The Cauchy Problem for a Tenth-Order Thin Film Equation I. Bifurcation of Oscillatory Fundamental Solutions

Fundamental global similarity solutions of the tenth-order thin film equation $$u_{t} = \nabla . (|u|^{n} \nabla \Delta^{4}u) \,\,\,\, {\rm in} \,\,\,\, \mathbb{R}^{N} \times \mathbb{R}_{+}$$ , where n >  0 are studied. The main approach consists in passing to the limit ${n \rightarrow 0^{+}}$ by using Hermitian non-self-adjoint spectral theory corresponding to the rescaled linear poly-harmonic equation $$u_{t} = \Delta^{5}u \,\,\,\, {\rm in} \,\,\,\, \mathbb{R}^{N} \times \mathbb{R}_{+}$$ .     

Existence of the generalized Fučík spectrum for nonhomogeneous elliptic operators

By variational methods and Morse theory, we prove the existence of uncountably many \((\alpha ,\beta )\in \mathbb R ^2\) for which the equation \(-\mathrm{div}\, A(x, \nabla u)=\alpha u_+^{p-1} -\beta u_-^{p-1}\) in \(\Omega \) , has a sign changing solution under the Neumann boundary condition, where a map \(A\) from \(\overline{\Omega }\times \mathbb R ^N\) to \(\mathbb R ^N\) satisfying certain regularity conditions. As a special case, the above equation contains the \(p\) -Laplace equation. However, the operator \(A\) is not supposed to be \((p-1)\) -homogeneous in the second variable. In particular, it is shown that generally the Fu?ík spectrum of the operator \(-\mathrm{div}\, A(x, \nabla u)\) on \(W^{1,p}(\Omega )\) contains some open unbounded subset of \(\mathbb R ^2\) .     

Well-posedness and blowup phenomena for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs

This paper deals with the Cauchy problem for a cross-coupled Camassa–Holm equation $$m_t=-(vm)_x-mv_x, n_t=-(un)_x-nu_x,$$ where \({n\doteq v-v_{xx}}\) , \({m\doteq u-u_{xx}+\omega}\) with a constant ω. The local well-posedness of solutions for the Cauchy problem of the cross-coupled Camassa–Holm equation in Sobolev space \({H^s(\mathbb{R})}\) with s > 5/2 is established. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and the blowup scenario of the solutions to the equation is also obtained.     

Biharmonic elliptic problems involving the 2nd Hessian operator

In this paper we will study the equation $$\begin{aligned} \Delta ^2 u=S_2(D^2u),\quad \Omega \subset \mathbb {R}^N, \end{aligned}$$ with \(N=3,\) where \( S_2(D^2u)(x)=\sum _{1\le i , being \(\lambda _i,\) the solutions to the equation $$\begin{aligned} \mathrm{det}\left( \lambda I-D^2u(x)\right) =0, \end{aligned}$$ \(i=1,\dots ,N,\) and \(\Omega \) is a bounded domain with smooth boundary. We deal with several boundary conditions looking for the appropriate framework to get existence and multiplicity of nontrivial solutions. This kind of equation is related to some models of growth, and for this reason it is natural to study the effect of zero order local reaction terms of the type \(F_{\lambda }(x,u)=\lambda |u|^{p-1}u\) , with \(\lambda \in \mathbb {R}\) , \(\lambda >0\) , and \(0 , and also the solvability of the boundary problems with a source term \(f\) satisfying some integrability hypotheses.     

Stability of a conditional Cauchy equation on a set of measure zero

In this paper, we prove the Hyers–Ulam stability theorem when \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy $$|f(x + y) - g(x) - h(y)| \leq \epsilon$$ in a set \({\Gamma \subset \mathbb{R}^{2}}\) of measure \({m(\Gamma) = 0}\) , which refines a previous result in Chung (Aequat Math 83:313–320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy the Pexider equation $$f(x + y) - g(x) - h(y) = 0$$ in \({\Gamma}\) , then the equation holds for all \({x, y \in \mathbb{R}}\) . Using our method of construction of the set, we can find a set \({\Gamma \subset \mathbb{R}^{2n}}\) of 2n-dimensional measure 0 and obtain the above result for the functions \({f, g, h : \mathbb{R}^{n} \to \mathbb{C}}\) .     

An Analog of the Cameron--Johnson Theorem for Linear ℂ-Analytic Equations in Hilbert Space

The well-known Cameron--Johnson theorem asserts that the equation $\dot x = \mathcal{A}\left( t \right)x$ with a recurrent (Bohr almost periodic) matrix $\mathcal{A}\left( t \right)$ can be reduced by a Lyapunov transformation to the equation $\dot y = \mathcal{B}\left( t \right)y$ with a skew-symmetric matrix $\mathcal{B}\left( t \right)$ , provided that all solutions of the equation $\dot x = \mathcal{A}\left( t \right)x$ and of all its limit equations are bounded on the whole line. In the note, a generalization of this result to linear $\mathbb{C}$ -analytic equations in a Hilbert space is presented.     

On Rankin triple product L-functions over function fields: central critical values

The aim of this article is to study the special values of Rankin triple product $L$ -functions associated to Drinfeld type newforms of equal square-free levels. The functional equation of these $L$ -functions is deduced from a Garrett-type integral representation and the functional equation of Eisenstein series on the group of similitudes of a symplectic vector space of dimension $6$ . When the associated root number is positive, we present a function field analogue of Gross–Kudla formula for the central critical value. This formula is then applied to the non-vanishing of $L$ -functions coming from elliptic curves over function fields.     

Metrics with prescribed Ricci curvature near the boundary of a manifold

Suppose $M$ is a manifold with boundary. Choose a point $o\in \partial M$ . We investigate the prescribed Ricci curvature equation $\mathop {\mathrm{Ric}}\nolimits (G)=T$ in a neighborhood of $o$ under natural boundary conditions. The unknown $G$ here is a Riemannian metric. The letter $T$ on the right-hand side denotes a (0,2)-tensor. Our main theorems address the questions of the existence and the uniqueness of solutions. We explain, among other things, how these theorems may be used to study rotationally symmetric metrics near the boundary of a solid torus $\mathcal{T }$ . The paper concludes with a brief discussion of the Einstein equation on $\mathcal{T }$ .     

Nonlinear Schrödinger equations near an infinite well potential

The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation where the potential \(V_\lambda :\mathbb {R}^N\rightarrow \mathbb {R}\) is close to an infinite well potential \(V_\infty :\mathbb {R}^N\rightarrow \mathbb {R}\) , i. e. \(V_\infty =\infty \) on an exterior domain \(\mathbb {R}^N\setminus \Omega \) , \(V_\infty |_\Omega \in L^\infty (\Omega )\) , and \(V_\lambda \rightarrow V_\infty \) as \(\lambda \rightarrow \infty \) in a sense to be made precise. The nonlinearity may be of Gross–Pitaevskii type. A standing wave solution of \((NLS_\lambda )\) with \(\lambda =\infty \) vanishes on \(\mathbb {R}^N\setminus \Omega \) and satisfies Dirichlet boundary conditions, hence it solves We investigate when a standing wave solution \(\Phi _\infty \) of the infinite well potential \((NLS_\infty )\) gives rise to nearby solutions \(\Phi _\lambda \) of the finite well potential \((NLS_\lambda )\) with \(\lambda \gg 1\) large. Considering \((NLS_\infty )\) as a singular limit of \((NLS_\lambda )\) we prove a kind of singular continuation type results.     

Layer solutions for the fractional Laplacian on hyperbolic space: existence,uniqueness and qualitative properties

We investigate the equation $$\begin{aligned} (-\Delta _{\mathbb{H }^n})^{\gamma } w=f(w)\quad \text{ in } \mathbb{H }^{n}, \end{aligned}$$ where \((-\Delta _{\mathbb{H }^n})^\gamma \) corresponds to the fractional Laplacian on hyperbolic space for \(\gamma \in (0,1)\) and \(f\) is a smooth nonlinearity that typically comes from a double well potential. We prove the existence of heteroclinic connections in the following sense; a so-called layer solution is a smooth solution of the previous equation converging to \(\pm 1\) at any point of the two hemispheres \(S_\pm \subset \partial _\infty \mathbb{H }^n\) and which is strictly increasing with respect to the signed distance to a totally geodesic hyperplane \(\Pi \) . We prove that under additional conditions on the nonlinearity uniqueness holds up to isometry. Then we provide several symmetry results and qualitative properties of the layer solutions. Finally, we consider the multilayer case, at least when \(\gamma \) is close to one.     

From pointwise to local regularity for solutions of Hamilton–Jacobi equations

It is well-known that solutions to the Hamilton–Jacobi equation $$\begin{aligned} u_{t}(t,x)+H(x,u_{x}(t,x))=0 \end{aligned}$$ fail to be everywhere differentiable. Nevertheless, suppose a solution $u$ turns out to be differentiable at a given point $(t,x)$ in the interior of its domain. May then one deduce that $u$ must be continuously differentiable in a neighborhood of $(t,x)$ ? Although this question has a negative answer in general, our main result shows that it is indeed the case when the proximal subdifferential of $u(t,\cdot )$ at $x$ is nonempty. Our approach uses the representation of $u$ as the value function of a Bolza problem in the calculus of variations, as well as necessary conditions for such a problem.     

Uniqueness of generalized p-area minimizers and integrability of a horizontal normal in the Heisenberg group

We study the uniqueness of generalized \(p\) -minimal surfaces in the Heisenberg group. The generalized \(p\) -area of a graph defined by \(u\) reads \(\int |\nabla u+\vec {F}|+Hu.\) If \(u\) and \(v\) are two minimizers for the generalized \(p\) -area satisfying the same Dirichlet boundary condition, then we can only get \(N_{\vec {F}}(u) = N_{\vec {F}}(v)\) (on the nonsingular set) where \(N_{\vec {F}}(w) := \frac{\nabla w+\vec {F}}{|\nabla w+\vec {F}|}.\) To conclude \(u = v\) (or \(\nabla u = \nabla v)\) , it is not straightforward as in the Riemannian case, but requires some special argument in general. In this paper, we prove that \(N_{\vec {F}}(u) = N_{ \vec {F}}(v)\) implies \(\nabla u = \nabla v\) in dimension \(\ge \) 3 under some rank condition on derivatives of \(\vec {F}\) or the nonintegrability condition of contact form associated to \(u\) or \(v\) . Note that in dimension 2 ( \(n=1),\) the above statement is no longer true. Inspired by an equation for the horizontal normal \(N_{\vec {F}}(u),\) we study the integrability for a unit vector to be the horizontal normal of a graph. We find a Codazzi-like equation together with this equation to form an integrability condition.     

On the supercritical KdV equation with time-oscillating nonlinearity

For the initial value problem (IVP) associated to the generalized Korteweg–de Vries (gKdV) equation with supercritical nonlinearity, $$u_{t}+\partial_x^3u+\partial_x(u^{k+1}) =0,\qquad k\geq 5,$$ numerical evidence [3] shows that, there are initial data ${\phi\in H^1(\mathbb{R})}$ such that the corresponding solution may blow-up in finite time. Also, with the evidence from numerical simulation [1, 18], it has been claimed that a periodic time dependent coefficient in the nonlinearity would disturb the blow-up solution, either accelerating or delaying it. In this work, we investigate the IVP associated to the gKdV equation $$u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^{k+1}) =0,$$ where g is a periodic function and ${k\geq 5}$ is an integer. We prove that, for given initial data ${\phi \in H^1(\mathbb{R})}$ , as ${|\omega|\to \infty}$ , the solution ${u_{\omega} }$ converges to the solution U of the initial value problem associated to $$U_{t}+\partial_x^3U+m(g)\partial_x(U^{k+1}) =0,$$ with the same initial data, where m(g) is the average of the periodic function g. Moreover, if the solution U is global and satisfies ${\|U\|_{L_x^{5}L_t^{10}}<\infty}$ , then we prove that the solution ${u_{\omega} }$ is also global provided ${|\omega|}$ is sufficiently large.     

The second order pullback equation

Let $f,g$ be two closed $k$ -forms over $\mathbb{R }^{n}.$ The pullback equation studies the existence of a diffeomorphism $\varphi :\mathbb{R }^{n} \rightarrow \mathbb{R }^{n}$ such that $$\begin{aligned} \varphi ^{*}(g)=f. \end{aligned}$$ We prove two types of results. The first one sharpens some of the existing regularity results. The second one discusses the possibility of choosing the map $\varphi $ as the gradient of a function $\Phi :\mathbb{R }^{n} \rightarrow \mathbb R .$ We show that this is a very rare event unless the two forms are constant.     

On cycles for the doubling map which are disjoint from an interval

Let \(T:[0,1]\rightarrow [0,1]\) be the doubling map and let \(0 . We say that an integer \(n\ge 3\) is bad for \((a,b)\) if all \(n\) -cycles for \(T\) intersect \((a,b)\) . Let \(B(a,b)\) denote the set of all \(n\) which are bad for \((a,b)\) . In this paper we completely describe the sets: $$\begin{aligned} D_2=\{(a,b) : B(a,b)\,\text {is finite}\} \end{aligned}$$ and $$\begin{aligned} D_3=\{(a,b) : B(a,b)=\varnothing \}. \end{aligned}$$ In particular, we show that if \(b-a<\frac{1}{6}\) , then \((a,b)\in D_2\) , and if \(b-a\le \frac{2}{15}\) , then \((a,b)\in D_3\) , both constants being sharp.     

On the compactness of the set of invariant Einstein metrics

Let $M = G/H$ be a connected simply connected homogeneous manifold of a compact, not necessarily connected Lie group $G$ . We will assume that the isotropy $H$ -module $\mathfrak{g/h }$ has a simple spectrum, i.e. irreducible submodules are mutually non-equivalent. There exists a convex Newton polytope $N=N(G,H)$ , which was used for the estimation of the number of isolated complex solutions of the algebraic Einstein equation for invariant metrics on $G/H$ (up to scaling). Using the moment map, we identify the space $\mathcal{M }_1$ of invariant Riemannian metrics of volume 1 on $G/H$ with the interior of this polytope $N$ . We associate with a point ${x \in \partial N}$ of the boundary a homogeneous Riemannian space (in general, only local) and we extend the Einstein equation to $\partial N$ . As an application of the Alekseevsksky–Kimel’fel’d theorem, we prove that all solutions of the Einstein equation associated with points of the boundary are locally Euclidean. We describe explicitly the set $T\subset \partial N$ of solutions at the boundary together with its natural triangulation. Investigating the compactification ${\overline{\mathcal{M }}}_{1} = N$ of $\mathcal{M }_1$ , we get an algebraic proof of the deep result by Böhm, Wang and Ziller about the compactness of the set $\mathcal{E }_1 \subset \mathcal{M }_1$ of Einstein metrics. The original proof by Böhm, Wang and Ziller was based on a different approach and did not use the simplicity of the spectrum. In Appendix, we consider the non-symmetric flag manifolds $G/H$ with the second Betti number $b_2=1$ . We calculate the normalized volumes $2,6,20,82,344$ of the corresponding Newton polytopes and discuss the number of complex solutions of the algebraic Einstein equation and the finiteness problem.     

The Hopf–Lax formula in Carnot groups: a control theoretic approach

The purpose of this paper is to bring a new light on the state-dependent Hamilton–Jacobi equation and its connection with the Hopf–Lax formula in the framework of a Carnot group $(\mathbf G ,\circ ).$ The equation we shall consider is of the form $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{t}+ \Psi (X_{1}u, \ldots , X_{m}u)=0\qquad &{}(x,t)\in \mathbf G \times (0,\infty ) \\ {u}(x,0)=g(x)&{}x\in \mathbf G , \end{array} \right. \end{aligned}$$ where $X_{1},\ldots , X_{m}$ are a basis of the first layer of the Lie algebra of the group $\mathbf G ,$ and $\Psi : \mathbb{R }^{m} \rightarrow \mathbb{R }$ is a superlinear, convex function. The main result shows that the unique viscosity solution of the Hamilton–Jacobi equation can be given by the Hopf–Lax formula $$\begin{aligned} u(x,t) = \inf _{y\in \mathbf G }\left\{ t \Psi ^\mathbf{G }\left( \delta _{\frac{1}{t}}(y^{-1}\circ x)\right) + g(y) \right\} , \end{aligned}$$ where $\Psi ^\mathbf{G }:\mathbf G \rightarrow \mathbb{R }$ is the $\mathbf G $ -Legendre–Fenchel transform of $\Psi ,$ defined by a control theoretical approach. We recover, as special cases, some known results like the classical Hopf–Lax formula in the Euclidean spaces $\mathbb{R }^n,$ showing that $\Psi ^{\mathbb{R }^n}$ is the Legendre–Fenchel transform $\Psi ^*$ of $\Psi ;$ moreover, we recover the result by Manfredi and Stroffolini in the Heisenberg group for particular Hamiltonian function $\Psi .$ In this paper we follow an optimal control problem approach and we obtain several properties for the value functions $u$ and $\Psi ^\mathbf G .$      

Sharp energy estimates for nonlinear fractional diffusion equations

We study the nonlinear fractional equation $(-\Delta )^su=f(u)$ in $\mathbb R ^n,$ for all fractions $0<s<1$ and all nonlinearities $f$ . For every fractional power $s\in (0,1)$ , we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional symmetry of bounded global minimizers and of bounded monotone solutions in dimension $n=3$ whenever $1/2\le s<1$ . This result is the analogue of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-\Delta u=f(u)$ in $\mathbb R ^n$ . It remains open for $n=3$ and $s<1/2$ , and also for $n\ge 4$ and all $s$ .