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We present a construction of a multi-scale Gaussian beam parametrix for the Dirichlet boundary value problem associated with the wave equation, and study its convergence rate to the true solution in the highly oscillatory regime. The construction elaborates on the wave-atom parametrix of Bao, Qian, Ying, and Zhang and extends to a multi-scale setting the technique of Gaussian beam propagation from a boundary of Katchalov, Kurylev and Lassas., Comment: 64 pages, 7 figures, minor update

International audience; A wave equation whose main coefficient is discontinuous models the evolution of waves amplitude in a media composed of at least two different materials, in which the propagation speed is different. In our mathematical setting, the spatial domain where the partial differential equation evolves is an open bounded subset of R^2 and the wave speed is assumed to be constant in each one of two sub-domains, separated by a smooth and possibly non-convex interface. This article is concerned with the construction of Carleman weights for this wave operator, allowing generalizations of previous results to the case of an interface that is not necessarily the boundary of a convex set. Indeed, using the orthogonal projection onto this interface, we define convex functions satisfying the transmission conditions imposed by the equation, such that, under usual hypothesis on the sign of the jump of the wave speed, can be used as Carleman weights.

Chemotaxis is a fundamental mechanism of cells and organisms, which is responsible for attracting microbes to food, embryonic cells into developing tissues, or immune cells to infection sites. Mathematically chemotaxis is described by the Patlak–Keller–Segel model. This macroscopic system of equations is derived from the microscopic model when limiting behaviour is studied. However, on taking the limit and passing from the microscopic equations to the macroscopic equations, fluctuations are neglected. Perturbing the system by a Gaussian random field restitutes the inherent randomness of the system. This gives us the motivation to study the classical Patlak–Keller–Segel system perturbed by random processes. We study a stochastic version of the classical Patlak–Keller–Segel system under homogeneous Neumann boundary conditions on an interval O = [ 0 , 1 ] . In particular, let W 1 , W 2 be two time-homogeneous spatial Wiener processes over a filtered probability space A . Let u and v denote the cell density and concentration of the chemical signal. We investigate the coupled system { d u − ( r u Δ u − χ div ( u ∇ v ) ) d t = u ∘ d W 1 , d v − ( r v Δ v − α v ) d t = β u d t + v ∘ d W 2 , with initial conditions ( u ( 0 ) , v ( 0 ) ) = ( u 0 , v 0 ) . The positive terms r u and r v are the diffusivity of the cells and chemoattractant, respectively, the positive value χ is the chemotactic sensitivity, α ≥ 0 is the so-called damping constant. The noise is interpreted in the Stratonovich sense. Given T > 0 , we will prove the existence of a martingale solution on [ 0 , T ] .

The aim of this paper is twofold. The first is to study the asymptotics of a parabolically scaled, continuous and space-time stationary in time version of the well-known Funaki-Spohn model in Statistical Physics. After a change of unknowns requiring the existence of a space-time stationary eternal solution of a stochastically perturbed heat equation, the problem transforms to the qualitative homogenization of a uniformly elliptic, space-time stationary, divergence form, nonlinear partial differential equation, the study of which is the second aim of the paper. An important step is the construction of correctors with the appropriate behavior at infinity.

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,\mu_X)$ satisfying a $2$-Poincar\'e inequality. Given a bounded domain $\Omega\subset X$ with $\mu_X(X\setminus\Omega)>0$, and a function $f$ in the Besov class $B^\theta_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^\theta_{2,2}(X)$ such that $u=f$ in $X\setminus\Omega$ and $\mathcal{E}_\theta(u,u)\le \mathcal{E}_\theta(h,h)$ whenever $h\in B^\theta_{2,2}(X)$ with $h=f$ in $X\setminus\Omega$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally H\"older continuous on $\Omega$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups., Comment: 42 pages, comments welcome, submitted. Revision to add crucial references and attributions to the introduction

We consider a class of cooperative reaction-diffusion systems with free boundaries in one space dimension, where the diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and they are allowed not to appear in some of the equations in the system. Such a system covers various models arising from mathematical biology, in particular a West Nile virus model and an epidemic model considered recently in [16] and [44] , respectively, where a “spreading-vanishing” dichotomy is known to govern the long time dynamical behaviour, but the question on spreading speed was left open. In this paper, we develop a systematic approach to determine the spreading profile of the system, and obtain threshold conditions on the kernel functions which decide exactly when the spreading has finite speed, or infinite speed (accelerated spreading). This relies on a rather complete understanding of both the associated semi-waves and travelling waves. When the spreading speed is finite, we show that the speed is determined by a particular semi-wave. This is Part 1 of a two part series. In Part 2, for some typical classes of kernel functions, we will obtain sharp estimates of the spreading rate for both the finite speed case, and the infinite speed case.

Linear first order Riemann-Liouville fractional differential equations are studied. These new equations unify and generalize the Riemann-Liouville, modified Caputo and Caputo fractional differential equations. The equivalences between the fractional differential equations and the corresponding perturbed Abel's integral equations are obtained. These results are useful not only to study the initial or boundary value problems for nonlinear first order Riemann-Liouville fractional differential equations but also to study the solutions of the perturbed Abel's integral equations arising in a problem of mechanics and many other physical problems. The well-known Tonelli's result on solvability of the Abel's integral equation is generalized. We exhibit that there are nonconstant equilibria for some first order Caputo fractional equations. This is different from nonlinear first order ordinary differential equations which have only constant equilibria. The equivalence results are applied to generalize the classical Mean Value Theorem to the first order Riemann-Liouville fractional derivatives.

Combinatorial vector fields on simplicial complexes as introduced by Robin Forman have found numerous and varied applications in recent years. Yet, their relationship to classical dynamical systems has been less clear. In recent work it was shown that for every combinatorial vector field on a finite simplicial complex one can construct a multivalued discrete-time dynamical system on the underlying polytope X which exhibits the same dynamics as the combinatorial flow in the sense of Conley index theory. However, Forman's original description of combinatorial flows appears to have been motivated more directly by the concept of flows, i.e., continuous-time dynamical systems. In this paper, it is shown that one can construct a semiflow on X which exhibits the same dynamics as the underlying combinatorial vector field. The equivalence of the dynamical behavior is established in the sense of Conley-Morse graphs and uses a tiling of the topological space X which makes it possible to directly construct isolating blocks for all involved isolated invariant sets based purely on the combinatorial information., Comment: 57 pages, 12 figures

In this work, a new perturbation approach is developed based on Bogdanov-Takens bifurcation theory, which enables the Poincare-Lyapunov method for switching systems with linear type centers to be applied for studying the center conditions of planar switching polynomial systems associated with a nilpotent equilibrium point. The new method is then applied to consider a class of quadratic switching nilpotent systems, and a complete classification is given on the conditions of the nilpotent equilibrium point to be a center. Moreover, based on one of the center conditions, an example is constructed to show the existence of seven small-amplitude limit cycles around the nilpotent equilibrium point, which is a new lower bound on the number of limit cycles in such systems.

In this paper, we study various aspects of the ODE's flow $X$ solution to the equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus $Y_d$, where $b$ is a regular $Z^d$-periodic vector field from $R^d$ in $R^d$. We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$: - the everywhere asymptotics of the flow $X$, - the almost-everywhere asymptotics of the flow $X$, - the global rectification of the vector field $b$ in $Y_d$, - the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure, - the unit set condition for Herman's rotation set $C_b$ composed of the means of $b$ related to the invariant probability measures, - the unit set condition for the subset $D_b$ of $C_b$ composed of the means of $b$ related to the invariant probability measures which are absolutely continuous with respect to Lebesgue's measure, - the homogenization of the linear transport equation with oscillating data and the oscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The main and surprising result of the paper is that the almost-everywhere asymptotics of the flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ is assumed to be non empty, and that the two conditions turn to be equivalent to the homogenization of the transport equation when $b$ is divergence free. In contrast, using an elementary approach based on classical tools of PDE's analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any $d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow may hold without satisfying the everywhere asymptotics of the flow.

Let Ω be a C 2 bounded domain in R N ( N ≥ 3 ), δ ( x ) = dist ( x , ∂ Ω ) and C H ( Ω ) be the best constant in the Hardy inequality with respect to Ω. We investigate positive solutions to a boundary value problem for Lane-Emden equations with Hardy potential of the form − Δ u − μ δ 2 u = u p in Ω , u = ρ ν on ∂ Ω , ( P ρ ) where 0 μ C H ( Ω ) , ρ is a positive parameter, ν is a positive Radon measure on ∂Ω with norm 1 and 1 p N μ , with N μ being a critical exponent depending on N and μ. It is known from [22] that there exists a threshold value ρ ⁎ such that problem ( P ρ ) admits a positive solution if 0 ρ ≤ ρ ⁎ , and no positive solution if ρ > ρ ⁎ . In this paper, we go further in the study of the solution set of ( P ρ ) . We show that the problem admits at least two positive solutions if 0 ρ ρ ⁎ and a unique positive solution if ρ = ρ ⁎ . We also prove the existence of at least two positive solutions for Lane-Emden systems { − Δ u − μ δ 2 u = v p in Ω , − Δ v − μ δ 2 v = u q in Ω , u = ρ ν , v = σ τ on ∂ Ω , under the smallness condition on the positive parameters ρ and σ.

This note relies mainly on a refined version of the main results of the paper by F. Catrina and D. Costa (J. Differential Equations 2009). We provide very short and self-contained proofs. Our results are sharp and minimizers are obtained in suitable functional spaces. As main tools we use the so-called \textit{expand of squares} method to establish sharp weighted $L^{2}$-Caffarelli-Kohn-Nirenberg (CKN) inequalities and density arguments., Comment: 13 pages

We prove L 2 stability estimates for entropic shocks among weak, possibly non-entropic, solutions of scalar conservation laws ∂ t u + ∂ x f ( u ) = 0 with strictly convex flux function f. This generalizes previous results by Leger and Vasseur, who proved L 2 stability among entropy solutions. Our main result, the estimate ∫ R | u ( t , ⋅ ) − u 0 s h o c k ( ⋅ − x ( t ) ) | 2 d x ≤ ∫ R | u 0 − u 0 s h o c k | 2 + C μ + ( [ 0 , t ] × R ) , for some Lipschitz shift x ( t ) , includes an error term accounting for the positive part of the entropy production measure μ = ∂ t ( u 2 / 2 ) + ∂ x q ( u ) , where q ′ ( u ) = u f ′ ( u ) . Stability estimates in this general non-entropic setting are of interest in connection with large deviation principles for the hydrodynamic limit of asymmetric interacting particle systems. Our proof adapts the scheme devised by Leger and Vasseur, where one constructs a shift x ( t ) which allows to bound from above the time-derivative of the left-hand side. The main difference lies in the fact that our solution u ( t , ⋅ ) may present a non-entropic shock at x = x ( t ) and new bounds are needed in that situation. We also generalize this stability estimate to initial data with bounded variation.

In this paper, complex integrability and linearizability of cubic Z 2 -equivariant systems with two 1:q resonant singular points are investigated, and the necessary and sufficient conditions on complex integrability and linearizability of the systems with two 1: ( − q ) resonant saddles are obtained for q = 1 , 2 , 3 , 4 . Moreover, for general positive integer q, the complex integrability and linearizability conditions are classified, and the sufficiency of the conditions is proved. Further, the linearizability conditions of cubic Z 2 -equivariant systems with two 1:q resonant node points are also classified.

The initial value problems of bipolar isentropic/non-isentropic compressible Navier-Stokes-Maxwell (CNS-M) systems arising from plasmas in R 3 are studied. The main difficulty of studying the bipolar isentropic/non-isentropic CNS-M systems lies in the appearance of the electromagnetic fields satisfying the hyperbolic Maxwell equations. The large time-decay rates of global smooth solutions with small amplitude in L q ( R 3 ) for 2 ≤ q ≤ ∞ are established. For the bipolar non-isentropic CNS-M system, the difference of velocities of two charged carriers decay at the rate ( 1 + t ) − 3 4 + 1 4 q which is faster than the rate ( 1 + t ) − 3 4 + 1 4 q ( ln ⁡ ( 3 + t ) ) 1 − 2 q of the bipolar isentropic CNS-M system, meanwhile, the magnetic field decay at the rate ( 1 + t ) − 3 4 + 3 4 q ( ln ⁡ ( 3 + t ) ) 1 − 2 q which is slower than the rate ( 1 + t ) − 3 4 + 3 4 q for the bipolar isentropic CNS-M system. The approach adopted is the classical energy method but with some new developments, where the techniques of choosing symmetrizers and the spectrum analysis on the linearized homogeneous system play the crucial roles.

It was proved by Karch and Pilarczyk that Landau solutions are asymptotically stable under any L 2 -perturbation. In our earlier work with L. Li, we have classified all ( − 1 ) -homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles. In this paper, we study the asymptotic stability of the least singular solutions among these solutions other than Landau solutions, and prove that such solutions are asymptotically stable under any L 2 -perturbation.

In (Euzebio et al., 2016 [10] ; Chen and Tang, 2020 [8] ), the bifurcation diagram and all global phase portraits of a degenerate planar piecewise linear differential system x ˙ = F ( x ) − y , y ˙ = g ( x ) − α with three zones were given completely for the non-extreme case. In this paper we deal with the system for the extreme case and find new nonlinear phenomena of bifurcation for this planar piecewise linear system, i.e., a generalized degenerate Hopf bifurcation occurs for points at infinity. Moreover, the bifurcation diagram and all global phase portraits in the Poincare disc are obtained, presenting scabbard bifurcation curves, grazing bifurcation curves for limit cycles, generalized supercritical (or subcritical) Hopf bifurcation curve for points at infinity, generalized degenerate Hopf bifurcation value for points at infinity and double limit cycle bifurcation curve.

We study the simplest parabolic-elliptic model of chemotaxis in space dimensions N ≥ 3 , and show the optimal conditions on the initial data for the finite time blow-up and the global existence of solutions in terms of stationary solutions. Our argument is based on the study of the Cauchy problem for the transformed equation involving the averaged mass of the solution.

In this paper we investigate the Sturmian theory for general (possibly uncontrollable) linear Hamiltonian systems by means of the Lidskii angles, which are associated with a symplectic fundamental matrix of the system. In particular, under the Legendre condition we derive formulas for the multiplicities of the left and right proper focal points of a conjoined basis of the system, as well as the Sturmian separation theorems for two conjoined bases of the system, in terms of the Lidskii angles. The results are new even in the completely controllable case. As the main tool we use the limit theorem for monotone matrix-valued functions by Kratz (1993). The methods allow to present a new proof of the known monotonicity property of the Lidskii angles. The results and methods can also be potentially applied in the singular Sturmian theory on unbounded intervals, in the oscillation theory of linear Hamiltonian systems without the Legendre condition, in the comparative index theory, or in linear algebra in the theory of matrices.

In this paper we introduce and study several mean forms of sensitive tuples. It is shown that the topological or measure-theoretical entropy tuples are correspondingly mean sensitive tuples under certain conditions (minimal in the topological setting or ergodic in the measure-theoretical setting). Characterizations of the question when every non-diagonal tuple is mean sensitive are presented. Among other results we show that under minimality assumption a topological dynamical system is weakly mixing if and only if every non-diagonal tuple is mean sensitive and so as a consequence every minimal weakly mixing topological dynamical system is mean n-sensitive for any integer n ≥ 2 . Moreover, the notion of weakly sensitive in the mean tuple is introduced and it turns out that this property has some special lift property. As an application we obtain that the maximal mean equicontinuous factor for any topological dynamical system can be induced by the smallest closed invariant equivalence relation containing all weakly sensitive in the mean pairs.

We study the convergence of the method of reflections for the Stokes equations in domains perforated by countably many spherical particles with boundary conditions typical for the suspension of rigid particles. We prove that a relaxed version of the method is always convergent in H ˙ 1 under a mild separation condition on the particles. Moreover, we prove optimal convergence rates of the method in W ˙ 1 , q , 1 q ∞ and in L ∞ in terms of the particle volume fraction under a stronger separation condition of the particles.

In this paper we consider a chemotaxis system coupling with the incompressible Euler equations in spatial dimension two, which describing the dynamics of chemotaxis in the inviscid fluid. We establish the regular solutions globally in time under some assumptions on the chemotactic sensitivity.

Our aim in this paper is to prove the existence of solutions to the Cahn–Hilliard equation with a general nonlinear source term. An essential difficulty is to obtain a global in time solution. Indeed, due to the presence of the source term, one cannot exclude the possibility of blow up in finite time when considering regular nonlinear terms and when considering an approximated scheme. Considering instead logarithmic nonlinear terms, we give sufficient conditions on the source term which ensure the existence of a global in time weak solution. These conditions are satisfied by several important models and applications which can be found in the literature.

We study the classical Mackey–Glass delay differential equation x ′ ( t ) = − a x ( t ) + b f n ( x ( t − 1 ) ) where a , b , n are positive reals, and f n ( ξ ) = ξ / [ 1 + ξ n ] for ξ ≥ 0 . As a limiting ( n → ∞ ) case we also consider the discontinuous equation x ′ ( t ) = − a x ( t ) + b f ( x ( t − 1 ) ) where f ( ξ ) = ξ for ξ ∈ [ 0 , 1 ) , f ( 1 ) = 1 / 2 , and f ( ξ ) = 0 for ξ > 1 . First, for certain parameter values b > a > 0 , an orbitally asymptotically stable periodic orbit is constructed for the discontinuous equation. Then it is shown that for large values of n, and with the same parameters a , b , the Mackey–Glass equation also has an orbitally asymptotically stable periodic orbit near to the periodic orbit of the discontinuous equation. Although the obtained periodic orbits are stable, their projections R ∋ t ↦ ( x ( t ) , ( x ( t − 1 ) ) ) ∈ R 2 can be complicated.

We consider the interior Holder regularity of spatial gradient of viscosity solution to the parabolic normalized p ( x , t ) -Laplace equation u t = ( δ i j + ( p ( x , t ) − 2 ) u i u j | D u | 2 ) u i j with some suitable assumptions on p ( x , t ) , which arises naturally from a two-player zero-sum stochastic differential game with probabilities depending on space and time.

In this paper, we construct a nontrivial solitary wave solution to Chern-Simons-Higgs system for non-self-dual case and verify its convergence of non-relativistic limit to a minimal mass solution to the non-relativistic Jackiw-Pi model. We also provide with an explicit rate and high regularity of the convergence, which is naturally obtained in process of construction.

Let $n\ge2$ and $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in $\Omega$. More precisely, let $p\in(n/(n-1),\infty)$. Using a real-variable argument, the authors obtain two necessary and sufficient conditions for $W^{1,p}$ estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse H\"older inequality with exponent $p$ or weighted $W^{1,q}$ estimates of solutions with $q\in(n/(n-1),p]$ and some Muckenhoupt weights. As applications, the authors establish some global regularity estimates for solutions to Robin boundary value problems of second order elliptic equations of divergence form with small $\mathrm{BMO}$ coefficients, respectively, on bounded Lipschitz domains, $C^1$ domains or (semi-)convex domains, in the scale of weighted Lebesgue spaces, via some quite subtle approach which is different from the existing ones and, even when $n=3$ in case of bounded $C^1$ domains, also gives an alternative correct proof of some know result. By this and some technique from harmonic analysis, the authors further obtain the global regularity estimates, respectively, in Morrey spaces, (Musielak--)Orlicz spaces and variable Lebesgue spaces, Comment: 54 pages; Submitted

The restricted elliptic isosceles three body problem (REI3BP) models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies called the primaries. The primaries of masses m1 = m2 move along a degenerate Keplerian elliptic collision orbit (on a line) under their gravitational attraction, whereas the third, massless particle, moves on the plane perpendicular to their line of motion and passing through the center of mass of the primaries. By symmetry, the component of the angular momentum G of the massless particle along the direction of the line of the primaries is conserved. We show the existence of symbolic dynamics in the REI3BP for large G by building a Smale horseshoe on a certain subset of the phase space. As a consequence we deduce that the REI3BP possesses oscillatory motions, namely orbits which leave every bounded region but return infinitely often to some fixed bounded region. The proof relies on the existence of transversal homoclinic connections associated to an invariant manifold at infinity. Since the distance between the stable and unstable manifolds of infinity is exponentially small, Melnikov theory does not apply. M. G. and J. P. have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 757802). T. M. S. has been also partly supported by the Spanish MINECO-FEDER Grant PGC2018-098676-B-100 (AEI/FEDER/UE) and the Catalan grant 2017SGR1049. M. G. and T. M. S. are supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2019. C. Vidal is supported by FONDECYT 1180288

A reaction-diffusion-ODE model for the Neolithic spread of farmers in Europe has been recently proposed in [7] . In this model, farmers are assumed to be divided into two subpopulations according to a mobility rule, namely, into sedentary and migrating farming populations. The conversion between the farming subpopulations depends on the total density of farmers and it is superimposed on the classical Lotka-Volterra competition model, so that it is described by a three-component reaction-diffusion-ODE system. In this article we consider a singular limit problem when the conversion rate tends to infinity and prove under appropriate conditions that solutions of the three component system converge to solutions of a two-component system with a linear diffusion and nonlinear degenerate diffusion.

For the semilinear Dirichlet problem Δ u + g ( u ) = f ( x ) for x ∈ Ω , u = 0 on ∂ Ω decompose f ( x ) = μ 1 φ 1 + e ( x ) , where φ 1 is the principal eigenfunction of the Laplacian with zero boundary conditions, and e ( x ) ⊥ φ 1 in L 2 ( Ω ) , and similarly write u ( x ) = ξ 1 φ i + U ( x ) , with U ⊥ φ 1 in L 2 ( Ω ) . We study properties of the solution curve ( u ( x ) , μ 1 ) ( ξ 1 ) , and in particular its section μ 1 = μ 1 ( ξ 1 ) , which governs the multiplicity of solutions. We consider both general nonlinearities, and some important classes of equations, and obtain detailed description of solution curves under the assumption g ′ ( u ) λ 2 . We obtain particularly detailed results in case of one dimension. This approach is well suited for numerical computations, which we perform to illustrate our results.

We study the following class of pseudo-relativistic Hartree equations − e 2 Δ + m 2 u + V ( x ) u = e μ − N ( | x | − μ ⁎ F ( u ) ) f ( u ) in R N , where the nonlinearity satisfies general hypotheses of Berestycki-Lions type. By using the method of penalization arguments, we prove the existence of a family of localized positive solutions that concentrate at the local minimum points of the indefinite potential V ( x ) , as e → 0 .

In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrodinger equation with spatial inhomogeneity coefficient K ( x ) behaves like | x | − b for 0 b min ⁡ { N 2 , 4 } . We show the local well-posedness in the whole H s -subcritical case, with 0 s ≤ 2 . The difficulties of this problem come from the singularity of K ( x ) and the lack of differentiability of the nonlinear term. To resolve this, we derive the bilinear Strichartz's type estimates for the nonlinear biharmonic Schrodinger equations in Besov spaces.

The global equi-continuity estimate on L p -viscosity solutions of parabolic bilateral obstacle problems with unbounded ingredients is established when obstacles are merely continuous. The existence of L p -viscosity solutions is established via an approximation of given data. The local Holder continuity estimate on the space derivative of L p -viscosity solutions is shown when the obstacles belong to C 1 , β , and p > n + 2 .

In the current paper we consider an inverse boundary value problem of electromagnetism with nonlinear Second Harmonic Generation (SHG) process. We show the unique determination of the electromagnetic material parameters and the SHG susceptibility parameter of the medium by making electromagnetic measurements on the boundary. We are interested in the case when a frequency is fixed.

We analyze a quasi-static Biot system of poroelasticity for both compressible and incompressible constituents. The main feature of this model is a nonlinear coupling of pressure and dilation through the system's permeability tensor. Such a model has been analyzed previously from the point of view of constructing weak solutions through a fully discretized approach. In this treatment, we consider simplified Dirichlet type boundary conditions in the elastic displacement and pressure variables and give a full treatment of weak solutions. Our construction of weak solutions for the nonlinear problem is natural and based on a priori estimates, a requisite feature in addressing the nonlinearity. This is in contrast to previous work which exploits linearity or monotonicity in the permeability, both of which are not available here. We utilize a spatial semi-discretization and employ a multi-valued fixed point argument in for a clear construction of weak solutions. We also provide regularity criteria for uniqueness of solutions.

Let Ω ⊂ R 3 be a Lipschitz domain. Consider a harmonic map v : Ω → S 2 with boundary data v | ∂ Ω = φ which minimises the Dirichlet energy. For p ≥ 2 , we show that any energy minimiser u whose boundary map ψ has a small W 1 , p -distance to φ is close to v in Holder norm modulo bi-Lipschitz homeomorphisms, provided that v is the unique minimiser attaining the boundary data. The index p = 2 is sharp: the above stability result fails for p 2 due to the constructions by Almgren–Lieb [2] and Mazowiecka–Strzelecki [15] .

Our first objective is to identify two-dimensional equations that model the displacement of a linearly elastic flexural shell subjected to the action of an external heat source. To this end, we embed the shell into a family of linearly elastic flexural shells, all sharing the same middle surface θ ( ω ‾ ) , where ω is a domain in R 2 and θ : ω ‾ → E 3 is a smooth enough immersion and whose thickness 2 e > 0 is considered as a “small” parameter approaching zero. We then identify, and justify by means of a rigorous asymptotic analysis as e approaches zero, the corresponding “limit” two-dimensional variational problem. Our second objective is to identify and justify a set of two-dimensional equations that are meant to approximate the original three-dimensional model in the case where the shell under consideration is either an elliptic membrane shell or a flexural shell.

The current article is to study the solvability of Nirenberg problem on S 2 through the so-called Gaussian curvature flow. We aim to propose a unified method to treat the problem for candidate functions without sign restriction and non-degenerate assumption. As a first step, we reproduce the following statement: suppose the critical points of a smooth function f with positive critical values are non-degenerate. Then the required solution exists, if the difference between the number of the local maximum points with positive values and the number of the saddle points with positive critical values as well as negative Laplace is not equal to 1. This statement has been proved for nearly thirty years through different methods.

In this work I show how a diffusion-advection equation in three space-dimensions may have its advection term weakly limited to a velocity field localized to a moving curve. This is rigorously accomplished through the technique of concentrated capacity, and the form of the concentrated capacity limit along with small time existence of solutions is determined. This problem is motivated by mathematical biology and the study of proteins in solvent where the latter is modeled as a diffusing quantity and the protein is taken to be a 1d object which advects the solvent by contact and its own motion. This work introduces a novel PDE's framework for that interaction., Comment: 23 pages

The paper is devoted to deriving necessary optimality conditions in a general optimal control problem for dynamical systems governed by controlled sweeping processes with hard-constrained control actions entering both polyhedral moving sets and additive perturbations. By using the first-order and mainly second-order tools of variational analysis and generalized differentiation, we develop a well-posed method of discrete approximations, obtain optimality conditions for solutions to discrete-time control systems, and then establish by passing to the limit verifiable necessary optimality conditions for local minimizers of the original controlled sweeping process that are expressed entirely in terms of its given data. The efficiency of the obtained necessary optimality conditions for the sweeping dynamics is illustrated by solving three nontrivial examples of their own interest., Comment: 33 pages, 3 figures

We study the existence and non-existence of classical solutions for inequalities of type ± Δ m u ≥ ( Ψ ( | x | ) ⁎ u p ) u q in R N ( N ≥ 1 ) . Here, Δ m ( m ≥ 1 ) is the polyharmonic operator, p , q > 0 and ⁎ denotes the convolution operator, where Ψ > 0 is a continuous non-increasing function. We devise new methods to deduce that solutions of the above inequalities satisfy the poly-superharmonic property. This further allows us to obtain various Liouville type results. Our study is also extended to the case of systems of simultaneous inequalities.

In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics (MHD) system in R 3 . Let ( v , B , p ) be a smooth solution to the stationary MHD equations in R 3 . We show that if there exist smooth matrix valued potential functions Φ, Ψ such that ∇ ⋅ Φ = v and ∇ ⋅ Ψ = B , whose L 6 mean oscillations have certain growth condition near infinity, namely ⨍ B ( r ) | Φ − Φ B ( r ) | 6 d x + ⨍ B ( r ) | Ψ − Ψ B ( r ) | 6 d x ≤ C r ∀ 1 r + ∞ , then v = B = 0 and p=constant. With additional assumption of r − 8 ∫ B ( r ) | B − B B ( r ) | 6 d x → 0 as r → + ∞ , similar result holds also for the Hall-MHD system.

We show that the stochastic 3D primitive equations with the Neumann boundary condition on the top, the lateral Dirichlet boundary condition and either the Dirichlet or the Neumann boundary condition on the bottom driven by multiplicative gradient-dependent white noise have unique maximal strong solutions both in the stochastic and PDE senses under certain assumptions on the growth of the noise. For the case of the Neumann boundary condition on the bottom, global existence is established by using the decomposition of the vertical velocity to the barotropic and baroclinic modes and an iterated stopping time argument. An explicit example of non-trivial infinite dimensional noise depending on the vertical average of the horizontal gradient of horizontal velocity is presented.

In this paper, we present necessary and sufficient conditions to have global analytic hypoellipticity for a class of first-order operators defined on T 1 × S 3 . In the case of real-valued coefficients, we prove that an operator in this class is conjugated to a constant-coefficient operator satisfying a Diophantine condition, and that such conjugation preserves the global analytic hypoellipticity. In the case where the imaginary part of the coefficients is non-zero, we show that the operator is globally analytic hypoelliptic if the Nirenberg-Treves condition ( P ) holds, in addition to an analytic Diophantine condition.

This work is concerned with the compressible micropolar fluids system in three–dimensional space. We consider the asymptotic behavior of the solution to the Cauchy problem near the constant equilibrium state provided that the initial perturbation is sufficiently small. Under some assumptions of the initial data, we show that the solution of the Cauchy problem converges to its constant equilibrium state at the exact same L 2 –decay rates as the linearized equations, which shows the convergence rates are optimal. The proof is based on the spectral analysis of the semigroup generated by the linearized equations and the nonlinear energy estimates.

In this paper, we study the Cauchy problem for the linear and semilinear Moore-Gibson-Thompson (MGT) equation in the dissipative case. Concerning the linear MGT model, by utilizing WKB analysis associated with Fourier analysis, we derive some L 2 estimates of solutions, which improve those in the previous research [51] . Furthermore, asymptotic profiles of the solution and an approximate relation in a framework of the weighted L 1 space are derived. Next, with the aid of the classical energy method and Hardy's inequality, we get singular limit results for an energy and the solution itself. Concerning the semilinear MGT model, basing on the obtained sharp L 2 estimates and constructing time-weighted Sobolev spaces, we investigate global (in time) existence of Sobolev solutions with different regularities. Finally, under a sign assumption on initial data, nonexistence of global (in time) weak solutions is proved by applying a test function method.

In this paper, we make a comprehensive study for the orbital stability of standing waves for the fractional Schrodinger equation with combined power-type nonlinearities (FNLS) i ∂ t ψ − ( − Δ ) s ψ + a | ψ | p 1 ψ + | ψ | p 2 ψ = 0 . We prove that when p 2 = 4 s N and a ( p 1 − 4 s N ) 0 , there exist the standing waves of (FNLS), which are orbitally stable. When a = 0 and 4 s N p 2 4 s N − 2 s , we present a new, simpler method to study the strong instability of standing waves. When a = − 1 , 0 p 1 p 2 and 4 s N ≤ p 2 4 s N − 2 s , or a = 1 and 4 s N ≤ p 1 p 2 4 s N − 2 s , or a = 1 , 0 p 1 4 s N p 2 4 s N − 2 s and ∂ λ 2 S ω ( u ω λ ) | λ = 1 ≤ 0 , we deduce that the ground state standing waves of (FNLS) are strongly unstable by blow-up.

For $N\geq 3$, by the seminal paper of Brezis and Veron (Arch. Rational Mech. Anal. 75(1):1--6, 1980/81), no positive solutions of $-\Delta u+u^q=0$ in $\mathbb R^N\setminus \{0\}$ exist if $q\geq N/(N-2)$; for $11$ and $\theta\in \mathbb R$, we prove that the nonlinear elliptic problem (*) $-\Delta u-\lambda \,|x|^{-2}\,u+|x|^{\theta}u^q=0$ in $\mathbb R^N\setminus \{0\}$ with $u>0$ has a $C^1(\mathbb R^N\setminus \{0\})$ solution if and only if $\lambda>\lambda^*$, where $\lambda^*=\Theta(N-2-\Theta) $ with $\Theta=(\theta+2)/(q-1)$. We show that (a) if $\lambda>(N-2)^2/4$, then $U_0(x)=(\lambda-\lambda^*)^{1/(q-1)}|x|^{-\Theta}$ is the only solution of (*) and (b) if $\lambda^* \max\{q_{N,\theta},1\}$, where $q_{N,\theta}=(N+2\theta+2)/(N-2)$. In addition, for $\theta\leq -2$ we settle the structure of the set of all positive solutions of (*) in $\Omega\setminus \{0\}$, subject to $u|_{\partial\Omega}=0$, where $\Omega$ is a smooth bounded domain containing zero, complementing the works of C\^irstea (Mem. Amer. Math. Soc. 227, 2014) and Wei--Du (J. Differential Equations 262(7):3864--3886, 2017).

We study quasilinear elliptic double obstacle problems with a variable exponent growth when the right-hand side is a measure. A global Calder\'{o}n-Zygmund estimate for the gradient of an approximable solution is obtained in terms of the associated double obstacles and a given measure, identifying minimal requirements for the regularity estimate., Comment: 26 pages, typos corrected

This paper is concerned with Friedrichs extensions for a class of Hamiltonian systems. The non-symmetric problems are usually complicated and have unexpected properties. Here, Friedrichs extensions of a class of singular Hamiltonian systems including non-symmetric cases are characterized by imposing some constraints on each element of domains D ( H ) of the maximal operators H. These characterizations are given independent of principal solutions. It is interesting that by the results in the paper the Friedrichs extension of each of a large class of non-symmetric Hamiltonian systems has similar form to that of a symmetric Hamiltonian system. In addition, Friedrichs extensions of regular Hamiltonian systems are characterized incidently, J -self-adjoint Friedrichs extensions are studied, and a result is given for elements of D ( H ) , which makes the expression of the Friedrichs extension domain simpler. All results for Hamiltonian systems are finally applied to Sturm-Liouville operators with matrix-valued coefficients.