Asymptotic solutions for large time of Hamilton–Jacobi equations in Euclidean n space

We study the large time behavior of solutions of the Cauchy problem for the Hamilton–Jacobi equation ut+H(x,Du)=0$u_t+H(x,Du)=0$ in Rn×(0,∞)${\mathbf{R}}^n\times (0,\infty )$, where H(x,p)$H(x,p)$ is continuous on Rn×Rn${\mathbf{R}}^n\times {\mathbf{R}}^n$ and convex in p  . We establish a general convergence result for viscosity solutions u(x,t)$u(x,t)$ of the Cauchy problem as t→∞$t\to \infty$.     

Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems

In this note we derive a maximum principle for an appropriate functional combination of u(x)$u(\mathbf{x})$ and |∇u|²${|\mathrm{\nabla }u|}^2$, where u(x)$u(\mathbf{x})$ is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in R^N+1${\mathbb{R}}^{N+1}$.     

Asymptotic behavior of the solution to the non-isothermal phase field equation

We consider the Penrose–Fife phase field model [Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43 (1990) 44–62] with homogeneous Neumann boundary condition to the nonlinear heat flux q=∇(1/θ)$\mathbf{q}=\nabla (1/\theta )$, i.e., q=0$\mathbf{q}=0$ on the boundary, where θ>0$\theta >0$ is the temperature. There is a unique H¹$H^1$ solution globally in time with the non-empty, connected, compact ω$\omega$-limit set composed of stationary solutions, and the linearized stable stationary solution is dynamically stable.     

Regularity in the obstacle problem for parabolic non-divergence operators of Hörmander type

In this paper we continue the study initiated in [15] concerning the obstacle problem for a class of parabolic non-divergence operators structured on a set of vector fields X={_X1,…,_Xq}$X=\{X_1,\dots ,X_q\}$ in Rⁿ${\mathbf{R}}^n$ with C^∞$C^{\infty }$-coefficients satisfying Hörmander?s finite rank condition, i.e., the rank of Lie[_X1,…,_Xq]$\mathrm{Lie}[X_1,\dots ,X_q]$ equals n   at every point in Rⁿ${\mathbf{R}}^n$. In [15] we proved, under appropriate assumptions on the operator and the obstacle, the existence and uniqueness of strong solutions to a general obstacle problem. The main result of this paper is that we establish further regularity, in the interior as well as at the initial state, of strong solutions. Compared to [15] we in this paper assume, in addition, that there exists a homogeneous Lie group G=(Rⁿ,°,_δλ)$\mathbf{G}=({\mathbf{R}}^n,°,{\delta }_{\lambda })$ such that _X1,…,_Xq$X_1,\dots ,X_q$ are left translation invariant on G and such that _X1,…,_Xq$X_1,\dots ,X_q$ are _δλ${\delta }_{\lambda }$-homogeneous of degree one.     

Stable extendibility of vector bundles over real projective spaces

Let F$\mathbb{F}$ be either the real number field R$\mathbb{R}$ or the complex number field C$\mathbb{C}$ and RPⁿ${\mathbb{RP}}^n$ the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given F$\mathbb{F}$-vector bundle over RPⁿ${\mathbb{RP}}^n$ to be stably extendible to RP^m${\mathbb{RP}}^m$ for every m?n$m?n$. In this paper, we simplify the theorems and apply them to the tangent bundle of RPⁿ${\mathbb{RP}}^n$, its complexification, the normal bundle associated to an immersion of RPⁿ${\mathbb{RP}}^n$ in R^n+r${\mathbb{R}}^{n+r}$(r>0)$(r>0)$, and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5].     

The topological structure of fuzzy sets with sendograph metric

For a non-degenerate convex subset Y of the n  -dimensional Euclidean space Rⁿ${\mathbb{R}}^n$, let F(Y)$\mathbb{F}(Y)$ be the family of all fuzzy sets of Rⁿ${\mathbb{R}}^n$ which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y  . We show that the space F(Y)$\mathbb{F}(Y)$ with the topology of sendograph metric is homeomorphic to the separable Hilbert space ?²${?}^2$ if Y   is compact; and the space F(Rⁿ)$\mathbb{F}({\mathbb{R}}^n)$ is also homeomorphic to ?²${?}^2$.     

Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.     

An improvement for the Trudinger–Moser inequality and applications

In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study the lack of compactness of Sobolev embedding of W^1,n(Rⁿ)$W^{1,n}({\mathbb{R}}^n)$, n?2$n?2$, into the Orlicz space _LΦα$L_{{\Phi }_{\alpha }}$ determined by the Young function _Φα(s)${\Phi }_{\alpha }(s)$ behaving like e^{α|s|n/(n−1)}−1$e^{\alpha {|s|}^{n/(n-1)}}-1$ as |s|→+∞$|s|\to +\infty$. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of the Trudinger–Moser type in the whole space Rⁿ${\mathbb{R}}^n$.     

Regularity properties of the solutions to the 3D Maxwell–Landau–Lifshitz system in weighted Sobolev spaces

This paper is concerned with special regularity properties of the solutions to the Maxwell–Landau–Lifshitz (MLL) system describing ferromagnetic medium. Besides the classical results on the boundedness of _∂tm,_∂tE${\mathrm{\partial }}_t\mathbf{m},{\mathrm{\partial }}_t\mathbf{E}$ and _∂tH${\mathrm{\partial }}_t\mathbf{H}$ in the spaces L^∞(I,L²(Ω))$L^{\infty }(I,L^2(\Omega ))$ and L²(I,W^1,2(Ω))$L^2(I,W^{1,2}(\Omega ))$ we derive also estimates in weighted Sobolev spaces. This kind of estimates can be used to control the Taylor remainder when estimating the error of a numerical scheme.     

On singular foliations on the solid torus

We study smooth foliations on the solid torus S¹×D²$S^1\times D^2$ having S¹×{0}$S^1\times \{\mathbf{0}\}$ and S¹×∂D²$S^1\times \partial D^2$ as the only compact leaves and S¹×{0}$S^1\times \{\mathbf{0}\}$ as singular set. We show that all other leaves can only be cylinders or planes, and give necessary conditions for the foliation to be a suspension of a diffeomorphism of the disc.     

Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential

We study the existence of solutions u:R³→R²$u:{\mathbb{R}}^3\to {\mathbb{R}}^2$ for the semilinear elliptic systems

equation(0.1)

−Δu(x,y,z)+∇W(u(x,y,z))=0,$-\mathrm{\Delta }u(x,y,z)+\mathrm{\nabla }W(u(x,y,z))=0,$

where W:R²→R$W:{\mathbb{R}}^2\to \mathbb{R}$ is a double well symmetric potential. We use variational methods to show, under generic non-degenerate properties of the set of one dimensional heteroclinic connections between the two minima _a±${\mathbf{a}}_{\pm }$ of W, that (0.1) has infinitely many geometrically distinct solutions u∈C²(R³,R²)$u\in C^2({\mathbb{R}}^3,{\mathbb{R}}^2)$ which satisfy u(x,y,z)→_a±$u(x,y,z)\to {\mathbf{a}}_{\pm }$ as x→±∞$x\to \pm \infty$ uniformly with respect to (y,z)∈R²$(y,z)\in {\mathbb{R}}^2$ and which exhibit dihedral symmetries with respect to the variables y and z  . We also characterize the asymptotic behavior of these solutions as |(y,z)|→+∞$|(y,z)|\to +\infty$.     

Generic global rigidity of body–bar frameworks

A basic geometric question is to determine when a given framework G(p)$G(\mathbf{p})$ is globally rigid in Euclidean space R^d${\mathbb{R}}^d$, where G is a finite graph and p is a configuration of points corresponding to the vertices of G  . G(p)$G(\mathbf{p})$ is globally rigid in  R^d${\mathbb{R}}^d$ if for any other configuration q for G   such that the edge lengths of G(q)$G(\mathbf{q})$ are the same as the corresponding edge lengths of G(p)$G(\mathbf{p})$, then p is congruent to q. A framework G(p)$G(\mathbf{p})$ is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.     

Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps

A quasiplane f(V)$f(V)$ is the image of an n-dimensional Euclidean subspace V   of R^N${\mathbb{R}}^N$ (1≤n≤N−1$1\le n\le N-1$) under a quasiconformal map f:R^N→R^N$f:{\mathbb{R}}^N\to {\mathbb{R}}^N$. We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz n  -manifold and for a quasiplane to have big pieces of bi-Lipschitz images of Rⁿ${\mathbb{R}}^n$. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension N−n$N-n$. To establish the big pieces criterion, we prove new extension theorems for “almost affine” maps, which are of independent interest. This work is related to investigations by Tukia and Väisälä on extensions of quasisymmetric maps with small distortion.     

On Markov operators preserving polynomials

The paper is concerned with a special class of positive linear operators acting on the space C(K)$C(K)$ of all continuous functions defined on a convex compact subset K   of R^d${\mathbf{R}}^d$, d?1$d?1$, having non-empty interior. Actually, this class consists of all positive linear operators T   on C(K)$C(K)$ which leave invariant the polynomials of degree at most 1 and which, in addition, map polynomials into polynomials of the same degree. Among other things, we discuss the existence of such operators in the special case where K is strictly convex by also characterizing them within the class of positive projections. In particular we show that such operators exist if and only if ∂K   is an ellipsoid. Furthermore, a characterization of balls of R^d${\mathbf{R}}^d$ in terms of a special class of them is furnished. Additional results and illustrative examples are presented as well.     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws

It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C¹$C^1$ solution u=u(t,x)$u=u(t,x)$ containing only n$n$ shock waves with small amplitude on t?0$t?0$ and this solution possesses a global structure similar to that of the similarity solution u=U(x/t)$u=U(x/t)$ of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data.