\theta^*_i-\theta^*_0)(\theta_{d-i+1}-\theta_0)$ $(1 \leq i \leq d)$; (PA5) $(\theta_{i-2}-\theta_{i+1})(\theta_{i-1}-\theta_i)^{-1}$, $(\theta^*_{i-2}-\theta^*_{i+1})(\theta^*_{i-1}-\theta^*_i)^{-1}$$ are equal and independent of i for $2 \leq i \leq d-1$ . In Terwilliger, J. Terwilliger, Linear Algebra Appl., Vol. 330(2001) p. 155 we showed the parameter arrays are in bijection with the isomorphism classes of Leonard systems. Using this bijection we obtain the following two characterizations of parameter arrays. Assume p satisfies PA1 and PA2. Let A, B,A^*, B^* denote the matrices in ${Mat}_{{\it d}+1}$ ( ${\mathbb K} $ ) which have entries A _ii =θ _i , B _ii =θ _d-i , A ^* _ii =θ^* _i , B ^* _ii =θ^* _i (0 ≤ i ≤ d), A _i,i-1=1, B _i,i-1=1, A ^* _i-1,i =φ _i , B ^* _i-1,i =? _i (1 ≤ i ≤ d), and all other entries 0. We show the following are equivalent: (i) p satisfies PA3–PA5; (ii) there exists an invertible G ∈Mat _d+1( ${\mathbb K} $ ) such that G ^?1 AG=B and G ^?1 A ^* G=B ^*; (iii) for 0 ≤ i ≤ d the polynomial $$ \sum_{n=0}^i \frac{ (\lambda-\theta_0) (\lambda-\theta_1) \cdots (\lambda-\theta_{n-1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1) \cdots (\theta^*_i-\theta^*_{n-1}) } {\varphi_1\varphi_2\cdots \varphi_n}$$ is a scalar multiple of the polynomial $$\sum_{n=0}^i \frac{ (\lambda-\theta_d) (\lambda-\theta_{d-1}) \cdots (\lambda-\theta_{d-n+1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1) \cdots (\theta^*_i-\theta^*_{n-1}) } {\phi_1\phi_2\cdots \phi_n}.$$ We display all the parameter arrays in parametric form. For each array we compute the above polynomials. The resulting polynomials form a class consisting of the q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk, affine q-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, Bannai/Ito, and Orphan polynomials. The Bannai/Ito polynomials can be obtained from the q-Racah polynomials by letting q tend to ?1. The Orphan polynomials have maximal degree 3 and exist for ( ${\mathbb K} $ )=2 only. For each of the polynomials listed above we give the orthogonality, 3-term recurrence, and difference equation in terms of the parameter array.     

Dimension Reduction for Finite Trees in \varvec{\ell _1}

We show that every $n$ -point tree metric admits a $(1+\varepsilon )$ -embedding into $\ell _1^{C(\varepsilon ) \log n}$ , for every $\varepsilon > 0$ , where $C(\varepsilon ) \le O\big ((\frac{1}{\varepsilon })^4 \log \frac{1}{\varepsilon })\big )$ . This matches the natural volume lower bound up to a factor depending only on $\varepsilon $ . Previously, it was unknown whether even complete binary trees on $n$ nodes could be embedded in $\ell _1^{O(\log n)}$ with $O(1)$ distortion. For complete $d$ -ary trees, our construction achieves $C(\varepsilon ) \le O\big (\frac{1}{\varepsilon ^2}\big )$ .     

A method for summing Fourier integrals of functions from H p (E 2n + ), 0 < p < ∞

Suppose that H ^p (E _2n ⁺ ) is the Hardy space for the first octant $$E_{2n}^ + = \{ z \in \mathbb{C}^n :\operatorname{Im} z_j > 0, j = 1, \ldots ,n\} $$ and P _? ^l (f, x), l > 0, is the generalized Abel-Poisson means of a function f ? H ^p (E _2n ⁺ ). In this paper, we prove the inequalities $$C_1 (l,p)\widetilde\omega _l (\varepsilon ,f)_p \leqslant \left\| {f(x) - P_\varepsilon ^l (f,x)} \right\|_p \leqslant C_2 (l,p)\omega _l (\varepsilon ,f)_p ,$$ where $\widetilde\omega _l (\varepsilon ,f)_p $ and ω _l (?, f) _p are the integral moduli of continuity of lth order. For n = 1 and an integer l, this result was obtained by Soljanik.     

Homeomorphisms of the annulus with a transitive lift

Let f be a homeomorphism of the closed annulus A that preserves the orientation, the boundary components and that has a lift ${\tilde{f}}$ to the infinite strip à which is transitive. We show that, if the rotation numbers of both boundary components of A are strictly positive, then there exists a closed nonempty unbounded set ${B^{-} \subset \tilde{A}}$ such that B ^? is bounded to the right, the projection of B ^? to A is dense, ${B^{-}-(1, 0) \subset B^{-}}$ and ${\tilde{f}(B^{-}) \subset B^{-}}$ . Moreover, if p ₁ is the projection on the first coordinate of Ã, then there exists d > 0 such that, for any ${\tilde z \in B^{-}}$ , $$\limsup_{n\to\infty}\frac{p_1(\tilde f^n(\tilde z))-p_1(\tilde z)}{n}<-d.$$ In particular, using a result of Franks, we show that the rotation set of any homeomorphism of the annulus that preserves orientation, boundary components, which has a transitive lift without fixed points in the boundary is an interval with 0 in its interior.     

Exact multiplicity for the perturbed Q-curvature problem in {\mathbb{R}^N,N \geq 5}

Let N ≥ 5 and \({{\mathcal{D}}^{2,2} (\mathbb{R}^N)}\) denote the closure of \({C_0^\infty (\mathbb{R}^N)}\) in the norm \({\|u\|_{{\mathcal{D}}^{2,2} (\mathbb{R}^N)}^2 := \int\nolimits_{\mathbb{R}^N} |\Delta u|^2.}\) Let \({K \in C^2 (\mathbb{R}^N).}\) We consider the following problem for ? ≥ 0: $$(P_\varepsilon) \left\{\begin{array}{llll}{\rm Find} \, u \in {\mathcal{D}}^{2, 2} (\mathbb{R}^N) \, \, {\rm solving} :\\ \left.\begin{array}{lll}\Delta^2 u = (1+ \varepsilon K (x)) u^{\frac{N+4}{N-4}}\\ u > 0 \end{array}\right\}{\rm in} \, \mathbb{R}^N.\end{array}\right.$$ We show an exact multiplicity result for (P _? ) for all small ? > 0.     

Characterization of the Bernoulli-polynomials,exp andΨ by Nörlund's multiplication formula

The system of functional equations $$\forall p\varepsilon N_ + \forall (x,y)\varepsilon D:f(x,y) = \frac{1}{p}\sum\limits_{k = 0}^{p - 1} {f(x + ky,py)}$$ is suited to characterize the functions $$(x,y) \mapsto y^m B_m \left( {\frac{x}{y}} \right),m\varepsilon N,$$ B _m means them-th Bernoulli-polynomial, $$(x,y) \mapsto \exp (x)y(\exp (y) - 1)^{ - 1}$$ (for these functionsD =R ×R ₊) and $$(x,y) \mapsto \log y + \Psi \left( {\frac{x}{y}} \right)(D = R_ + \times R_ + )$$ as those continuous solutions of this system which allow a certain separation of variables and take on some prescribed function values.     

Random subgraphs of the 2D Hamming graph: the supercritical phase

We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is the Cartesian product of two complete graphs on n vertices. Let p be the edge probability, and write ${p={(1+\varepsilon)}/{(2(n-1))}}$ for some ${\varepsilon \in \mathbb{R}}$ . In Borgs et al. (Random Struct Alg 27:137–184, 2005; Ann Probab 33:1886–1944, 2005), the size of the largest connected component was estimated precisely for a large class of graphs including H(2,n) for ${\varepsilon \leq \Lambda V^{-1/3}}$ , where Λ >  0 is a constant and V = n ² denotes the number of vertices in H(2,n). Until now, no matching lower bound on the size in the supercritical regime has been obtained. In this paper we prove that, when ${\varepsilon \gg (\log{V})^{1/3} V^{-1/3}}$ , then the largest connected component has size close to ${2 \varepsilon V}$ with high probability. We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of p are supercritical. Barring the factor ${(\log{V})^{1/3}}$ , this identifies the size of the largest connected component all the way down to the critical p window.     

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets ${K\subset H}$ of a Hilbert space H by the metric entropy of the set K where the covering numbers ${N(K, \varepsilon)}$ of K by ${\varepsilon}$ -balls of H satisfy the Lorentz condition $$ \int\limits_{0}^{\infty} \left(\log N(K,\varepsilon) \right)^{r/s}\, d\varepsilon^{s} < \infty $$ for some fixed ${0 < r, s \le \infty}$ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 < r < ∞, s = ∞. The integral here is an improper Stieltjes integral. Moreover, we obtain optimal estimates of Gelfand numbers of absolutely convex hulls if the metric entropy satisfies the entropy condition $$\sup_{\varepsilon >0 }\varepsilon \left(\log N(K,\varepsilon) \right)^{1/r}\left(\log(2+\log N(K,\varepsilon))\right)^\beta < \infty$$ for some fixed 0 < r < ∞, ?∞ < β < ∞. Using inequalities between Gelfand and entropy numbers we also get optimal estimates of the metric entropy of the absolutely convex hull cov(K). As an interesting feature of the estimates, a sudden jump of the asymptotic behavior of Gelfand numbers as well as of the metric entropy of absolutely convex hulls occurs for fixed s if the parameter r crosses the point r = 2 and, if r = 2 is fixed, if the parameter β crosses the point β = 1. The results established in Hilbert spaces extend and recover corresponding results of several authors.     

Asymptotics for the minimization of a Ginzburg-Landau functional

LetΩ ? ?² be a smooth bounded simply connected domain. Consider the functional $$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$ on the classH _g ¹ ={u εH ¹(Ω; ?);u=g on ?Ω} whereg:?Ω? → ? is a prescribed smooth map with ¦g¦=1 on ?Ω? and deg(g, ?Ω)=0. Let uu _ε be a minimizer for E_ε onH _g ¹ . We prove that u_ε → u₀ in \(C^{1,\alpha } (\bar \Omega )\) as ε → 0, where u₀ is identified. Moreover \(\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 \) .