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The reflection principle in the control problem of the heat equationFeb 21 2019We consider the control problem for the generalized heat equation for a Schr\"odinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on its respective ... More
The structure of the singular set in the thin obstacle problem for degenerate parabolic equationsFeb 20 2019We study the singular set in the thin obstacle problem (1.1) for degenerate parabolic equations with weight $|y|^a$ for $a \in (-1,1)$. Such problem arises as the local extension of the obstacle problem for the fractional heat operator $(\partial_t - ... More
Hopf's lemma for viscosity solutions to a class of non-local equations with applicationsFeb 20 2019We consider a large family of non-local equations featuring Markov generators of L\'evy processes, and establish a non-local Hopf's lemma and a variety of maximum principles for viscosity solutions. We then apply these results to study the principal eigenvalue ... More
On weak (measure-valued)-strong uniqueness for compressible Navier-Stokes system with non-monotone pressure lawFeb 19 2019In this paper our goal is to define a renormalised dissipative measure--valued (rDMV) solution of compressible Navier--Stokes system for fluids with non-monotone pressure--density relation. We prove existence of rDMV solutions and establish a suitable ... More
Global Strong Solution With BV Derivatives to Singular Solid-on-Solid model With Exponential NonlinearityFeb 19 2019In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearity $$h_t = \nabla \cdot (\frac{1}{|\nabla h|} \nabla e^{\frac{\delta E}{\delta ... More
Onsager's conjecture in bounded domains for the conservation of entropy and other companion lawsFeb 19 2019We show that weak solutions of general conservation laws in bounded domains conserve their generalized entropy, and other respective companion laws, if they possess a certain fractional differentiability of order 1/3 in the interior of the domain, and ... More
Similarity Solutions For The Complex Burgers' HierarchyFeb 19 2019A detailed analysis of the invariant point transformations for the first four partial differential equations which belong to the Complex Burgers` Hierarchy is performed. Moreover, a detailed application of the reduction process through the Lie point symmetries ... More
Fractional Gaussian estimates and holomorphy of semigroupsFeb 19 2019Let $\Omega\subset\R^N$ be an arbitrary open set and denote by $(e^{-t(-\Delta)_{\RR^N}^s})_{t\ge 0}$ (where $0<s<1$) the semigroup on $L^2(\RR^N)$ generated by the fractional Laplace operator. In the first part of the paper we show that if $T$ is a self-adjoint ... More
Geometric wave propagator on Riemannian manifoldsFeb 19 2019We study the propagator of the wave equation on a closed Riemannian manifold $M$. We propose a geometric approach to the construction of the propagator as a single oscillatory integral global both in space and in time with a distinguished complex-valued ... More
Well-posedness for Fractional Dissipative Benjamin-Ono EquationsFeb 19 2019This paper is devoted to study the Cauchy problem for the fractional dissipative BO equations $u_t+\mathcal{H}u_{xx}-(D_x^{\alpha}-D_x^{\beta})u+uu_x=0$, $0< \alpha < \beta$. When $1<\beta <2$, we prove GWP in $H^s(\mathbb{R})$, $s>-\beta/4$. For $\beta\geq ... More
A dichotomy for minimal hypersurfaces in manifolds thick at infinityFeb 18 2019Let $(M^{n+1},g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2\leq n\leq 6$. Our main theorem generalizes the solution of Yau's conjecture for minimal surfaces and builds on a result of Gromov. Suppose that $(M,g)$ is thick at infinity, ... More
The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov spaceFeb 18 2019In this work, we investigate the Cauchy problem for the spatially inhomogeneous non-cutoff Kac equation. If the initial datum belongs to the spatially critical Besov space, we can prove the well-posedness of weak solution under a perturbation framework. ... More
Averaging along degenerate flows on the annulusFeb 18 2019This paper focuses on the study of flows on the annulus that do not possess a spectral gap. Estimates for the so-called density of states are obtained for small values of the spectrum. Those estimates lead to rates of decay for the averaging dynamic of ... More
Limit behaviour of a singular perturbation problem for the biharmonic operatorFeb 18 2019We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic ... More
Near-critical reflection of internal wavesFeb 18 2019Internal waves describe the (linear) response of an incompressible stably stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection ... More
Microscopic validation of a variational model of epitaxially strained crystalline filmFeb 18 2019A discrete-to-continuum analysis for free-boundary problems related to crystalline films deposited on substrates is performed by $\Gamma$-convergence. The discrete model here introduced is characterized by an energy with two contributions, the surface ... More
A unified model for stress-driven rearrangement instabilitiesFeb 18 2019A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is characterized ... More
Unexpected differences between fundamental solutions of general higher-order elliptic operators and of products of second-order operatorsFeb 18 2019We study fundamental solutions of elliptic operators of order $2m\geq4$ with constant coefficients in large dimensions $n>2m$, where their singularities become unbounded. For compositions of second-order operators these can be chosen as convolution products ... More
Well-posedness of a non-local model for material flow on conveyor beltsFeb 18 2019In this paper, we focus on finite volume approximation schemes to solve a non-local material flow model in two space dimensions. Based on the numerical discretisation with dimensional splitting, we prove the convergence of the approximate solutions, where ... More
Local well-posedness for fourth order Benjamin-Ono type equationsFeb 18 2019We continue to study the local well-posedness for higher order Benjamin-Ono type equations, especially fourth order equations. The proof is based on the energy methods with correction terms. Although one of correction terms can eliminate the highest order ... More
(In)Stability of Travelling Waves in a Model of HaptotaxisFeb 18 2019We examine the spectral stability of travelling waves of the haptotaxis model studied previously by Harley et al. (2014). In the process we apply Li\'enard coordinates to the linearised stability problem and develop a new method for numerically computing ... More
2-parameter $τ$-function for the first Painlevé equation -Topological recursion and direct monodromy problem via exact WKB analysis-Feb 18 2019We show that a 2-parameter family of $\tau$-functions for the first Painlev\'e equation can be constructed by the discrete Fourier transform of the topological recursion partition function for a family of elliptic curves. We also perform an exact WKB ... More
Approximate convexity principles and applications to PDEs in convex domainsFeb 18 2019We obtain approximate convexity principles for solutions to some classes of nonlinear elliptic partial differential equations in convex domains involving approximately concave nonlinearities. Furthermore, we provide some applications to some meaningful ... More
Inverse coefficient problems for a transport equation by local Carleman estimateFeb 17 2019We consider the transport equation $\ppp_tu(x,t) + (H(x)\cdot \nabla u(x,t)) + p(x)u(x,t) = 0$ in $\OOO \times (0,T)$ where $\OOO \subset \R^n$ is a bounded domain, and discuss two inverse problems which consist of determining a vector-valued function ... More
Extended lifespan of the fractional BBM equationFeb 17 2019For $0<\alpha<1$ and with initial data $\vert\vert u_0\vert\vert_{H^{N+\frac{\alpha}{2}}}=\varepsilon$, sufficently small, we show that the existence time for solutions of the fractional BBM equation $\partial_tu+\partial_xu+u\partial_xu+\vert\mathrm{D}\vert^\alpha\partial_tu=0$, ... More
On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flowsFeb 17 2019The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \begin{align*} \partial_t u + \nabla_u u &= - \mathrm{grad}_g p \\ \mathrm{div}_g u &= 0, \end{align*} where $u: [0,T] \to \Gamma(T M)$ is the velocity field and ... More
Repartition of the quasi-stationary distribution and first exit point density for a double-well potentialFeb 17 2019Let $f: \mathbb R^{d} \to \mathbb R$ be a smooth function and $(X_t)_{t\ge 0}$ be the stochastic process solution to the overdamped Langevin dynamics $$d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t.$$ Let $\Omega\subset \mathbb R^d$ be a smooth bounded ... More
Blowup for the nonlinear heat equation with small initial data in scale-invariant Besov normsFeb 17 2019We consider the Cauchy problem of the nonlinear heat equation $u_t -\Delta u= u^{b},\ u(0,x)=u_0$, with $b\geq 2$ and $b\in \mathbb{N}$. We prove that initial data $u_0\in \mathcal{S}(\mathbb{R}^{n})$ (the Schwartz class)arbitrarily small in the scale ... More
Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited modelFeb 17 2019We are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In ... More
The Stokes limit in a three-dimensional chemotaxis-Navier-Stokes systemFeb 17 2019We consider initial-boundary value problems for the $\kappa$-dependent family of chemotaxis-(Navier--)Stokes systems \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=\Delta n-\nabla\!\cdot(n\nabla ... More
Solving the 4NLS with white noise initial dataFeb 16 2019We construct global-in-time singular dynamics for the (renormalized) cubic fourth order nonlinear Schr\"odinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the "random-resonant / nonlinear ... More
Homogenization of the Poisson equation in a non-periodically perforated domainFeb 16 2019We study the Poisson equation in a perforated domain with homogeneous Dirichlet boundary conditions. The size of the perforations is denoted by $\epsilon$ > 0, and is proportional to the distance between neighbouring perforations. In the periodic case, ... More
On the $C^1$ and $C^2$-convergence to weak K.A.M. solutionsFeb 16 2019We introduce a notion of upper Green regular solutions to the Lax-Oleinik semi-group that is defined on the set of $C^0$ functions of a closed manifold via a Tonelli Lagrangian. Then we prove some weak $C^2$ convergence results to such a solution for ... More
Asymptotic behaviour of a structured population model on a space of measuresFeb 16 2019In this paper we consider a physiologically structured population model with distributed states at birth, formulated on the space of non-negative Radon measures. Using a characterisation of the pre-dual space of bounded Lipschitz functions, we show how ... More
Local $L^{p}$-solution for semilinear heat equation with fractional noiseFeb 16 2019We study the $L^{p}$-solutions for the semilinear heat equation with unbounded coefficients and driven by a infinite dimensional fractional Brownian motion with self-similarity parameter $H > 1/2$. Existence and uniqueness of local mild solutions are ... More
Pairings between bounded divergence-measure vector fields and BV functionsFeb 16 2019We introduce a family of pairings between a bounded divergence-measure vector field $\boldsymbol{A}$ and a function $u$ of bounded variation, depending on the choice of the pointwise representative of $u$. We prove that these pairings inherit from the ... More
Polarizing Anisotropic Heisenberg GroupsFeb 15 2019We expand the class of polarizable Carnot groups by implementing a technique to polarize anisotropic Heisenberg groups.
Existence, uniqueness and regularity for the stochastic Ericksen-Leslie equationFeb 15 2019We investigate existence and uniqueness for the stochastic liquid crystal flow driven by colored noise on the two-dimensional torus. After giving a natural uniqueness criterion, we prove local solvability in $L^p$-based spaces, for every $p>2.$ Thanks ... More
Global Logarithmic Stability Of The Cauchy Problem For Anisotropic Wave EquationsFeb 15 2019We discuss the Cauchy problem for anisotropic wave equations. Precisely, we address the question to know which kind of Cauchy data on the lateral boundary are necessary to guarantee uniqueness of solutions of an anisotropic wave equation. In the case ... More
Quantitative analysis of a singularly perturbed shape optimization problem in a polygonFeb 15 2019We carry on our study of the connection between two shape optimization problems with spectral cost. On the one hand, we consider the optimal design problem for the survival threshold of a population living in a heterogenous habitat $\Omega$; this problem ... More
A bulk-surface reaction-diffusion system for cell polarizationFeb 15 2019We propose a model for cell polarization as a response to an external signal which results in a system of PDEs for different variants of a protein on the cell surface and interior respectively. We study stationary states of this model in certain parameter ... More
Matrix solitons solutions of the modified Korteweg-de Vries equationFeb 15 2019Nonlinear non-Abelian Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations and their links via Baecklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified Korteweg-de Vries ... More
On nodal and generalized singular structures of Laplacian eigenfunctions and applicationsFeb 15 2019In this paper, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviours of the eigenfunctions. We consider the nodal lines and also introduce the notion ... More
Quenched invariance principle for random walks among random degenerate conductancesFeb 15 2019We consider the random conductance model in a stationary and ergodic environment. Under suitable moment conditions on the conductances and their inverse, we prove a quenched invariance principle for the random walk among the random conductances. The moment ... More
Stochastic homogenization of the Landau-Lifshitz-Gilbert equationFeb 15 2019Following the ideas of V. V. Zhikov and A. L. Pyatnitski, and more precisely the stochastic two-scale convergence, this paper establishes a homogenization theorem in a stochastic setting for two nonlinear equations : the equation of harmonic maps into ... More
Detection of Hermitian connections in wave equations with cubic non-linearityFeb 15 2019We consider the geometric non-linear inverse problem of recovering a Hermitian connection $A$ from the source-to-solution map of the cubic wave equation $\Box_{A}\phi+\kappa |\phi|^{2}\phi=f$, where $\kappa\neq 0$ and $\Box_{A}$ is the connection wave ... More
Superposition principle and schemes for Measure Differential EquationsFeb 14 2019Measure Differential Equations (MDE) describe the evolution of probability measures driven by probability velocity fields, i.e. probability measures on the tangent bundle. They are, on one side, a measure-theoretic generalization of ordinary differential ... More
On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactionsFeb 14 2019We study the following nonlocal mixed order Gross-Pitaevskii equation $$i\,\partial_t \psi=-\frac{1}{2}\,\Delta \psi+V_{ext}\,\psi+\lambda_1\,|\psi|^2\,\psi+\lambda_2\,(K*|\psi|^2)\,\psi+\lambda_3\,|\psi|^{p-2}\,\psi,$$ where $K$ is the classical dipole-dipole ... More
The Strong Maximum Principle for Schrödinger operators on fractalsFeb 14 2019We prove a strong maximum principle for Schr\"odinger operators defined on a class of fractal sets and their blowups without boundary. Our primary interest is in weaker regularity conditions than have previously appeared in the literature; in particular ... More
Analysis of a time-discrete scheme for the Navier-Stokes/Allen-Cahn modelFeb 14 2019This paper address the approximation of the dynamic of two fluids with non matching densities and viscosities modeled by the Allen-Cahn equation coupled with the time dependent Navier-Stokes equations. Existence, uniqueness and a maximum principle are ... More
Point interactions for 3D sub-LaplaciansFeb 14 2019In this paper we show that, for a sub-Laplacian $\Delta$ on a $3$-dimensional manifold $M$, no point interaction centered at a point $q_0\in M$ exists. When $M$ is complete w.r.t. the associated sub-Riemannian structure, this means that $\Delta$ acting ... More
Solitary waves for weakly dispersive equations with inhomogeneous nonlinearitiesFeb 14 2019We show existence of solitary-wave solutions to the equation \begin{equation*} u_t+ (Lu - n(u))_x = 0\,, \end{equation*} for weak assumptions on the dispersion $L$ and the nonlinearity $n$. The symbol $m$ of the Fourier multiplier $L$ is allowed to be ... More
Regularizing effect of the lower-order terms in elliptic problems with Orlicz growthFeb 14 2019Under various conditions on the data we analyse how appearence of lower order terms affects the gradient estimates on solutions to a general nonlinear elliptic equation of the form \[-{\rm div}\, a(x,Du)+b(x,u)=\mu\] with data $\mu$ not belonging to the ... More
Smooth Solutions of the Surface Semi-Geostrophic EquationsFeb 14 2019The semi-geostrophic equations have attracted the attention of the physical and mathematical communities since the work of Hoskins in the 1970s owing to their ability to model the formation of fronts in rotation-dominated flows, and also to their connection ... More
Shrinking scale equidistribution for monochromatic random waves on compact manifoldsFeb 14 2019We prove equidistribution at shrinking scales for the monochromatic ensemble on a compact Riemannian manifold of any dimension. This ensemble on an arbitrary manifold takes a slowly growing spectral window in order to synthesize a random function. With ... More
Existence of weak solution for mean curvature flow with transport term and forcing termFeb 14 2019We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove a global-in-time existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn ... More
Extraction formulae for an inverse boundary value problem for the equation $\nabla\cdot(σ-iωε)\nabla u=0$Feb 14 2019We consider an inverse boundary value problem for the equation $\nabla\cdot(\sigma-i\omega\epsilon)\nabla u=0$ in a given bounded domain $\Omega$ at a fixed $\omega>0$. $\sigma$ and $\epsilon$ denote the conductivity and permittivity of the material forming ... More
Global existence of weak solution to compressible two-fluid model without any domination condition in three dimensionsFeb 14 2019We consider the Dirichlet problem for a compressible two-fluid model in three dimensions, and obtain the global existence of weak solution with large initial data and independent adiabatic constants \Gamma,\gamma>=9/5. The pressure functions are of two ... More
Liouville theorems on the upper half spaceFeb 14 2019In this paper we shall establish some Liouville theorems for solutions bounded from below to certain linear elliptic equations on the upper half space. In particular, we show that for $a \in (0, 1)$ constants are the only $C^1$ up to the boundary positive ... More
On reconstruction from a partial knowledge of the Neumann-to-Dirichlet operatorFeb 14 2019We give formulae that yield an information about the location of an unknown polygonal inclusion having unknown constant conductivity inside a known conductive material having known constant conductivity from a partial knowledge of the Neumann -to-Dirichlet ... More
On reconstruction in the inverse conductivity problem with one measurementFeb 14 2019We consider an inverse problem for electrically conductive material occupying a domain $\Omega$ in $\Bbb R^2$. Let $\gamma$ be the conductivity of $\Omega$, and $D$ a subdomain of $\Omega$. We assume that $\gamma$ is a positive constant $k$ on $D$, $k\not=1$ ... More
Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniquenessFeb 14 2019We consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows to define global solutions, even in the presence of blow-ups of the freezing rate. ... More
Fractional Operators Applied to Geophysical ElectromagneticsFeb 13 2019A growing body of applied mathematics literature in recent years has focussed on the application of fractional calculus to problems of anomalous transport. In these analyses, the anomalous transport (of charge, tracers, fluid, etc.) is presumed attributable ... More
A Continuum Model for Cities Based on the Macroscopic Fundamental Diagram: a Semi-Lagrangian Solution MethodFeb 13 2019This paper presents a formulation of the reactive dynamic user equilibrium problem in continuum form using a network-level Macroscopic Fundamental Diagram (MFD). Compared to existing continuum models for cities -- all based in Hughes' pedestrian model ... More
Prescribing the center of mass of a multi-soliton solution for a perturbed semilinear wave equationFeb 13 2019We construct a finite-time blow-up solution for a class of strongly perturbed semilinear wave equation with an isolated characteristic point in one space dimension. Given any integer $k\ge 2$ and $\zeta_0 \in \mathbb{R}$, we construct a blow-up solution ... More
Generalized ergodic problems: existence and uniqueness structures of solutionsFeb 13 2019We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat $n$-dimensional torus. We first obtain existence of solutions to this problem under quite general assumptions. Various examples are presented ... More
The Fujita-Kato Theorem for a Corotational Johnson-Segalman ModelFeb 13 2019In this paper, we investigate the Cauchy problem associated to a system of PDE's of Johnson-Segalman type. The considered model describes the evolution of certain viscoelastic fluids within a corotational framework. We show that some widespread results ... More
Self-Adjointness of two dimensional Dirac operators on corner domainsFeb 13 2019We study the self-adjointenss of the two-dimensional Dirac operator with Quantum-dot and Lorentz-scalar $\delta$-shell boundary conditions, on piecewise $C^2$ domains with finitely many corners. For both models, we prove the existence of a unique self-adjoint ... More
Sharp Hardy-Littlewood-Sobolev inequalities on compact CR manifoldFeb 13 2019Assume that $M$ is a CR compact manifold without boundary and CR Yamabe invariant $\mathcal{Y}(M)$ is positive. Here, we devote to study a class of sharp Hardy-Littlewood-Sobolev inequality as follows \begin{equation*} \Bigl| \int_M\int_M [G_\xi^\theta(\eta)]^{\frac{Q-\alpha}{Q-2}} ... More
Symmetrization with respect to mixed volumesFeb 13 2019In this paper we introduce new symmetrization with respect to mixed volumes or anisotropic curvature integral, which generalizes the one with respect to quermassintegral due to Talenti and Tso. We show that such symmetrization diminishes the anisotropic ... More
Curvature stabilized Skyrmions with angular momentumFeb 13 2019We examine skyrmionic field configurations on a spherical ferromagnet with large normal anisotropy. Exploiting variational concepts of angular momentum we find a new family of localized solutions to the Landau-Lifshitz equation that are topologically ... More
On the classification of incompressible fluids and a mathematical analysis of the equations that govern their motionFeb 13 2019In the first part of the paper we provide a new classification of incompressible fluids characterized by a continuous monotone relation between the velocity gradient and the Cauchy stress. The considered class includes Euler fluids, Navier-Stokes fluids, ... More
A Partial Data Problem in Linear ElasticityFeb 13 2019We discuss the determination of the Lam\'e parameters of an elastic material by the means of boundary measurements. We will combine previous results of Eskin-Ralston and Isakov to prove inverse results in the case of bounded domains with partial data. ... More
Waves Interacting With A Partially Immersed Obstacle In The Boussinesq RegimeFeb 13 2019This paper is devoted to the derivation and mathematical analysis of a wave-structure interaction problem which can be reduced to a transmission problem for a Boussinesq system. Initial boundary value problems and transmission problems in dimension d= ... More
Mass-conserving self-similar solutions to coagulation-fragmentation equationsFeb 13 2019Existence of mass-conserving self-similar solutions with a sufficiently small total mass is proved for a specific class of homogeneous coagulation and fragmentation coefficients. The proof combines a dynamical approach to construct such solutions for ... More
Existence of Weak Solutions for $p(.)$-Laplacian Equation via Compact Embeddings of the Double Weighted Variable Exponent Sobolev SpacesFeb 13 2019In this study, we define double weighted variable exponent Sobolev spaces $W^{1,q(.),p(.)}\left( \Omega ,\vartheta _{0},\vartheta \right) $ with respect to two different weight functions. Also, we investigate the basic properties of this spaces. Moreover, ... More
From delayed minimization to the harmonic map heat equationFeb 13 2019In the context of cell motility modelling and more particularly related to the Filament Based Lamelipodium Model [Manhart et al 2015 & 2017], this work deals with a rigorous mathematical proof of convergence between an adhesion delay non-linear space-dependent ... More
Weighted Stochastic Field Exponent Sobolev Spaces and Nonlinear Degenerated Elliptic ProblemFeb 13 2019In this study, we consider weighted stochastic field exponent function spaces $L_{\vartheta }^{p(.,.)}\left( D\times \Omega \right) $ and $W_{\vartheta }^{k,p(.,.)}\left( D\times \Omega \right) $. Also, we investigate some basic properties and embeddings ... More
Weak solutions of semilinear elliptic equations with Leray-Hardy potential and measure dataFeb 13 2019We study existence and stability of solutions of (E 1) --$\Delta$u + $\mu$ |x| 2 u + g(u) = $\nu$ in $\Omega$, u = 0 on $\partial$$\Omega$, where $\Omega$ is a bounded, smooth domain of R N , N $\ge$ 2, containing the origin, $\mu$ $\ge$ -- (N --2) 2 ... More
Global existence and blowup for Choquard equations with an inverse-square potentialFeb 13 2019In this paper, the Choquard equation with an inverse-square potential and both focusing and defocusing nonlinearities in the energy-subcritical regime is investigated. For all the cases, the local well-posedness result in $H^1(\mathbb{R}^N)$ is established. ... More
Regularity of extremal solutions of nonlocal elliptic systemsFeb 12 2019We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problems with an integro-differential operator, including the fractional Laplacian, of the form of $$ \mathcal L(u (x))= \lim_{\epsilon\to 0} \int_{\mathbb R^n\setminus B_\epsilon(x) ... More
Asymptotic expansion for the eigenvalues of a perturbed anharmonic oscillatorFeb 12 2019In this article, we study the spectral properties of the perturbation of the generalized anharmonic oscillator. We consider a piecewise H\"older continuous perturbation and investigate how the H\"older constant can affect the eigenvalues. More precisely, ... More
Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problemsFeb 12 2019Let us consider the following minimum problem \[ \lambda_\alpha(p,r)= \min_{\substack{u\in W_{0}^{1,p}(-1,1)\\ u\not\equiv0}}\dfrac{\displaystyle\int_{-1}^{1}|u'|^{p}dx+\alpha\left|\int_{-1}^{1}|u|^{r-1}u\, dx\right|^{\frac pr}}{\displaystyle\int_{-1}^{1}|u|^{p}dx}, ... More
Boundary null-controllability of two coupled parabolic equations : simultaneous condensation of eigenvalues and eigenfunctionsFeb 12 2019Let the matrix operator L = D$\partial$xx + q(x)A0, with D = diag(1, $\nu$), $\nu$ = 1, q $\in$ L $\infty$ (0, $\pi$), and A0 is a Jordan block of order 1. We analyze the boundary null controllability for system yt -- Ly = 0. When v \notin Q * + and q(x) ... More
Degenerate nonlocal Cahn-Hilliard equations: well-posedness, regularity and local asymptoticsFeb 12 2019Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection ... More
Stable determination of polygonal inclusions in Calderón's problem by a single partial boundary measurementFeb 12 2019We are concerned with the Calder\'on problem of determining an unknown conductivity of a body from the associated boundary measurement. We establish a logarithmic type stability estimate in terms of the Hausdorff distance in determining the support of ... More
Quadratic differential equations : partial Gelfand-Shilov smoothing effect and null-controllabilityFeb 12 2019We study the partial Gelfand-Shilov regularizing effect and the exponential decay for the solutions to evolution equations associated to a class of accretive non-selfadjoint quadratic operators, which fail to be globally hypoelliptic on the whole phase ... More
Global solutions of continuous coagulation-fragmentation equations with unbounded coefficientsFeb 12 2019In this paper we prove the existence of global classical solutions to continuous coagulation-fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus ... More
Pointwise gradient estimates for a class of singular quasilinear equation with measure dataFeb 12 2019Local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data $-\operatorname{div}(A(x,\nabla u))=\mu$ in a bounded and possibly nonsmooth domain $\Omega$ in $\mathbb{R}^n$. Here $\operatorname{div}(A(x,\nabla ... More
Periodic Traveling-wave solutions for regularized dispersive equations: Sufficient conditions for orbital stability with applicationsFeb 12 2019In this paper, we establish a new criterion for the orbital stability of periodic waves related to a general class of regularized dispersive equations. More specifically, we present sufficient conditions for the stability without knowing the positiveness ... More
Wellposedness and convergence of solutions to a class of forced non-diffusive equations with applicationsFeb 12 2019This paper considers a family of non-diffusive active scalar equations where a viscosity type parameter enters the equations via the constitutive law that relates the drift velocity with the scalar field. The resulting operator is smooth when the viscosity ... More
On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic spaceFeb 12 2019In this paper, we study nonlinear Helmholtz equations (NLH) $-\Delta_{\mathbb{H}^N} u - \frac{(N-1)^2}{4} u -\lambda^2 u = \Gamma|u|^{p-2}u$ in $\mathbb{H}^N$, $N\geq 2$ where $\Delta_{\mathbb{H}^N}$ denotes the Laplace-Beltrami operator in the hyperbolic ... More
Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equationsFeb 12 2019In this note we prove that, under some minimal regularity assumptions on the initial datum, solutions to the spatially homogenous Boltzmann and Landau equations for hard potentials uniformly propagate the Fisher information. The proof of such a result ... More
The Dirichlet problem for elliptic equation with several singular coefficientsFeb 12 2019Recently found all the fundamental solutions of a multidimensional singular elliptic equation are expressed in terms of the well-known Lauricella hypergeometric function in many variables. In this paper, we find a unique solution of the Dirichlet problem ... More
A trajectory map for the pressureless Euler equationsFeb 12 2019We consider the dynamics of a collection of particles that interact pairwise and are restricted to move along the real line. Moreover, we focus on the situation in which particles undergo perfectly inelastic collisions when they collide. The equations ... More
Resonance in rarefaction and shock curves: local analysis and numerics of the continuation methodFeb 11 2019In this paper, we describe certain crucial steps in the development of an algorithm for finding the Riemann solution in systems of conservation laws. We relax the classical hypotheses of strict hyperbolicity and genuine nonlinearity of Lax. First, we ... More
Green's function for Poisson's equation and the EEG equation with Neumann boundary condition on $n$-ballsFeb 11 2019We provide an elementary derivation of the Green's function for Poisson's equation with Neumann boundary data on balls of arbitrary dimension, which was recently found in [Sadybekov et al., Eurasian Math. J. 7(2):100-105, 2016]. The underlying idea consists ... More
Renormalizing the Kardar-Parisi-Zhang equation in $d\geq 3$ in weak disorderFeb 11 2019We study Kardar-Parisi-Zhang equation in spatial dimension 3 or larger driven by a Gaussian space-time white noise with a small convolution in space. When the noise intensity is small, it is known that the solutions converge to a random limit as the smoothing ... More
Variational methods for the kinetic Fokker-Planck equationFeb 11 2019We develop a functional analytic approach to the study of the Kramers and kinetic Fokker-Planck equations which parallels the classical $H^1$ theory of uniformly elliptic equations. In particular, we identify a function space analogous to $H^1$ and develop ... More
Blow-up for the 3-dimensional axially symmetric harmonic map flow into S2Feb 11 2019We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) ... More