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Relaxation of Functionals in the Space of Vector-Valued Functions of Bounded HessianFeb 08 2018In this paper it is shown that if $\Omega \subset \mathbb{R}^N$ is an open, bounded Lipschitz set, and if $f: \Omega \times \mathbb{R}^{d \times N \times N} \rightarrow [0, \infty)$ is a continuous function with $f(x, \cdot)$ of linear growth for all ... More

Stability of Ricci de Turck flow on Singular SpacesFeb 08 2018In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical ... More

Weak and strong connectivity regimes for a general time elapsed neuron network modelFeb 08 2018For large fully connected neuron networks, we study the dynamics of homogenous assemblies of interacting neurons described by time elapsed models. Under general assumptions on the firing rate which include the ones made in previous works [7, 8, 6], we ... More

Existence and uniqueness of solutions to singular Cahn-Hilliard equations with nonlinear viscosity terms and dynamic boundary conditionsFeb 08 2018We prove global existence and uniqueness of solutions to a Cahn-Hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions. The problem is highly singular, characterized by four nonlinearities and two separate diffusive terms, ... More

Convergence as period goes to infinity of spectra of periodic traveling waves toward essential spectra of a homoclinic limitFeb 08 2018We revisit the analysis by R.A. Gardner of convergence of spectra of periodic traveling waves in the homoclinic, or infinite-period limit, extending his results to the case of essential rather than point spectra of the limiting homoclinic wave. Notably, ... More

On a polynomial scalar perturbation of a Schrödinger system in $L^p$-spacesFeb 08 2018In the paper \cite{KLMR} the $L^p$-realization $L_p$ of the matrix Schr\"odinger operator $\mathcal{L}u=div(Q\nabla u)+Vu$ was studied. The generation of a semigroup in $L^p(\R^d,\C^m)$ and characterization of the domain $D(L_p)$ has been established. ... More

Nonlinear diffusion equations with Robin boundary conditions as asymptotic limits of Cahn-Hilliard systemsFeb 08 2018Condition imposed on the nonlinear terms of a nonlinear diffusion equation with {R}obin boundary condition is the main focus of this paper. The degenerate parabolic equations, such as the {S}tefan problem, the {H}ele--{S}haw problem, the porous medium ... More

Ground states and concentration of mass in stationary Mean Field Games with superlinear HamiltoniansFeb 08 2018In this paper we provide the existence of classical solutions to stationary mean field game systems in the whole space $\mathbb{R}^N$, with coercive potential, aggregating local coupling, and under general conditions on the Hamiltonian, completing the ... More

The dynamics of disappearing pulses in a singularly perturbed reaction-diffusion system with parameters that vary in time and spaceFeb 08 2018We consider the evolution of multi-pulse patterns in an extended Klausmeier equation with parameters that change in time and/or space. We formally show that the full PDE dynamics of a $N$-pulse configuration can be reduced to a $N$-dimensional dynamical ... More

Representation and Characterization of Non-Stationary Processes by Dilation Operators and Induced Shape Space ManifoldsFeb 08 2018We have introduce a new vision of stochastic processes through the geometry induced by the dilation. The dilation matrices of a given processes are obtained by a composition of rotations matrices, contain the measure information in a condensed way. Particularly ... More

On the asymptotic behaviour of nonlocal perimetersFeb 07 2018We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K, which might be singular. In the first part of the ... More

Uniqueness of a Potential from Boundary Data in Locally Conformally Transversally Anisotropic GeometriesFeb 07 2018Let $(\Omega^3,g)$ be a compact smooth Riemannian manifold with smooth boundary and suppose that $U$ is a an open set in $\Omega$ such that $g|_U$ is the Euclidean metric. Let $\Gamma= \overline{U} \cap \partial \Omega$ be connected and suppose that $U$ ... More

Sharp operator-norm asymptotics for linearised elastic plates with rapidly oscillating periodic propertiesFeb 07 2018We analyse a system of partial differential equations describing the behaviour of an elastic plate with periodic moduli in the two planar directions. We assume that the displacement gradients of the points of the plate are small enough for the equations ... More

A general existence result for stationary solutions to the Keller-Segel systemFeb 07 2018We consider a Liouville-type PDE on a smooth bounded planar domain, which is related to stationary solutions of the Keller-Segel's model for chemotaxis. We prove existence of solutions under some algebraic conditions on the parameters. In particular, ... More

Multiple positive bound states for critical Schrödinger-Poisson systemsFeb 07 2018Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schr\"odinger-Poisson system: $$ \left\{ \begin{array}{l} \vspace{2mm} -\Delta u+V(x)u+K(x)\phi(x)u=u^5 -\Delta \phi =K(x)u^2\qquad ... More

Exponential equilibration of genetic circuits using entropy methodsFeb 07 2018We analyse a continuum model for genetic circuits based on a partial integro-differential equation initially proposed in Friedman, Cai \& Xie (2006) as an approximation of a chemical master equation. We use entropy methods to show exponentially fast convergence ... More

The Hydrostatic Stokes Semigroup and Well-Posedness of the Primitive Equations on Spaces of Bounded FunctionsFeb 07 2018Consider the $3$-d primitive equations in a layer domain $\Omega=G \times (-h,0)$, $G=(0,1)^2$, subject to mixed Dirichlet and Neumann boundary conditions at $z=-h$ and $z=0$, respectively, and the periodic lateral boundary condition. It is shown that ... More

Classification of traveling waves for a quadratic Szeg{ö} equationFeb 07 2018We give a complete classification of the traveling waves of the following quadratic Szeg{\"o} equation : $i \partial\_t u = 2J\Pi(|u|^2)+\bar{J}u^2, \quad u(0, \cdot)=u\_0$, and we show that they are given by two families of rational functions, one of ... More

On Strichartz estimates for a dispersion modulated by a time-dependent deterministic noiseFeb 07 2018We address the Cauchy problem for a nonlinear Schr{\"o}dinger equation where the dispersion is modulated by a deterministic noise. The noise is understood as the derivative of a self-affine function of order H $\in$ (0, 1). Due to the self-similarity ... More

On the influence of gravity on density-dependent incompressible periodic fluidsFeb 07 2018The present work is devoted to the analysis of density-dependent, incompressible fluids in a 3D torus, when the Froude number $\varepsilon$ goes to zero. We consider the very general case where the initial data do not have a zero horizontal average, where ... More

On fractional Hardy inequalities in convex setsFeb 07 2018We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodecki\u{\i} spaces of order $(s,p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result ... More

Turbulent Cascade Direction and Lagrangian Time-AsymmetryFeb 07 2018We establish Lagrangian formulae for energy conservation anomalies involving the discrepancy between short-time two-particle dispersion forward and backward in time. These results are facilitated by a rigorous version of the Ott-Mann-Gaw\c{e}dzki relation, ... More

Learning interacting particle systems: diffusion parameter estimation for aggregation equationsFeb 07 2018In this article, we study the parameter estimation of interacting particle systems subject to the Newtonian aggregation. Specifically, we construct an estimator $\widehat{\nu}$ with partial observed data to approximate the diffusion parameter $\nu$, and ... More

On blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equationFeb 06 2018We study dynamical properties of blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schr\"odinger equation \[ i\partial_t u -(-\Delta)^s u = -|u|^\alpha u, \quad u(0) = u_0, \quad \text{on } [0,\infty) \times \mathbb{R}^d, \] where ... More

The nodal set of solutions to some elliptic problems: sublinear equations, and unstable two-phase membrane problemFeb 06 2018We are concerned with the nodal set of solutions to equations of the form \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- > 0$, $q \in [1,2)$, ... More

Loss of regularity for the continuity equation with non-Lipschitz velocity fieldFeb 06 2018We consider the Cauchy problem for the continuity equation in space dimension ${d \geq 2}$. We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces $W^{1,p}$, for $1 \leq p<\infty$, and a smooth compactly supported initial ... More

Second order differentiation formula on $RCD^*(K,N)$ spacesFeb 06 2018Aim of this paper is to prove the second order differentiation formula for $H^{2,2}$ functions along geodesics in $RCD^*(K,N)$ spaces with $N < \infty$. This formula is new even in the context of Alexandrov spaces, where second order differentiation is ... More

Finite speed of propagation for the thin film equation in spherical geometryFeb 06 2018We show that a double degenerate thin film equation, which originated from modeling of viscous coating flow on a spherical surface, has finite speed of propagation for nonnegative strong solutions and hence there exists an interface or free boundary separating ... More

Stable solutions to some elliptic problems: minimal cones, the Allen-Cahn equation, and blow-up solutionsFeb 06 2018These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19-23, 2017. The notes contain the proofs of several results on the classification of stable solutions to some nonlinear elliptic ... More

When does a perturbed Moser-Trudinger inequality admit an extremal?Feb 06 2018In this paper, we are interested in several questions raised mainly in [17]. We consider the perturbed Moser-Trudinger inequality $I\_\alpha^g(\Omega)$ below, at the critical level $\alpha=4\pi$, where $g$, satisfying $g(t)\to 0$ as $t\to +\infty$, can ... More

Sharp quantization for Lane-Emden problems in dimension twoFeb 06 2018In this short note, we prove a sharp quantization for positive solutions of Lane-Emden problems in a bounded planar domain. This result has been conjectured by De Marchis, Ianni and Pacella [6, Remark 1.2].

Propagation of chaos for the VPFP equation with a polynomial cut-offFeb 06 2018We consider a $N$-particle system interacting through the Newtonian potential with a polynomial cut-off in the presence of noise in velocity. We rigorously prove the propagation of chaos for this interacting stochastic particle system. Taking the cut-off ... More

Minimization of a Ginzburg-Landau type energy with weight and with potential having a zero of infinite orderFeb 06 2018In this paper, we study the asymptotic behaviour of minimizing solutions of a Ginzburg-Landau type functional with potential having a zero at 1 of infinite order and we estimate the energy. We generalize in this case a lower bound for the energy of unit ... More

Unified Models for Second-Order TV-Type Regularisation in Imaging - A New Perspective Based on Vector OperatorsFeb 06 2018We introduce a novel regulariser based on natural vector field operations. For suitable choices of the weighting parameters it generalises several well-known first- and second-order TV-type regularisation methods like for example the total variation (TV), ... More

Improved parametrix in the glancing region for the interior Dirichlet-to-Neumann mapFeb 06 2018We study the semi-classical microlocal structure of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a non-empty smooth boundary. We build a new, improved parametrix in the glancing region compaired with that one built in ... More

Blow-up profile of rotating 2D focusing Bose gasesFeb 06 2018We consider the Gross-Pitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation $\Omega$. First we study the behavior of the ground state when ... More

Euclidean Triangles Have No Hot SpotsFeb 06 2018We show that a second Neumann eigenfunction u of a Euclidean triangle has at most one (nonvertex) critical point p, and if p exists, then it is a non-degenerate critical point of Morse index 1. Using this we deduce that (1) the extremal values of u are ... More

Gagliardo-Nirenberg-Sobolev inequalities for convex domains in $\mathbb{R}^d$Feb 06 2018A special type of Gagliardo-Nirenberg-Sobolev (GNS) inequalities in $\mathbb{R}^d$ has played a key role in several proofs of Lieb-Thirring inequalities. Recently, a need for GNS inequalities in convex domains of $\mathbb{R}^d$, in particular for cubes, ... More

Multiplicity Phenomena for Fully Nonlinear Equations with Quadratic Growth in the GradientFeb 05 2018We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as $$ -F(x,u,Du,D^2u) =\lambda c(x)u+\langle M(x)D u, D u \rangle +h(x) $$ in a bounded domain with a Dirichlet boundary condition; here $\lambda \in\mathbb{R}$, ... More

Remarks on the self-shrinking Clifford torusFeb 05 2018On the one hand, we prove that the Clifford torus in $\mathbb{C}^2$ is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian $F$-stable and locally area minimising under Hamiltonian ... More

The convexity of inclusions and gradient's concentration for Lamé systems with partially infinite coefficientsFeb 05 2018It is interesting to study the stress concentration between two adjacent stiff inclusions in composite materials, which can be modeled by the Lam\'e system with partially infinite coefficients. To overcome the difficulty from the lack of maximum principle ... More

Simulation of multiphase porous media flows with minimizing movement and finite volume schemesFeb 05 2018The Wasserstein gradient flow structure of the PDE system governing multiphase flows in porous media was recently highlighted in [C. Canc\`es, T. O. Gallou\"et, and L. Monsaingeon, {\it Anal. PDE} 10(8):1845--1876, 2017]. The model can thus be approximated ... More

Asymptotic properties for second-order linear evolution problems with fractional laplacian operatorsFeb 04 2018In this work we study the asymptotic behavior of solutions for a general linear second-order evolution differential equation in time with fractional Laplace operators in $\mathbb{R}^n$. We obtain improved decay estimates with less demand on the initial ... More

Regularity and quantitative gradient estimate of p-harmonic mappings between Riemannian manifoldsFeb 03 2018Let $M$ be a $C^2$-smooth Riemannian manifold with boundary and $N$ a complete $C^2$-smooth Riemannian manifold. We show that each minimizing $p$-harmonic mapping $u\colon M\to N$ is locally $C^{1,\alpha}$ for some $\alpha\in (0,1)$, provided either $N$ ... More

The $L_p$ dual Minkowski problem for $p>1$ and $q>0$Feb 03 2018Feb 07 2018General $L_p$ dual curvature measures have recently been introduced by Lutwak, Yang and Zhang. These new measures unify several other geometric measures of the Brunn-Minkowski theory and the dual Brunn-Minkowski theory. $L_p$ dual curvature measures arise ... More

The $L_p$ dual Minkowski problem for $p>1$ and $q>0$Feb 03 2018General $L_p$ dual curvature measures have recently been introduced by Lutwak, Yang and Zhang. These new measures unify several other geometric measures of the Brunn-Minkowski theory and the dual Brunn-Minkowski theory. $L_p$ dual curvature measures arise ... More

Invariant measure construction at a fixed massFeb 03 2018In this paper we analyze the derivative nonlinear Schr\"odinger equation on $\mathbb{T}$ with randomized initial data in $\cap_{s < \frac{1}{2}} H^{s}(\mathbb{T})$ according to a Wiener measure. We construct an invariant measure at each sufficiently small, ... More

A continuous time tug-of-war game for parabolic $p(x,t)$-Laplace type equationsFeb 02 2018We formulate a stochastic differential game in continuous time that represents the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the normalized $p(x,t)$-Laplace operator. Our game is formulated ... More

Analysis and computation of some tumor growth models with nutrient: from cell density models to free boundary dynamicsFeb 02 2018In this paper, we study the tumor growth equation along with various models for the nutrient component, including the \emph{in vitro} model and the \emph{in vivo} model. At the cell density level, the spatial availability of the tumor density $n$ is governed ... More

Stable CMC integral varifolds of codimension $1$: regularity and compactnessFeb 01 2018We give two structural conditions on a codimension $1$ integral $n$-varifold with first variation locally summable to an exponent $p>n$ that imply the following: whenever each orientable portion of the $C^{1}$-embedded part of the varifold (which is non-empty ... More

The fractional Schrödinger equation with Hardy-type potentials and sign-changing nonlinearitiesFeb 01 2018We look for solutions to a fractional Schr\"odinger equation of the following form $$ (-\Delta)^{\alpha / 2} u + \left( V(x) - \frac{\mu}{|x|^{\alpha}} \right) u = f(x,u)-K(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N \setminus \{0\}, $$ where $V$ is bounded and ... More

Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifoldsFeb 01 2018Let $(M^n,g)$ be simply connected, complete, with non-positive sectional curvatures, and $\Sigma$ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in $M$. Let $S$ be an area minimising integral 3-current (resp.~flat chain ... More

Hot spots of solutions to the heat equation with inverse square potentialFeb 01 2018We investigate the large time behavior of the hot spots of the solution to the Cauchy problem for the heat equation with a potential $\partial_t u-\Delta u+V(|x|)u=0$, where $V=V(r)$ decays quadratically as $r\to\infty$. In this paper, based on the arguments ... More

Gradient estimates for nonlinear elliptic equations with a gradient-dependent nonlinearityFeb 01 2018In this paper, we obtain gradient estimates of the positive solutions to weighted $p$-Laplacian type equations with a gradient-dependent nonlinearity of the form \begin{equation} \label{one} {\rm div} (|x|^{\sigma}|\nabla u|^{p-2} \nabla u)= |x|^{-\tau} ... More

Stochastic Differential Equations with Critical DriftsJan 31 2018We construct a strong solution to the stochastic differential equation with additive noise when drift term belongs to $L^{q,1}([0,T],L^p_x)$ for $p,q\in (1,\infty)$ satisfying $\frac{2}{q}+\frac{d}{p} =1$. We also prove the Sobolev regularity of the stochastic ... More

Continuity Properties for Divergence Form Boundary Data Homogenization ProblemsJan 31 2018We study the continuity/discontinuity of the effective boundary condition for periodic homogenization of oscillating Dirichlet data for nonlinear divergence form equations and linear systems. For linear systems we show continuity, for nonlinear equations ... More

Heat content in non-compact Riemannian manifoldsJan 31 2018Let $\Omega$ be an open set in a complete, smooth, non-compact, $m$-dimensional Riemannian manifold $M$ without boundary, where $M$ satisfies a two-sided Li-Yau gaussian heat kernel bound. It is shown that if $\Omega$ has infinite measure, and if $\Omega$ ... More

Fractional $p\&q$ Laplacian problems in $\mathbb{R}^{N}$ with critical growthJan 31 2018We deal with the following nonlinear problem involving fractional $p$ and $q$-Laplacians: \begin{equation*} (-\Delta)^{s}_{p}u+(-\Delta)^{s}_{q}u+|u|^{p-2}u+|u|^{q-2}u=\lambda f(u)+|u|^{q^{*}_{s}-2}u \mbox{ in } \mathbb{R}^{N} \end{equation*} where $s\in ... More

Composition Operators on Sobolev Spaces and Neumann EigenvaluesJan 31 2018In this paper we discuss applications of the geometric theory of composition operators on Sobolev spaces to the spectral theory of non-linear elliptic operators. The lower estimates of the first non-trivial Neumann eigenvalues of the $p$-Laplace operator ... More

Existence and Uniqueness of Boundary Value Problems for Hilfer-Hadamard-Type Fractional Differential EquationsJan 31 2018In this paper, we used some theorems of fixed point for studying the results of Existence and Uniqueness For Hilfer-Hadamard-Type Fractional Differential Equations, \[_{H}D^{\alpha,\beta}x(t)+f(t,x(t))=0, ~~~~~~ on~~the~~ interval~~ J:=[1,e]\] with Boundary ... More

Uniqueness of axisymmetric viscous flows originating from positive linear combinations of circular vortex filamentsJan 31 2018Following the recent papers [9] and [10] by T. Gallay and V. \u{S}ver\'ak, in the line of work initiated by H. Feng and V. \u{S}ver\'ak in their paper [3], we prove the uniqueness of a solution of the axisymmetric Navier-Stokes equations without swirl ... More

On correctors for linear elliptic homogenization in the presence of local defectsJan 31 2018We consider the corrector equation associated, in homogenization theory , to a linear second-order elliptic equation in divergence form --$\partial$i(aij$\partial$ju) = f , when the diffusion coefficient is a locally perturbed periodic coefficient. The ... More

On correctors for linear elliptic homogenization in the presence of local defects: the case of advection-diffusionJan 31 2018We follow-up on our works devoted to homogenization theory for linear second-order elliptic equations with coefficients that are perturbations of periodic coefficients. We have first considered equations in divergence form in [6, 7, 8]. We have next shown, ... More

Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operatorJan 30 2018We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a $m$-dimensional submanifold $M$ in $R^d$ as the sample size $n$ increases and the neighborhood size $h$ tends to zero. We show that eigenvalues ... More

Phaseless inverse problems with interference wavesJan 30 2018We consider two phaseless inverse problems for elliptic equation. The statements of these problems differ from have considered. Namely, instead of given information about modulus of scattering waves, we consider the information related to modulus of full ... More

Ecological invasion in competition-diffusion systems when the exotic species is either very strong or very weakJan 30 2018Reaction-diffusion systems with a Lotka-Volterra-type reaction term, also known as competition-diffusion systems, have been used to investigate the dynamics of the competition among $m$ ecological species for a limited resource necessary to their survival ... More

The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEsJan 30 2018We present the Maple package TDDS (Thomas Decomposition of Differential Systems) for decomposition of polynomially nonlinear differential systems, which in addition to equations may contain inequations, into a finite set of differentially triangular and ... More

Nonlinear stability of 2-solitons of the Sine-Gordon equation in the energy spaceJan 30 2018In this article we prove that 2-soliton solutions of the sine-Gordon equation (SG) are orbitally stable in the natural energy space of the problem. The solutions that we study are the {\it 2-kink, kink-antikink and breather} of SG. In order to prove this ... More

Multiplicity results for $(p,\, q)$ fractional elliptic equations involving critical nonlinearitiesJan 30 2018In this paper we prove the existence of infinitely many nontrivial solutions for the class of $(p,\, q)$ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of ... More

Linear Elasticity and Homogenization in the absence of very strong ellipticityJan 30 2018Homogenization in linear elliptic problems usually assumes coercivity of the accompanying Dirichlet form. In linear elasticity, coercivity is not ensured through mere (strong) ellipticity so that the usual estimates that render homogenization meaningful ... More

Positivity preserving along a flow over projective bundleJan 30 2018In this paper, we introduce a flow over the projective bundle $p:P(E^*)\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and K\"ahler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ ... More

On conormal and oblique derivative problem for elliptic equations with Dini mean oscillation coefficientsJan 30 2018We show that weak solutions to conormal derivative problem for elliptic equations in divergence form are continuously differentiable up to the boundary provided that the mean oscillations of the leading coefficients satisfy the Dini condition, the lower ... More

Regularity and continuity of the multilinear strong maximal operatorsJan 30 2018Jan 31 2018Let $m\ge 1$, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operator $${\mathscr{M}}_{\mathcal{R}}(\vec{f})(x)=\sup_{\substack{R \ni x R\in\mathcal{R}}}\prod\limits_{i=1}^m\frac{1}{|R|}\int_{R}|f_i(y)|dy,$$ ... More

Time domain boundary elements for dynamic contact problemsJan 29 2018This article considers a unilateral contact problem for the wave equation. The problem is reduced to a variational inequality for the Dirichlet-to-Neumann operator for the wave equation on the boundary, which is solved in a saddle point formulation using ... More

Counterexamples in Calculus of Variations in $L^\infty$ through the vectorial Eikonal equationJan 29 2018We show that for any regular bounded domain $\Omega\subseteq \mathbb R^n$, there exist infinitely many global diffeomorphisms equal to the identity on $\partial \Omega$ which solve the Eikonal equation. We also provide explicit examples of such maps on ... More

Boundary elements with mesh refinements for the wave equationJan 29 2018The solution of the wave equation in a polyhedral domain in $\mathbb{R}^3$ admits an asymptotic singular expansion in a neighborhood of the corners and edges. In this article we formulate boundary and screen problems for the wave equation as equivalent ... More

A multi-scale limit of a randomly forced rotating $3$-D compressible fluidJan 29 2018We study a singular limit of a scaled compressible Navier--Stokes--Coriolis system driven by both a deterministic and stochastic forcing terms in three dimensions. If the Mach number is comparable to the Froude number with both proportional to say $\varepsilon\ll ... More

Global Sobolev inequalities and Degenerate P-Laplacian equationsJan 29 2018We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of $p$-Laplacian equations. Given $\Omega\subset\mathbb{R}^n$, let $\rho$ be a quasi-metric on $\Omega$, and let $Q$ be ... More

A BDF2-Approach for the Non-linear Fokker-Planck EquationJan 29 2018We prove convergence of a variational formulation of the BDF2 method applied to the non-linear Fokker-Planck equation. Our approach is inspired by the JKO-method and exploits the differential structure of the underlying $L^2$-Wasserstein space. The technique ... More

Ground states of some coupled nonlocal fractional dispersive PDEsJan 29 2018Jan 31 2018We show the existence of ground state solutions to the following stationary system coming from some coupled fractional dispersive equations such as: nonlinear fractional Schr\"odinger (NLFS) equations (for dimension $n=1,\, 2,\, 3$) or NLFS and fractional ... More

Grow-up for a quasilinear heat equation with a localized reaction in higher dimensionsJan 29 2018We study the behaviour of nonnegative solutions to the quasilinear heat equation with a reaction localized in a ball $$ u_t=\Delta u^m+a(x)u^p, $$ for $m>0$, $0<p\le\max\{1,m\}$, $a(x)=\mathds{1}_{B_L}(x)$, $0<L<\infty$ and $N\ge2$. We study when solutions, ... More

Grow-up for a quasilinear heat equation with a localized reactionJan 29 2018We study the behaviour of global solutions to the quasilinear heat equation with a reaction localized $$ u_t=(u^m)_{xx}+a(x) u^p, $$ $m, p>0$ and $a(x)$ being the characteristic function of an interval. we prove that there exists $p_0=\max\{1,\frac{m+1}2\}$ ... More

Estimates of Green and Martin kernels for Schrödinger operators with singular potential in Lipschitz domainsJan 29 2018Consider operators of the form $L^{\gamma V}:=\Delta +\gamma V$ in a bounded Lipschitz domain $\Omega\subset \mathbb{R}^N$. Assume that $V\in C^1(\Omega)$ satisfies $|V(x)| \leq \bar a \,\mathrm{distance}\,(x,\partial\Omega)^{-2}$ for every $x\in \Omega$ ... More

Non-local elasticity theory as a continuous limit of 3D networks of pointwise interacting massesJan 29 2018Small oscillations of an elastic system of point masses (particles) with a nonlocal interaction are considered. We study the asymptotic behavior of the system, when number of particles tends to infinity, and the distances between them and the forces of ... More

Global well-posedness of the 1D compressible Navier-Stokes equations with constant heat conductivity and nonnegative densityJan 29 2018Jan 30 2018In this paper we consider the initial-boundary value problem to the one-dimensional compressible Navier-Stokes equations for idea gases. Both the viscous and heat conductive coefficients are assumed to be positive constants, and the initial density is ... More

Singularity formation to the Cauchy problem of the two-dimensional non-baratropic magnetohydrodynamic equations without heat conductivityJan 28 2018We study the breakdown of strong solutions to the two-dimensional (2D) Cauchy problem of the non-baratropic compressible magnetohydrodynamic equations without heat conductivity. It is proved that the strong solution exists globally if the gradient of ... More

Oscillations and integrability of the vorticity in the 3D NS flowsJan 27 2018In the studies of the Navier-Stokes (NS) regularity problem, it has become increasingly clear that a more realistic path to improved a priori bounds is to try to break away from the scaling of the energy-level estimates in the realm of the blow-up-type ... More

Non reflection and perfect reflection via Fano resonance in waveguidesJan 26 2018We investigate a time-harmonic wave problem in a waveguide. By means of asymptotic analysis techniques, we justify the so-called Fano resonance phenomenon. More precisely, we show that the scattering matrix considered as a function of a geometrical parameter ... More

An ergodic problem for Mean Field Games: qualitative properties and numerical simulationsJan 26 2018This paper is devoted to some qualitative descriptions and some numerical results for ergodic Mean Field Games systems which arise, e.g., in the homogenization with a small noise limit. We shall consider either power type potentials or logarithmic type ... More

Deterministic particle approximation for nonlocal transport equations with nonlinear mobilityJan 26 2018We construct a deterministic, Lagrangian many-particle approximation to a class of nonlocal transport PDEs with nonlinear mobility arising in many contexts in biology and social sciences. The approximating particle system is a nonlocal version of the ... More

Regularity for an anisotropic equation in the planeJan 26 2018We present a simple proof of the $C^1$ regularity of $p$-anisotropic functions in the plane for $2\leq p<\infty$. We achieve a logarithmic modulus of continuity for the derivatives. The monotonicity (in the sense of Lebesgue) of the derivatives is used. ... More

Multiplicity of positive solutions for a quasilinear Schrödinger equation with an almost critical nonlinearityJan 25 2018In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem \begin{equation*} \left\{ \begin{array}[c]{ll} -\Delta u - \Delta (u^2)u = |u|^{p-2}u & \mbox{ in } \Omega u= 0 &\mbox{ on } \partial\Omega, ... More

Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical ImplementationsJan 25 2018We study convex integration solutions in the context of the modelling of shape-memory alloys. The purpose of the article is two-fold, treating both rigidity and flexibility properties: Firstly, we relate the maximal regularity of convex integration solutions ... More

Small-amplitude static periodic patterns at a fluid-ferrofluid interfaceJan 25 2018We establish the existence of static doubly periodic patterns (in particular rolls, rectangles and hexagons) on the free surface of a ferrofluid near onset of the Rosensweig instability, assuming a general (nonlinear) magnetisation law. A novel formulation ... More

Global existence of solutions to 2-D Navier-Stokes flow with non-decaying initial data in half-planeJan 25 2018We investigate the Navier-Stokes initial boundary value problem in the half-plane $R^2_+$ with initial data $u_0 \in L^\infty(R^2_+)\cap J_0^2(R^2_+)$ or with non decaying initial data $u_0\in L^\infty(R^2_+) \cap J_0^p(R^2_+), p > 2$ . We introduce a ... More

Generation of semigroup for symmetric matrix Schrödinger operators in $L^p$-spacesJan 25 2018In this paper we establish generation of analytic strongly continuous semigroup in $L^p$--spaces for the symmetric matrix Schr\"odinger operator $div(Q\nabla u)-Vu$, where, for every $x\in\mathbb{R}^d$, $V(x)=(v_{ij}(x))$ is a semi-definite positive and ... More

Another proof of the existence of self-similar solution of the inverse mean curvature flowJan 25 2018We will give a new proof of a recent result of P.Daskalopoulos and G.Huisken [DH] on the existence of self-similar solution of the inverse mean curvature flow which is the graph of a radially symmetric solution in $\mathbb{R}^n$, $n\ge 2$, of the form ... More

Existence and concentration phenomena for a class of indefinite variational problems with critical growthJan 24 2018In this paper we are interested to prove the existence and concentration of ground state solution for the following class of problems $$ -\Delta u+V(x)u=A(\epsilon x)f(u), \quad x \in \R^{N}, \eqno{(P)_{\epsilon}} $$ where $N \geq 2$, $\epsilon>0$, $A:\R^{N}\rightarrow\R$ ... More

The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonancesJan 24 2018We consider a variety of boundary value problems (BVPs) for the heterogeneous Helmholtz equation, i.e. the equation $\nabla\cdot(A \nabla u ) + k^2 n u =-f$ where both $A$ and $n$ are functions of position. We prove new a priori bounds on the solution ... More

Boundary regularity for the porous medium equationJan 24 2018We study the boundary regularity of solutions to the porous medium equation $u_t = \Delta u^m$ in the degenerate range $m>1$. In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the parabolic boundary ... More