Weak KAM theory for discounted Hamilton–Jacobi equations and its application

Weak KAM theory for discounted Hamilton–Jacobi equations and corresponding discounted Lagrangian/Hamiltonian dynamics is developed. Then it is applied to error estimates for viscosity solutions in the vanishing discount process. The main feature is to introduce and investigate the family of \(\alpha \)-limit points of minimizing curves, with some details in terms of minimizing measures. In error estimates, the family of \(\alpha \)-limit points is effectively exploited with properties of the corresponding dynamical systems.     

Compact $$\lambda $$-translating Solitons with Boundary

A \(\lambda \)-translating soliton with density vector \(\mathbf {v}\) is a surface \(\varSigma \) in Euclidean space \(\mathbb {R}^3\) whose mean curvature H satisfies \(2H=2\lambda +\langle N,\mathbf {v}\rangle \), where N is the Gauss map of \(\varSigma \). In this article, we study the shape of a compact \(\lambda \)-translating soliton in terms of its boundary. If \(\varGamma \) is a given closed curve, we deduce under what conditions on \(\lambda \) there exists a compact \(\lambda \)-translating soliton \(\varSigma \) with boundary \(\varGamma \) and we provide estimates of the surface area depending on the height of \(\varSigma \). Finally, we study the shape of \(\varSigma \) related with the geometry of \(\varGamma \), in particular, we give conditions that assert that \(\varSigma \) inherits the symmetries of its boundary \(\varGamma \).     

Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time-Dependent Setting

We obtain lower bounds on blow-up of solutions for the 3D magneto-micropolar equations. More precisely, we establish some estimates for the solution \((\mathbf{u},\mathbf{w},\mathbf{b}) (t)\) in its maximal interval \([0,T^{*})\) provided that \(T^{*}<\infty\), which show for \(\delta\in(0,1)\) that \(\|(\mathbf{u},\mathbf{w},\mathbf{b})(t)\|_{\dot{H}^{s}}\) is at least of the order \((T^{*}-t)^{-(\delta s)/(1+2\delta)}\) for \(s\geq1/2+\delta\). In particular, by choosing a suitable \(\delta\), one concludes that \(\|(\mathbf{u},\mathbf{w},\mathbf{b})(t)\|_{\dot{H}^{s}}\) is at least of the order \((T^{*}-t)^{-s/4}\), and \((T^{*}-t)^{1/4-s/2}\) for \(s\geq1\), and \(1/2< s<3/2\), respectively. We also show that \((T^{*}-t)^{-s/3}\) is a lower rate for \(\|(\mathbf{u},\mathbf{w},\mathbf{b})(t)\|_{\dot{H}^{s}}\) if \(s>3/2\).     

A characterization of cut locus for $$C^1$$ hypersurfaces

Let \(\Omega \) be an open set in \(\mathbb {R}^n\) with \(C^1\)-boundary and \(\Sigma \) be the skeleton of \(\Omega \), which consists of points where the distance function to \(\partial \Omega \) is not differentiable. This paper characterizes the cut locus (ridge) \(\overline{\Sigma }\), which is the closure of the skeleton, by introducing a generalized radius of curvature and its lower semicontinuous envelope. As an application we give a sufficient condition for vanishing of the Lebesgue measure of \(\overline{\Sigma }\).     

Height estimates and topology at infinity of hypersurfaces immersed in a certain class of warped products

Height estimates are given for hypersurfaces immersed in a class of warped products of the type \(\mathbb {R}\times _{\rho } M^n\), under the assumption that some higher order mean curvatures are linearly related. When the fiber \(M^n\) is compact and such a hypersurface \(\Sigma ^n\) is noncompact, two-sided and properly immersed, we apply our height estimates in order to get information concerning the topology at infinity of \(\Sigma ^n\). Furthermore, when \(M^n\) is not necessarily compact, using a generalized version of the Omori–Yau maximum principle we establish new half-space theorems for these hypersurfaces.     

Global $$W^{1,p}$$ estimates for solutions to the linearized Monge–Ampère equations

In this paper, we investigate regularity for solutions to the linearized Monge–Ampère equations when the nonhomogeneous term has low integrability. We establish global \(W^{1,p}\) estimates for all \(p<\frac{nq}{n-q}\) for solutions to the equations with right-hand side in \(L^q\) where \(n/2<q\le n\). These estimates hold under natural assumptions on the domain, Monge–Ampère measures, and boundary data. Our estimates are affine invariant analogues of the global \(W^{1,p}\) estimates of N. Winter for fully nonlinear, uniformly elliptic equations.     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

Selection problems of $${{\mathbb {Z}}}^2$$-periodic entropy solutions and viscosity solutions

\({{\mathbb {Z}}}^2\)-periodic entropy solutions of hyperbolic scalar conservation laws and \({{\mathbb {Z}}}^2\)-periodic viscosity solutions of Hamilton–Jacobi equations are not unique in general. However, uniqueness holds for viscous scalar conservation laws and viscous Hamilton–Jacobi equations. Bessi (Commun Math Phys 235:495–511, 2003) investigated the convergence of approximate \({{\mathbb {Z}}}^2\)-periodic solutions to an exact one in the process of the vanishing viscosity method, and characterized this physically natural \({{\mathbb {Z}}}^2\)-periodic solution with the aid of Aubry–Mather theory. In this paper, a similar problem is considered in the process of the finite difference approximation under hyperbolic scaling. We present a selection criterion different from the one in the vanishing viscosity method, which may depend on the approximation parameter.     

Reflecting Diffusion Processes on Manifolds Carrying Geometric Flow

Let \(L_t:=\Delta _t+Z_t\) for a \(C^{\infty }\)-vector field Z on a differentiable manifold M with boundary \(\partial M\), where \(\Delta _t\) is the Laplacian operator, induced by a time dependent metric \(g_t\) differentiable in \(t\in [0,T_\mathrm {c})\). We first establish the derivative formula for the associated reflecting diffusion semigroup generated by \(L_t\). Then, by using parallel displacement and reflection, we construct the couplings for the reflecting \(L_t\)-diffusion processes, which are applied to gradient estimates and Harnack inequalities of the associated heat semigroup. Finally, as applications of the derivative formula, we present a number of equivalent inequalities for a new curvature lower bound and the convexity of the boundary. These inequalities include the gradient estimates, Harnack inequalities, transportation-cost inequalities and other functional inequalities for diffusion semigroups.     

Flatness implies smoothness for solutions of the porous medium equation

One of the major problems in the theory of the porous medium equation \(\partial _t\rho =\Delta _x\rho ^m,\,m > 1\), is the regularity of the solutions \(\rho (t,x)\ge 0\) and the free boundaries \(\Gamma =\partial \{(t,x): \rho >0\}\). Here we assume flatness of the solution and derive \(C^\infty \) regularity of the interface after a small time, as well as \(C^\infty \) regularity of the solution in the positivity set and up to the free boundary for some time interval. The proof starts from Caffarelli’s blueprint of an improvement of flatness by rescaling, and combines it with the Carleson measure approach applied to the degenerate subelliptic equation satisfied by the pressure of the porous medium equation in transformed coordinates. The improvement of flatness finally hinges on Gaussian estimates for the subelliptic problem. We use these facts to prove the following eventual regularity result: solutions defined in the whole space with compactly supported initial data are smooth after a finite time \(T_r\) that depends on \(\rho _0\). More precisely, we prove that for \(t \ge T_r\) the pressure \(\rho ^{m-1}\) is \(C^\infty \) in the positivity set and up to the free boundary, which is a \(C^\infty \) hypersurface. Moreover, \(T_r\) can be estimated in terms of only the initial mass and the initial support radius. This regularity result eliminates the assumption of non-degeneracy on the initial data that has been carried on for decades in the literature. Let us recall that regularization for small times is false, and that as \(t\rightarrow \infty \) the solution increasingly resembles a Barenblatt function and the support looks like a ball.     

Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

We prove that any weak space-time \(L^2\) vanishing viscosity limit of a sequence of strong solutions of Navier–Stokes equations in a bounded domain of \({\mathbb R}^2\) satisfies the Euler equation if the solutions’ local enstrophies are uniformly bounded. We also prove that \(t-a.e.\) weak \(L^2\) inviscid limits of solutions of 3D Navier–Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second-order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a priori excluded in the limit.     

On the regularity of weak solutions to Burgers’ equation with finite entropy production

Bounded weak solutions of Burgers’ equation \(\partial _tu+\partial _x(u^2/2)=0\) that are not entropy solutions need in general not be BV. Nevertheless it is known that solutions with finite entropy productions have a BV-like structure: a rectifiable jump set of dimension one can be identified, outside which u has vanishing mean oscillation at all points. But it is not known whether all points outside this jump set are Lebesgue points, as they would be for BV solutions. In the present article we show that the set of non-Lebesgue points of u has Hausdorff dimension at most one. In contrast with the aforementioned structure result, we need only one particular entropy production to be a finite Radon measure, namely \(\mu =\partial _t (u^2/2)+\partial _x(u^3/3)\). We prove Hölder regularity at points where \(\mu \) has finite \((1+\alpha )\)-dimensional upper density for some \(\alpha >0\). The proof is inspired by a result of De Lellis, Westdickenberg and the second author : if \(\mu _+\) has vanishing 1-dimensional upper density, then u is an entropy solution. We obtain a quantitative version of this statement: if \(\mu _+\) is small then u is close in \(L^1\) to an entropy solution.     

Normalized bound states for the nonlinear Schrödinger equation in bounded domains

Given \(\rho >0\), we study the elliptic problem

$$\begin{aligned} \text {find } (U,\lambda )\in H^1_0(\Omega )\times {\mathbb {R}}\text { such that } {\left\{ \begin{array}{ll} -\Delta U+\lambda U=|U|^{p-1}U\\ \int _{\Omega } U^2\, dx=\rho , \end{array}\right. } \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain and \(p>1\) is Sobolev-subcritical, searching for conditions (about \(\rho \), N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is \(L^2\)-subcritical, i.e. \(1<p<1+4/N\), the problem admits solutions for every \(\rho >0\). In the \(L^2\)-critical and supercritical case, i.e. when \(1+4/N \le p < 2^*-1\), we show that, for any \(k\in {\mathbb {N}}\), the problem admits solutions having Morse index bounded above by k only if \(\rho \) is sufficiently small. Next we provide existence results for certain ranges of \(\rho \), which can be estimated in terms of the Dirichlet eigenvalues of \(-\Delta \) in \(H^1_0(\Omega )\), extending to changing sign solutions and to general domains some results obtained in Noris et al. in Anal. PDE 7:1807–1838, 2014 for positive solutions in the ball.

     

On partial disjunction properties of theories containing Peano arithmetic

Let \(\varGamma \) be a class of formulas. We say that a theory T in classical logic has the \(\varGamma \)-disjunction property if for any \(\varGamma \) sentences \(\varphi \) and \(\psi \), either \(T \vdash \varphi \) or \(T \vdash \psi \) whenever \(T \vdash \varphi \vee \psi \). First, we characterize the \(\varGamma \)-disjunction property in terms of the notion of partial conservativity. Secondly, we prove a model theoretic characterization result for \(\varSigma _n\)-disjunction property. Thirdly, we investigate relationships between partial disjunction properties and several other properties of theories containing Peano arithmetic. Finally, we investigate unprovability of formalized partial disjunction properties.     

Rigidity of Free Boundary Surfaces in Compact 3-Manifolds with Strictly Convex Boundary

In this paper, we obtain an analogue of Toponogov theorem in dimension 3 for compact manifolds \(M^3\) with nonnegative Ricci curvature and strictly convex boundary \(\partial M\). Here we obtain a sharp upper bound for the length \(L(\partial \Sigma )\) of the boundary \(\partial \Sigma \) of a free boundary minimal surface \(\Sigma ^2\) in \(M^3\) in terms of the genus of \(\Sigma \) and the number of connected components of \(\partial \Sigma \), assuming \(\Sigma \) has index one. After, under a natural hypothesis on the geometry of M along \(\partial M\), we prove that if \(L(\partial \Sigma )\) saturates the respective upper bound, then \(M^3\) is isometric to the Euclidean 3-ball and \(\Sigma ^2\) is isometric to the Euclidean disk. In particular, we get a sharp upper bound for the area of \(\Sigma \), when \(M^3\) is a strictly convex body in \(\mathbb {R}^3\), which is saturated only on the Euclidean 3-balls (by the Euclidean disks). We also consider similar results for free boundary stable CMC surfaces.     

On the maximality of a set of mutually orthogonal Sudoku Latin Squares

The maximum number of mutually orthogonal Sudoku Latin squares (MOSLS) of order \(n=m^2\) is \(n-m\). In this paper, we construct for \(n=q^2\), q a prime power, a set of \(q^2-q-1\) MOSLS of order \(q^2\) that cannot be extended to a set of \(q^2-q\) MOSLS. This contrasts to the theory of ordinary Latin squares of order n, where each set of \(n-2\) mutually orthogonal Latin Squares (MOLS) can be extended to a set of \(n-1\) MOLS (which is best possible). For this proof, we construct a particular maximal partial spread of size \(q^2-q+1\) in \(\mathrm {PG}(3,q)\) and use a connection between Sudoku Latin squares and projective geometry, established by Bailey, Cameron and Connelly.     

The Raising Steps Method: Applications to the $$\bar{\partial }$$ Equation in Stein Manifolds

In order to have estimates on the solutions of the equation \(\bar{\partial }u=\omega \) on a Stein manifold, we introduce a new method, the “raising steps method”, to get global results from local ones. In particular, it allows us to transfer results from open sets in \({\mathbb {C}}^{n}\) to open sets in a Stein manifold. Using it, we get \(\displaystyle L^{r}-L^{s}\) results for solutions of the equation \(\bar{\partial }u=\omega \) with a gain, \(\displaystyle s>r\), in strictly pseudo convex domains in Stein manifolds. We also get \(\displaystyle L^{r}-L^{s}\) results for domains in \({\mathbb {C}}^{n}\) locally biholomorphic to convex domains of finite type.     

A Polynomial-Type Jost Solution and Spectral Properties of a Self-Adjoint Quantum-Difference Operator

In this paper, we find a polynomial-type Jost solution of a self-adjoint \(q\)-difference equation of second order. Then we investigate the analytical properties and asymptotic behavior of the Jost solution. We prove that the self-adjoint operator \(L\) generated by the \(q\)-difference expression of second order has essential spectrum filling the segment \([-2\sqrt{q},2\sqrt{q}]\), \(q>1\). Finally, we examine the properties of the eigenvalues of \(L\).     

Estimating Averages of Order Statistics of Bivariate Functions

We prove uniform estimates for the expected value of averages of order statistics of bivariate functions in terms of their largest values by a direct analysis. As an application, uniform estimates for the expected value of averages of order statistics of sequences of independent random variables in terms of Orlicz norms are obtained. In the case where the bivariate functions are matrices, we provide a “minimal” probability space which allows us to C-embed certain Orlicz spaces \(\ell _M^n\) into \(\ell _1^{cn^3}\), with \(c,C>0\) being absolute constants.