Subjects -> MATHEMATICS (Total: 1013 journals)     - APPLIED MATHEMATICS (92 journals)    - GEOMETRY AND TOPOLOGY (23 journals)    - MATHEMATICS (714 journals)    - MATHEMATICS (GENERAL) (45 journals)    - NUMERICAL ANALYSIS (26 journals)    - PROBABILITIES AND MATH STATISTICS (113 journals) MATHEMATICS (714 journals)                  1 2 3 4 | Last

1 2 3 4 | Last

Similar Journals
 Calculus of Variations and Partial Differential EquationsJournal Prestige (SJR): 3.352 Citation Impact (citeScore): 2Number of Followers: 0      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1432-0835 - ISSN (Online) 0944-2669 Published by Springer-Verlag  [2469 journals]
• Stable determination of an elastic medium scatterer by a single far-field
measurement and beyond

Abstract: Abstract We are concerned with the time-harmonic elastic scattering due to an inhomogeneous elastic material inclusion located inside a uniformly homogeneous isotropic medium. We establish a sharp stability estimate of logarithmic type in determining the support of the elastic scatterer, independent of its material content, by a single far-field measurement when the support is a convex polyhedral domain in $${\mathbb {R}}^n$$ , $$n=2,3$$ . Our argument in establishing the stability result is localized around a corner of the medium scatterer. This enables us to further establish a byproduct result by proving that if a generic medium scatterer, not necessary to be a polyhedral shape, possesses a corner, then there exists a positive lower bound of the scattered far-field patterns. The latter result indicates that if an elastic material object possesses a corner on its support, then it scatters every incident wave stably and invisibility phenomenon does not occur.
PubDate: 2022-06-24

• A flow approach to the generalized Loewner-Nirenberg problem of the
$$\sigma _k$$ σ k -Ricci equation

Abstract: Abstract We introduce a flow approach to the generalized Loewner-Nirenberg problem (1.5)-(1.7) of the $$\sigma _k$$ -Ricci equation on a compact manifold $$(M^n,g)$$ with boundary. We prove that for initial data $$u_0\in C^{4,\alpha }(M)$$ which is a subsolution to the $$\sigma _k$$ -Ricci equation (1.5), the Cauchy-Dirichlet problem (3.1)-(3.3) has a unique solution u which converges in $$C^4_{loc}(M^{\circ })$$ to the solution $$u_{\infty }$$ of the problem (1.5)-(1.7), as $$t\rightarrow \infty$$ .
PubDate: 2022-06-24

• Domain walls in the coupled Gross–Pitaevskii equations with the
harmonic potential

Abstract: Abstract We study the existence and variational characterization of steady states in a coupled system of Gross–Pitaevskii equations modeling two-component Bose-Einstein condensates with the magnetic field trapping. The limit with no trapping has been the subject of recent works where domain walls have been constructed and several properties, including their orbital stability have been derived. Here we focus on the full model with the harmonic trapping potential and characterize minimizers according to the value of the coupling parameter $$\gamma$$ . We first establish a rigorous connection between the two problems in the Thomas-Fermi limit via $$\Gamma$$ -convergence. Then, we identify the ranges of $$\gamma$$ for which either the symmetric states $$(\gamma < 1)$$ or the uncoupled states $$(\gamma > 1)$$ are minimizers. Domain walls arise as minimizers in a subspace of the energy space with a certain symmetry for some $$\gamma > 1$$ . We study bifurcation of the domain walls and furthermore give numerical illustrations of our results.
PubDate: 2022-06-22

• Minimal measures on the level sets below Mañé critical value

Abstract: Abstract Minimal measure was introduced for positive definite Lagrangian systems to study their dynamics. For autonomous system, the support lies in the level set of the Hamiltonian not below the Mañé critical value. It is found in this paper that, for mechanical system, there also exist some minimal measures on the level sets below the Mañé critical value.
PubDate: 2022-06-22

• $$L^\infty$$ L ∞ -estimates in optimal transport for non quadratic
costs

Abstract: Abstract For cost functions $$c(x,y)=h(x-y)$$ , with $$h\in C^2\left( {{\mathbb {R}}}^n\setminus \{0\}\right) \cap C^1\left( {{\mathbb {R}}}^n\right)$$ homogeneous of degree $$p>1$$ , we show $$L^\infty$$ -estimates of $$Tx-x$$ on balls, where T is an h-monotone map. Estimates for the interpolating mappings $$T_t=t(T-I)+I$$ are deduced from this.
PubDate: 2022-06-22

• On $$W^{2,p}$$ W 2 , p -estimates for solutions of obstacle problems for
fully nonlinear elliptic equations with oblique boundary conditions

Abstract: Abstract This paper concerns fully nonlinear elliptic obstacle problems with oblique boundary conditions. We investigate the existence, uniqueness and $$W^{2,p}$$ -regularity results by finding approximate non-obstacle problems with the same oblique boundary condition and then making a suitable limiting process.
PubDate: 2022-06-22

• On the cauchy problem of 3D compressible, viscous, heat-conductive
navier-stokes-Poisson equations subject to large and non-flat doping
profile

Abstract: Abstract In this paper, we study an initial value problem of the Navier-Stokes-Poisson equations for compressible, viscous, heat-conducting flows on the whole space $$\mathrm{{{\mathbb {R}}}^3}$$ . The global well-posedness of strong solutions subject to large and non-flat doping profile is established. The initial data is of small energy but possible large oscillations, and the initial density is allowed to contain vacuum states.
PubDate: 2022-06-22

• Kellogg’s theorem for diffeomophic minimizers of Dirichlet energy
between doubly connected Riemann surfaces

Abstract: Abstract We extend the celebrated theorem of Kellogg for conformal diffeomorphisms to the minimizers of Dirichlet energy. Namely we prove that a diffeomorphic minimizer of Dirichlet energy of Sobolev mappings between doubly connected Riemannian surfaces $$({\mathbb {X}},\sigma )$$ and $$({\mathbb {Y}},\rho )$$ having $${\mathscr {C}}^{n,\alpha }$$ boundary, $$0<\alpha <1$$ , is $${\mathscr {C}}^{n,\alpha }$$ up to the boundary, provided the metric $$\rho$$ is smooth enough. Here n is a positive integer. It is crucial that every diffeomorphic minimizer of Dirichlet energy is a harmonic mapping with a very special Hopf differential and this fact is used in the proof. This improves and extends a recent result by the author and Lamel in Kalaj and Lamel (Math Ann 377:1643–1672, 2020), where the authors proved a similar result for doubly-connected domains in the complex plane but for $$\alpha '$$ which is $$\le \alpha$$ and $$\rho \equiv 1$$ . This is a complementary result of an existence result proved by T. Iwaniec et al. in Iwaniec et al. (Invent Math 186:667–707, 2011) and the author in Kalaj (Var Partial Differ Equ 51:465–494, 2014)
PubDate: 2022-06-22

• Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric

Abstract: Abstract This paper introduces and studies a metamorphosis framework for geometric measures known as varifolds, which extends the diffeomorphic registration model for objects such as curves, surfaces and measures by complementing diffeomorphic deformations with a transformation process on the varifold weights. We consider two classes of cost functionals to penalize those combined transformations, in particular the LDDMM-Fisher-Rao energy which, as we show, leads to a well-defined Riemannian metric on the space of varifolds with existence of corresponding geodesics. We further introduce relaxed formulations of the respective optimal control problems, study their well-posedness and derive optimality conditions for the solutions. From these, we propose a numerical approach to compute optimal metamorphoses between discrete varifolds and illustrate the interest of this model in the situation of partially missing data.
PubDate: 2022-06-22

• Boundary regularity estimates in Hölder spaces with variable exponent

Abstract: Abstract We present a general blow-up technique to obtain local regularity estimates for solutions, and their derivatives, of second order elliptic equations in divergence form in Hölder spaces with variable exponent. The procedure allows to extend the estimates up to a portion of the boundary where Dirichlet or Neumann boundary conditions are prescribed and produces a Schauder theory for partial derivatives of solutions of any order $$k\in {\mathbb {N}}$$ . The strategy relies on the construction of a class of suitable regularizing problems and an approximation argument. The given data of the problem are taken in Hölder and Lebesgue spaces, both with variable exponent.
PubDate: 2022-06-22

• Partial existence result for homogeneous quasilinear parabolic problems
beyond the duality pairing

Abstract: Abstract In this paper, we study the existence of distributional solutions solving (1.3) on a bounded domain $$\Omega$$ satisfying a uniform capacity density condition where the nonlinear structure $$\mathcal {A}(x,t,\nabla u)$$ is modelled after the standard parabolic p-Laplace operator. In this regard, we need to prove a priori estimates for the gradient of the solution below the natural exponent and a higher integrability result for very weak solutions at the initial boundary. The elliptic counterpart to these two estimates is fairly well developed over the past few decades, but no analogous theory exists in the quasilinear parabolic setting. Two important features of the estimates proved here are that they are non-perturbative in nature and we are able to take non-zero boundary data. As a consequence, our a priori estimates are new even for the heat equation on bounded domains. This partial existence result is a nontrivial extension of the existence theory of very weak solutions from the elliptic setting to the quasilinear parabolic setting. Even though we only prove partial existence result, nevertheless we establish the necessary framework that when proved would lead to obtaining the full result for the homogeneous problem.
PubDate: 2022-06-18

• Limit profiles for singularly perturbed Choquard equations with local
repulsion

Abstract: Abstract We study Choquard type equation of the form where $$N\ge 3$$ , $$I_\alpha$$ is the Riesz potential with $$\alpha \in (0,N)$$ , $$p>1$$ , $$q>2$$ and $$\varepsilon \ge 0$$ . Equations of this type describe collective behaviour of self-interacting many-body systems. The nonlocal nonlinear term represents long-range attraction while the local nonlinear term represents short-range repulsion. In the first part of the paper for a nearly optimal range of parameters we prove the existence and study regularity and qualitative properties of positive groundstates of $$(P_0)$$ and of $$(P_\varepsilon )$$ with $$\varepsilon >0$$ . We also study the existence of a compactly supported groundstate for an integral Thomas–Fermi type equation associated to $$(P_{\varepsilon })$$ . In the second part of the paper, for $$\varepsilon \rightarrow 0$$ we identify six different asymptotic regimes and provide a characterisation of the limit profiles of the groundstates of $$(P_\varepsilon )$$ in each of the regimes. We also outline three different asymptotic regimes in the case $$\varepsilon \rightarrow \infty$$ . In one of the asymptotic regimes positive groundstates of $$(P_\varepsilon )$$ converge to a compactly supported Thomas–Fermi limit profile. This is a new and purely nonlocal phenomenon that can not be observed in the local prototype case of $$(P_\varepsilon )$$ with $$\alpha =0$$ . In particular, this provides a justification for the Thomas–Fermi approximation in astrophysical models of self-gravitating Bose–Einstein condensate.
PubDate: 2022-06-18

• Scattering for the non-radial inhomogenous biharmonic NLS equation

Abstract: Abstract We consider the focusing inhomogeneous biharmonic nonlinear Schrödinger equation in $$H^2(\mathbb {R}^N)$$ , \begin{aligned} iu_t + \Delta ^2 u - x ^{-b} u ^{\alpha }u=0, \end{aligned} when $$b > 0$$ and $$N \ge 5$$ . We first obtain a small data global result in $$H^2$$ , which, in the five-dimensional case, improves a previous result from Pastor and the second author. In the sequel, we show the main result, scattering below the mass-energy threshold in the intercritical case, that is, $$\frac{8-2b}{N}< \alpha <\frac{8-2b}{N-4}$$ , without assuming radiality of the initial data. The proof combines the decay of the nonlinearity with Virial-Morawetz-type estimates to avoid the radial assumption, allowing for a much simpler proof than the Kenig-Merle roadmap.
PubDate: 2022-06-14

• Nonlinear approximation of 3D smectic liquid crystals: sharp lower bound
and compactness

Abstract: Abstract We consider the 3D smectic energy \begin{aligned} {\mathcal {E}}_{\epsilon }\left( u\right) =\frac{1}{2}\int _{\Omega }\frac{1}{\varepsilon } \left( \partial _{z}u-\frac{(\partial _{x}u)^{2}+(\partial _{y}u)^{2}}{2}\right) ^{2} +\varepsilon \left( \partial _{x}^{2}u+\partial _{y}^{2}u\right) ^{2}dx\,dy\,dz. \end{aligned} The model contains as a special case the well-known 2D Aviles-Giga model. We prove a sharp lower bound on $${\mathcal {E}}_{\varepsilon }$$ as $$\varepsilon \rightarrow 0$$ by introducing 3D analogues of the Jin–Kohn entropies Jin and Kohn (J Nonlinear Sci 10:355–390, 2000). The sharp bound corresponds to an equipartition of energy between the bending and compression strains and was previously demonstrated in the physics literature only when the approximate Gaussian curvature of each smectic layer vanishes. Also, for $$\varepsilon _{n}\rightarrow 0$$ and an energy-bounded sequence $$\{u_n \}$$ with $$\Vert \nabla u_n\Vert _{L^{p}(\Omega )},\, \Vert \nabla u_n\Vert _{L^2(\partial \Omega )}\le C$$ for some $$p>6$$ , we obtain compactness of $$\nabla u_{n}$$ in $$L^{2}$$ assuming that $$\Delta _{xy}u_{n}$$ has constant sign for each n.
PubDate: 2022-06-14

• Monotonicity of positive solutions to quasilinear elliptic equations in
half-spaces with a changing-sign nonlinearity

Abstract: Abstract In this paper we prove the monotonicity of positive solutions to $$-\Delta _p u = f(u)$$ in half-spaces under zero Dirichlet boundary conditions, for $$(2N+2)/(N+2)< p < 2$$ and for a general class of regular changing-sign nonlinearities f. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.
PubDate: 2022-06-14

• Reducibility of quantum harmonic oscillator on $$\mathbb {R}^d$$ R d
perturbed by a quasi: periodic potential with logarithmic decay

Abstract: Abstract We prove the reducibility of quantum harmonic oscillators in $${\mathbb {R}}^d$$ perturbed by a quasi-periodic in time potential $$V(x,\omega t)$$ with $$\textit{logarithmic decay}$$ . By a new estimate built for solving the homological equation we improve the reducibility result by Grébert-Paturel(Annales de la Faculté des sciences de Toulouse : Mathématiques. $$\mathbf{28}$$ , 2019).
PubDate: 2022-06-14

• Critical Hardy inequality on the half-space via the harmonic
transplantation

Abstract: Abstract We prove a critical Hardy inequality on the half-space $${\mathbb {R}}^N_+$$ by using the harmonic transplantation for functions in $${\dot{W}}_0^{1,N}({\mathbb {R}}^N_+)$$ . Also we give an improvement of the subcritical Hardy inequality on $${\dot{W}}_0^{1,p}({\mathbb {R}}^N_+)$$ for $$p \in [2, N)$$ , which converges to the critical Hardy inequality when $$p \nearrow N$$ . Sobolev type inequalities are also discussed.
PubDate: 2022-06-14

• Classification of solutions for some elliptic system

Abstract: Abstract In this paper, we classify the solution of the following elliptic system \begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u(x)=e^{3v(x)}, &{}\quad x\in {\mathbb {R}}^4, \\ \\ \displaystyle (-\Delta )^2v(x)=u(x)^4, &{}\quad x\in {\mathbb {R}}^4. \end{array} \right. \end{aligned} Under some assumptions, we will show that the solution has the following form \begin{aligned} u(x)=\frac{C_1(\varepsilon )}{\varepsilon ^2+ x-x_0 ^2},\ v(x)=\ln \frac{C_2(\varepsilon )}{\varepsilon ^2+ x-x_0 ^2}, \end{aligned} where $$C_1,C_2$$ are two positive constants depending only on $$\varepsilon$$ and $$x_0$$ is a fixed point in $${\mathbb {R}}^4.$$
PubDate: 2022-06-10

• Global Existence, Regularity and Boundedness in a Higher-dimensional
Chemotaxis-Navier-Stokes System with Nonlinear Diffusion and General
Sensitivity

Abstract: Abstract We consider an incompressible chemotaxis-Navier-Stokes system with nonlinear diffusion and rotational flux \begin{aligned} \left\{ \begin{array}{l} n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c),\quad x\in \Omega , t>0,\\ c_t+u\cdot \nabla c=\Delta c-nc,\quad x\in \Omega , t>0,\\ u_t+\kappa (u \cdot \nabla )u+\nabla P=\Delta u+n\nabla \phi ,\quad x\in \Omega , t>0,\\ \nabla \cdot u=0,\quad x\in \Omega , t>0\\ \end{array}\right. \end{aligned} in a bounded domain $$\Omega \subset {\mathbb {R}}^N(N=2,3)$$ with smooth boundary $$\partial \Omega$$ , where $$\kappa \in {\mathbb {R}}$$ . The chemotaxtic sensitivity S is a given tensor-valued function fulfilling $$S(x,n,c) \le S_0(c)$$ for all $$(x,n,c)\in {\bar{\Omega }} \times [0, \infty )\times [0, \infty )$$ with $$S_0(c)$$ nondecreasing on $$[0,\infty )$$ . By introducing some new methods (see Sect. 4 and Sect. 5), we prove that under the condition $$m >1$$ and some other proper regularity hypotheses on initial data, the corresponding initial-boundary problem possesses at least one global weak solution. The present work also shows that the weak solution could be bounded provided that $$N= 2$$ . Since S is tensor-valued, it is easy to see that the restriction on m here is optimal, which answers the left question in Bellomo-Belloquid-Tao-Winkler (Math Models Methods Appl Sci 25:1663–1763, 2015) and Tao-Winkler (Ann Inst H Poincaré Anal Non Linéaire 30:157–178, 2013). And obviously, this work improves previous results of several other authors (see Remark 1.1).
PubDate: 2022-06-10

• Generic existence of multiplicity-1 minmax minimal hypersurfaces via
Allen–Cahn

Abstract: Abstract In Guaraco (J. Differential Geom. 108(1):91–133, 2018) a new proof was given of the existence of a closed minimal hypersurface in a compact Riemannian manifold $$N^{n+1}$$ with $$n\ge 2$$ . This was achieved by employing an Allen–Cahn approximation scheme and a one-parameter minmax for the Allen–Cahn energy (relying on works by Hutchinson, Tonegawa, Wickramasekera to pass to the limit as the Allen-Cahn parameter tends to 0). The minimal hypersurface obtained may a priori carry a locally constant integer multiplicity. Here we modify the minmax construction of Guaraco (J. Differential Geom. 108(1):91–133, 2018), by allowing an initial freedom on the choice of the valley points between which the mountain pass construction is carried out, and then optimising over said choice. We then prove that, when $$2\le n\le 6$$ and the metric is bumpy, this minmax leads to a (smooth closed) minimal hypersurface with multiplicity 1. (When $$n=2$$ this conclusion also follows from Chodosh and Mantoulidis (Ann. Math. 191(1):213–328, 2020).) As immediate corollary we obtain that every compact Riemannian manifold of dimension $$n+1$$ , $$2\le n\le 6$$ , endowed with a bumpy metric, admits a two-sided smooth closed minimal hypersurface (this existence conclusion also follows from Zhou X (Ann. Math. (2), 192(3):767–820, 2020) for minmax constructions via Almgren–Pitts theory).
PubDate: 2022-06-10

JournalTOCs
School of Mathematical and Computer Sciences
Heriot-Watt University
Edinburgh, EH14 4AS, UK
Email: journaltocs@hw.ac.uk
Tel: +00 44 (0)131 4513762