We consider non-interacting particles subject to a fixed external potential $V$ and a self-generated magnetic field $B$. The total energy includes the field energy $\beta \int B^2$ and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter $\beta$ tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and...

We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy $E_{\epsilon }\left(v\right)=\epsilon {\int }_{\Omega }|{\nabla }^2{v|}^{2}dx+{\epsilon }^{-1}{\int }_{\Omega }{\left(1-\right|\nabla v|}^2{)}^{2}dx$ over $v\in H^2\left(\Omega \right)$, where $\epsilon >0$ is a small parameter. Given $v\in W^{1,\infty }\left(\Omega \right)$ such that $\nabla v\in BV$ and $\left|\nabla v\right|=1$ a.e., we construct a family $\left\{v_{\epsilon }\right\}$ satisfying: $v_{\epsilon }\to v$ in $W^{1,p}\left(\Omega \right)$ and $E_{\epsilon }\left(v_{\epsilon }\right)\to \frac{1}{3}{\int }_{J_{\nabla v}}{\left|{\nabla }^{+}v-{\nabla }^{-}v\right|}^{3}d{\mathscr{H}}^{N-1}$ as $\epsilon$ goes to 0.

In a cholesteric liquid crystal the director field $n(x,y,z)$ tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known...

The distributional $k$-dimensional Jacobian of a map $u$ in the Sobolev space $W^{1,k-1}$ which takes values in the sphere $S^{k-1}$ can be viewed as the boundary of a rectifiable current of codimension $k$ carried by (part of) the singularity of $u$ which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary $M$ of codimension $k$ can be realized as Jacobian of a Sobolev map valued in $S^{k-1}$. In case $M$ is polyhedral, the...

We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data $u_0\in H^{s-1+\epsilon }$, $w_0\in H^{s-1}$ and $b_0\in H^s$ for $s>\frac{3}{2}$ and any $0<\epsilon <1$. The initial regularity of the micro-rotational velocity $w$ is weaker than velocity of the fluid $u$.

We consider solutions of quasilinear equations $u_t=\Delta u^m+u^p$ in ${ℝ}^N$ with the initial data $u_0$ satisfying $0<u_0<M$ and ${lim}_{\left|x\right|\to \infty }{u}_0\left(x\right)=M$ for some constant $M>0$. It is known that if $0<m<p$ with $p>1$, the blow-up set is empty. We find solutions $u$ that blow up throughout ${ℝ}^N$ when $m>p>1$.

We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{-\beta }$, with $\beta \in [0,\infty )$. We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $\beta$. If $\beta \in [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $\beta =1$, it is given by a parabolic equation involving an operator...

As observed by Yamazaki, the third component $b_3$ of the magnetic field can be estimated by the corresponding component $u_3$ of the velocity field in $L^{\lambda }$ $(2\le \lambda \le 6)$ norm. This leads him to establish regularity criterion involving $u_3,j_3$ or $u_3,{\omega }_3$. Noticing that $\lambda$ can be greater than 6 in this paper, we can improve previous results.

We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from $T^1$ to $S^2.$ This estimate yields some continuity properties of the flow map for the topology of $L^2(T^1,S^2)$, provided one takes its quotient by the continuous group action of $T^1$ given by translations. We also prove that without taking this quotient, for any $t\&gt;0$ the flow map at time $t$ is discontinuous as a map from ${𝒞}^{\infty }(T^1,S^2)$, equipped with the weak topology of $H^{1/2},$ to the space of distributions ${\left({𝒞}^{\infty }(T^1,{ℝ}^3)\right)}^{*}.$ The argument relies...

We consider the quasilinear equation ${\Delta }_{p}u+K\left(\right|x\left|\right)u^q=0$, and present the proof of the local existence of positive radial solutions near $0$ under suitable conditions on $K$. Moreover, we provide a priori estimates of positive radial solutions near $\infty$ when $r^{-\ell }K\left(r\right)$ for $\ell \ge -p$ is bounded near $\infty$.

We present a novel approach for bounding the resolvent of $$H=-\Delta +i(A·\nabla +\nabla ·A)+V=:-\Delta +L1$$ for large energies. It is shown here that there exist a large integer $m$ and a large number ${\lambda }_0$ so that relative to the usual weighted $L^2$-norm, $$\parallel {\left(L{(-\Delta +(\lambda +i0))}^{-1}\right)}^m\parallel <\frac{1}{2}2$$ for all $\lambda >{\lambda }_0$. This requires suitable decay and smoothness conditions on $A,V$. The estimate (2) is trivial when $A=0$, but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...

The energy class ${}_p$ is studied for 0 < p < 1. A characterization of certain bounded plurisubharmonic functions in terms of ${ℱ}_p$ and its pluricomplex p-energy is proved.

For a fixed positive integer $k$ and $G=(V,E)$ a connected graph of order $n$, whose minimum vertex degree is at least $k$, a set $S\subseteq V$ is a total $k$-dominating set, also known as a $k$-tuple total dominating set, if every vertex $v\in V$ has at least $k$ neighbors in $S$. The minimum size of a total $k$-dominating set for $G$ is called the total $k$-domination number of $G$, denoted by ${\gamma }_{kt}\left(G\right)$. The total $k$-domination problem is to determine a minimum total $k$-dominating set of $G$. Since the exact problem is in general quite difficult...

We consider the Yamabe type family of problems $\left(P_{\epsilon }\right):-\Delta u_{\epsilon }=u_{\epsilon }^{(n+2)/\left(n-2\right)}$, $u_{\epsilon }>0$ in $A_{\epsilon }$, $u_{\epsilon }=0$ on $\partial A_{\epsilon }$, where $A_{\epsilon }$ is an annulus-shaped domain of ${ℝ}^n$, $n\ge 3$, which becomes thinner as $\epsilon \to 0$. We show that for every solution $u_{\epsilon }$, the energy ${\int }_{A_{\epsilon }}|\nabla u_{|}^2$ as well as the Morse index tend to infinity as $\epsilon \to 0$. This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on ${ℝ}^n$, a half-space or an infinite strip. Our argument also involves a Liouville...

We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ($P_{\lambda }$) ⎩ $u_{\mid \partial \Omega }=0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ($P_{\lambda }$) admits a non-zero, non-negative strong solution $u_{\lambda }\in {\bigcap }_{p\ge 2}{W}^{2,p}\left(\Omega \right)$ such that $li{m}_{\lambda \to 0⁺}\left|\right|u_{\lambda }{\left|\right|}_{W^{2,p}\left(\Omega \right)}=0$ for all p ≥ 2. Moreover, the function $\lambda \mapsto I_{\lambda }\left(u_{\lambda }\right)$ is negative and decreasing in ]0,λ*[, where $I_{\lambda }$ is the energy functional related to ($P_{\lambda }$). ...

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $\Delta u-u+f\left(u\right)=0$ in ${ℝ}^N$, $u\in H^1\left({ℝ}^N\right)$, where $N\ge 2$. Under natural conditions on the nonlinearity $f$, we prove the existence of $\mathrm{𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠}$ in any dimension $N\ge 2$. Our result complements earlier works of Bartsch and Willem $(N=4\mathrm{𝚘𝚛}N\ge 6)$ and Lorca-Ubilla $(N=5)$ where solutions invariant under the action of $O\left(2\right)\times O(N-2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_k\times O(N-2)$ where $D_k\subset O\left(2\right)$ denotes the dihedral...

We prove some results in the context of isoperimetric inequalities with quantitative terms. In the $2$-dimensional case, our main contribution is a method for determining the optimal coefficients $c_1,...,c_m$ in the inequality $\delta P\left(E\right)\ge {\sum }_{k=1}^m{c}_k\alpha {\left(E\right)}^k+o\left(\alpha {\left(E\right)}^m\right)$, valid for each Borel set $E$ with positive and finite area, with $\delta P\left(E\right)$ and $\alpha \left(E\right)$ being, respectively, the $\mathrm{𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡}$ and the $\mathrm{𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦}$ of $E$. In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of $\mathrm{𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠}$ including the lower semicontinuous extension of $\frac{\delta P\left(E\right)}{\alpha {\left(E\right)}^2}$, we...

Let $G$ be a finite group $G$, $K$ a field of characteristic $p\ge 17$ and let $U$ be the group of units in $KG$. We show that if the derived length of $U$ does not exceed $4$, then $G$ must be abelian.

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula to determine the values of $R(P_n,K_{1,m})$, where $P_n$ is a path on $n$ vertices and $K_{1,m}$ is a star on $m+1$ vertices. In this note, we study the Ramsey numbers $R(P_n,K_1\vee F_m)$, where $F_m$ is a linear forest on $m$ vertices. We determine the exact values of $R(P_n,K_1\vee F_m)$ for the cases $m\le n$ and $m\ge 2n$, and for the case that $F_m$ has no odd component. Moreover, we...

We consider the systems of hyperbolic equations ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+\Delta \left(b(t,x)v\right)+{h(t,x)\left|v\right|}^p$, t > 0, $x\in {ℝ}^N$, (S1) ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{k(t,x)\left|u\right|}^q$, t > 0, $x\in {ℝ}^N$ ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+{h(t,x)\left|v\right|}^p$, t > 0, $x\in {ℝ}^N$, (S2) ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{l(t,x)\left|v\right|}^m+k(t,x){\left|u\right|}^q$, t > 0, $x\in {ℝ}^N$, (S3) ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+\Delta \left(b(t,x)v\right)+{h(t,x)\left|u\right|}^p$, t > 0, $x\in {ℝ}^N$, ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{k(t,x)\left|v\right|}^q$, t > 0, $x\in {ℝ}^N$, in $(0,\infty )\times {ℝ}^N$ with u(0,x) = u₀(x), v(0,x) = v₀(x), uₜ(0,x) = u₁(x), vₜ(0,x) = v₁(x). We show that, in each case, there exists a bound B on N such that for 1 ≤ N ≤ B solutions to the systems blow up in finite time.

Let ${\ell }_j:=-d²/dx²+k²q_j\left(x\right)$, k = const > 0, j = 1,2, $0<essinf{q}_j\left(x\right)\le esssup{q}_j\left(x\right)<\infty$. Suppose that (*) ${\int }_0^{1}p\left(x\right)u₁(x,k)u₂(x,k)dx=0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_j$ solves the problem ${\ell }_j{u}_j=0$, 0 ≤ x ≤ 1, $u_j^{\text{'}}(0,k)=0$, $u_j(0,k)=1$. It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.