Approximation of Flat <Emphasis Type="Italic">W</Emphasis><Superscript>2,2</Superscript> Isometric Immersions by Smooth Ones

Let \({S\subset\mathbb{R}^2}\) be a bounded Lipschitz domain and denote by \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\) the set of mappings \({u\in W^{2,2}(S;\mathbb{R}^3)}\) which satisfy \({(\nabla u)^T(\nabla u) = Id}\) almost everywhere. Under an additional regularity condition on the boundary \({\partial S}\) (which is satisfied if \({\partial S}\) is piecewise continuously differentiable), we prove that the strong W ^2,2 closure of \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)\cap C^{\infty}(\overline{S};\mathbb{R}^3)}\) agrees with \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\).     

Exponential Decay of the Vorticity in the Steady-State Flow of a Viscous Liquid Past a Rotating Body

We construct a Sobolev homeomorphism in dimension \({n \geqq 4,\,f \in W^{1,1}((0, 1)^n,\mathbb{R}^n)}\) such that \({J_f = {\rm det} Df > 0}\) on a set of positive measure and J _f  < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) f _k such that \({f_k\to f}\) in \({W^{1,1}_{\rm loc}}\).     

Steady Nearly Incompressible Vector Fields in Two-Dimension: Chain Rule and Renormalization

Given bounded vector field \({b : {\mathbb{R}^{d}} \to {\mathbb{R}^{d}}}\), scalar field \({u : {\mathbb{R}^{d}} \to {\mathbb{R}}}\), and a smooth function \({\beta : {\mathbb{R}} \to {\mathbb{R}}}\), we study the characterization of the distribution \({{\rm div}(\beta(u)b)}\) in terms of div b and div(ub). In the case of BV vector fields b (and under some further assumptions), such characterization was obtained by L. Ambrosio, C. De Lellis and J. Malý, up to an error term which is a measure concentrated on the so-called tangential set of b. We answer some questions posed in their paper concerning the properties of this term. In particular, we construct a nearly incompressible BV vector field b and a bounded function u for which this term is nonzero. For steady nearly incompressible vector fields b (and under some further assumptions), in the case when d = 2, we provide complete characterization of div(\({\beta(u)b}\)) in terms of div b and div(ub). Our approach relies on the structure of level sets of Lipschitz functions on \({{\mathbb{R}^{2}}}\) obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique, we obtain new sufficient conditions when any bounded weak solution u of \({\partial_t u + b \cdot \nabla u=0}\) is renormalized, that is when it also solves \({\partial_t \beta(u) + b \cdot \nabla \beta(u)=0}\) for any smooth function \({\beta \colon{\mathbb{R}} \to {\mathbb{R}}}\). As a consequence, we obtain new a uniqueness result for this equation.     

SBV Regularity for Hamilton–Jacobi Equations in $${{\mathbb R}^n}$$

In this paper we study the regularity of viscosity solutions to the following Hamilton–Jacobi equations

$\partial_{t}u+H(D_{x}u)=0\quad\hbox{in }\Omega\subset{\mathbb R}\times{\mathbb R}^{n}.$

In particular, under the assumption that the Hamiltonian \({H\in C^2({\mathbb R}^n)}\) is uniformly convex, we prove that D _x u and ? _t u belong to the class SBV _loc (Ω).

     

Stability of Traveling Waves of Nonlinear Schrödinger Equation with Nonzero Condition at Infinity

We study effective elastic behavior of the incompatibly prestrained thin plates, where the prestrain is independent of thickness and uniform through the plate’s thickness h. We model such plates as three-dimensional elastic bodies with a prescribed pointwise stress-free state characterized by a Riemannian metric G, and seek the limiting behavior as \({h \to 0}\). We first establish that when the energy per volume scales as the second power of h, the resulting \({\Gamma}\) -limit is a Kirchhoff-type bending theory. We then show the somewhat surprising result that there exist non-immersible metrics G for whom the infimum energy (per volume) scales smaller than h². This implies that the minimizing sequence of deformations carries nontrivial residual three-dimensional energy but it has zero bending energy as seen from the limit Kirchhoff theory perspective. Another implication is that other asymptotic scenarios are valid in appropriate smaller scaling regimes of energy. We characterize the metrics G with the above property, showing that the zero bending energy in the Kirchhoff limit occurs if and only if the Riemann curvatures R₁₂₁₃, R₁₂₂₃ and R₁₂₁₂ of G vanish identically. We illustrate our findings with examples; of particular interest is an example where \({G_{2 \times 2}}\), the two-dimensional restriction of G, is flat but the plate still exhibits the energy scaling of the Föppl–von Kármán type. Finally, we apply these results to a model of nematic glass, including a characterization of the condition when the metric is immersible, for \({G = Id_{3} + \gamma n \otimes n}\) given in terms of the inhomogeneous unit director field distribution \({ n \in \mathbb{R}^3}\).     

Fine Level Set Structure of Flat Isometric Immersions

A result by Pogorelov asserts that C ¹ isometric immersions u of a bounded domain \({S \subset \mathbb R^2}\) into \({\mathbb {R}^3}\) whose normal takes values in a set of zero area enjoy the following regularity property: the gradient \({f := \nabla u}\) is ‘developable’ in the sense that the nondegenerate level sets of f consist of straight line segments intersecting the boundary of S at both endpoints. Motivated by applications in nonlinear elasticity, we study the level set structure of such f when S is an arbitrary bounded Lipschitz domain. We show that f can be approximated by uniformly bounded maps with a simplified level set structure. We also show that the domain S can be decomposed (up to a controlled remainder) into finitely many subdomains, each of which admits a global line of curvature parametrization.     

New Derived from Anosov Diffeomorphisms with Pathological Center Foliation

In this paper we focused our study on derived from Anosov diffeomorphisms (DA diffeomorphisms ) of the torus \(\mathbb {T}^3,\) it is, an absolute partially hyperbolic diffeomorphism on \(\mathbb {T}^3\) homotopic to a linear Anosov automorphism of the \(\mathbb {T}^3.\) We can prove that if \(f: \mathbb {T}^3 \rightarrow \mathbb {T}^3 \) is a volume preserving DA diffeomorphism homotopic to a linear Anosov A,  such that the center Lyapunov exponent satisfies \(\lambda ^c_f(x) > \lambda ^c_A > 0,\) with x belongs to a positive volume set, then the center foliation of f is non absolutely continuous. We construct a new open class U of non Anosov and volume preserving DA diffeomorphisms, satisfying the property \(\lambda ^c_f(x) > \lambda ^c_A > 0\) for \(m-\)almost everywhere \(x \in \mathbb {T}^3.\) Particularly for every \(f \in U,\) the center foliation of f is non absolutely continuous.     

Local Fixed Point Indices of Iterations of Planar Maps

Let \({f: U\rightarrow {\mathbb R}^2}\) be a continuous map, where U is an open subset of \({{\mathbb R}^2}\). We consider a fixed point p of f which is neither a sink nor a source and such that {p} is an isolated invariant set. Under these assumption we prove, using Conley index methods and Nielsen theory, that the sequence of fixed point indices of iterations \({\{{\rm ind}(f^n,p)\}_{n=1}^\infty}\) is periodic, bounded from above by 1, and has infinitely many non-positive terms, which is a generalization of Le Calvez and Yoccoz theorem (Annals of Math., 146, 241–293 (1997)) onto the class of non-injective maps. We apply our result to study the dynamics of continuous maps on 2-dimensional sphere.     

Multiple Solutions for a Class of Nonhomogeneous Fractional Schrödinger Equations in $$\mathbb {R}^{N}$$

This paper is concerned with the following fractional Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s} u+u= k(x)f(u)+h(x) \text{ in } \mathbb {R}^{N}\\ u\in H^{s}(\mathbb {R}^{N}), \, u>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$

where \(s\in (0,1),N> 2s, (-\Delta )^{s}\) is the fractional Laplacian, k is a bounded positive function, \(h\in L^{2}(\mathbb {R}^{N}), h\not \equiv 0\) is nonnegative and f is either asymptotically linear or superlinear at infinity. By using the s-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that \(|h|_{2}\) is sufficiently small.

     

Convergence of Radially Symmetric Solutions for (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian Elliptic Equations with a Damping Term

This study considers the quasilinear elliptic equation with a damping term,

$$\begin{aligned} \text {div}(D(u)\nabla u) + \frac{k(|{\mathbf {x}}|)}{|{\mathbf {x}}|}\,{\mathbf {x}}\cdot (D(u)\nabla u) + \omega ^2\big (|u|^{p-2}u + |u|^{q-2}u\big ) = 0, \end{aligned}$$

where \({\mathbf {x}}\) is an N-dimensional vector in \(\big \{{\mathbf {x}} \in \mathbb {R}^N: |{\mathbf {x}}| \ge \alpha \big \}\) for some \(\alpha > 0\) and \(N \in {\mathbb {N}}\setminus \{1\}\); \(D(u) = |\nabla u|^{p-2} + |\nabla u|^{q-2}\) with \(1 < q \le p\); k is a nonnegative and locally integrable function on \([\alpha ,\infty )\); and \(\omega \) is a positive constant. A necessary and sufficient condition is given for all radially symmetric solutions to converge to zero as \(|{\mathbf {x}}|\rightarrow \infty \). Our necessary and sufficient condition is expressed by an improper integral related to the damping coefficient k. The case that k is a power function is explained in detail.

     

Convexity Criteria and Uniqueness of Absolutely Minimizing Functions

We show that an absolutely minimizing function with respect to a convex Hamiltonian \({H : \mathbb{R}^{n} \rightarrow \mathbb{R}}\) is uniquely determined by its boundary values under minimal assumptions on H. Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in L^∞.     

On Some Nonlinear Fractional Equations Involving the Bessel Operator

Under different assumptions on the potential functions b and c, we study the fractional equation \(\left( I-\varDelta \right) ^{\alpha } u = \lambda b(x) |u|^{p-2}u+c(x)|u|^{q-2}u\) in \(\mathbb {R}^N\). Our existence results are based on compact embedding properties for weighted spaces.     

Complex Dynamics in a ODE Model Related to Phase Transition

Motivated by some recent studies on the Allen–Cahn phase transition model with a periodic nonautonomous term, we prove the existence of complex dynamics for the second order equation

$$\begin{aligned} -\ddot{x} + \left( 1 + \varepsilon ^{-1} A(t)\right) G'(x) = 0, \end{aligned}$$

where A(t) is a nonnegative T-periodic function and \(\varepsilon > 0\) is sufficiently small. More precisely, we find a full symbolic dynamics made by solutions which oscillate between any two different strict local minima \(x_0\) and \(x_1\) of G(x). Such solutions stay close to \(x_0\) or \(x_1\) in some fixed intervals, according to any prescribed coin tossing sequence. For convenience in the exposition we consider (without loss of generality) the case \(x_0 =0\) and \(x_1 = 1\).

     

Stationary Solutions and Connecting Orbits for <Emphasis Type="Italic">p</Emphasis>-Laplace Equation

We deal with one dimensional p-Laplace equation of the form

$$\begin{aligned} u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, \end{aligned}$$

under Dirichlet boundary condition, where \(p>2\) and \(f:[0,l]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function with \(f(x,0)=0\). We will prove that if there is at least one eigenvalue of the p-Laplace operator between \(\lim _{u\rightarrow 0} f(x,u)/|u|^{p-2}u\) and \(\lim _{|u|\rightarrow +\infty } f(x,u)/|u|^{p-2}u\), then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are based on Conley index and detect stationary states even when those based on fixed point theory do not apply. In order to compute the Conley index for nonlinear semiflows deformation along p is used.

     

Uniqueness of Positive Ground State Solutions of the Logarithmic Schrödinger Equation

We prove the uniqueness of positive ground state solutions of the problem \({ {\frac {d^{2}u}{dr^{2}}} + {\frac {n-1}{r}}{\frac {du}{dr}} + u \ln(|u|) = 0}\), \({u(r) > 0~\forall r \ge 0}\), and \({(u(r),u'(r)) \to (0, 0)}\) as \({r \to \infty}\). This equation is derived from the logarithmic Schrödinger equation \({{\rm i}\psi_{t} = {\Delta} \psi + u \ln \left(|u|^{2}\right)}\), and also from the classical equation \({{\frac {\partial u}{\partial t}} = {\Delta} u +u \left(|u|^{p-1}\right) -u}\). For each \({n \ge 1}\), a positive ground state solution is \({ u_{0}(r) = \exp \left(-{\frac{r^2}{4}} + {\frac{n}{2}}\right),~0 \le r < \infty}\). We combine \({u_{0}(r)}\) with energy estimates and associated Ricatti equation estimates to prove that, for each \({n \in \left[1, 9 \right]}\), \({u_{0}(r)}\) is the only positive ground state. We also investigate the stability of \({u_{0}(r)}\). Several open problems are stated.     

Global Well-Posedness for Navier–Stokes Equations with Small Initial Value in $${B^{0}_{n,\infty}(\Omega)}$$

We prove global well-posedness for instationary Navier–Stokes equations with initial data in Besov space \({B^{0}_{n,\infty}(\Omega)}\) in whole and half space, and bounded domains of \({{\mathbb R}^{n}}\), \({n \geq 3}\). To this end, we prove maximal \({L^{\infty}_{\gamma}}\) -regularity of the sectorial operators in some Banach spaces and, in particular, maximal \({L^{\infty}_{\gamma}}\) -regularity of the Stokes operator in little Nikolskii spaces \({b^{s}_{q,\infty}(\Omega)}\), \({s \in (-1, 2)}\), which are of independent significance. Then, based on the maximal regularity results and \({b^{s_{1}}_{q_{1},\infty}-B^{s_{2}}_{q_{2,1}}}\) estimates of the Stokes semigroups, we prove global well-posedness for Navier–Stokes equations under smallness condition on \({\|u_{0}\|_{B^{0}_{n,\infty}(\Omega)}}\) via a fixed point argument using Banach fixed point theorem.     

Global Well-Posedness of the 2D Boussinesq Equations with Vertical Dissipation

We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data \({(u_0,\theta_0)}\) are required to be only in the space \({X=\{f\in L^2(\mathbb{R}^2)\,|\,\partial_{x} f \in L^2(\mathbb{R}^2)\}}\), and thus our result generalizes that of Cao and Wu (Arch Rational Mech Anal, 208:985–1004, 2013), where the initial data are assumed to be in \({H^2(\mathbb{R}^2)}\). The assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions. A logarithmic type limiting Sobolev embedding inequality for the \({L^\infty(\mathbb{R}^2)}\) norm, in terms of anisotropic Sobolev norms, and a logarithmic type Gronwall inequality are established to obtain the global in time a priori estimates, which guarantee the local solution to be a global one.     

New Periodic Solutions for Newtonian n-Body Problems with Dihedral Group Symmetry and Topological Constraints

In this paper, we prove the existence of a family of new non-collision periodic solutions for the classical Newtonian n-body problems. In our assumption, the \({n=2l \geqq 4}\) particles are invariant under the dihedral rotation group D_l in \({\mathbb{R}^3}\) such that, at each instant, the n particles form two twisted l-regular polygons. Our approach is the variational minimizing method and we show that the minimizers are collision-free by level estimates and local deformations.