We consider a curvature flow \(V=H\) in the band domain \(\Omega :=[-1,1]\times \mathbb {R}\), where, for a graphic curve \(\Gamma _t\), V denotes its normal velocity and H denotes its curvature. If \(\Gamma _t\) contacts the two boundaries \(\partial _\pm \Omega \) of \(\Omega \) with constant slopes, in 1993, Altschular and Wu (Math Ann 295:761–765, 1993) proved that \(\Gamma _t\) converges to a

We consider the Dirichlet problem for a compressible two-fluid model in multi-dimensions. It consists of the continuity equations for each fluid and the momentum equations for the mixture. This model can be derived from the compressible two-fluid model with equal velocities (Bresch et al., in Arch Rational Mech Anal 196:599–629, 2010) and from a scaling limit of the Vlasov-Fokker-Planck/compressible

We prove pointwise and \(L^{p}\)-gradient comparison results for solutions to elliptic Dirichlet problems defined on open subsets of a (possibly non-smooth) space with positive Ricci curvature (more precisely of an \({{\,\mathrm{RCD}\,}}(K,N)\) metric measure space, with \(K>0\) and \(N\in (1,\infty )\)). The obtained Talenti-type comparison is sharp, rigid and stable with respect to \(L^{2}\)/mea

We establish Hölder continuity for locally bounded weak solutions to certain parabolic systems of porous medium type, i.e. $$\begin{aligned} \partial _t \mathbf{u}-\mathrm{div}(m|\mathbf{u}|^{m-1}D\mathbf{u})=0,\quad m>0. \end{aligned}$$ As a consequence of our local Hölder estimates, a Liouville type result for bounded global solutions is also established. In addition, sufficient conditions are given

A correction to this paper has been published: https://doi.org/10.1007/s00526-021-02026-1

We study the Bose–Einstein condensates (BEC) in two or three dimensions with attractive interactions, described by \(L^{2}\) constraint Gross-Pitaevskii energy functional. First, we give the precise description of the chemical potential of the condensate \(\mu \) and the attractive interaction a. Next, for a class of degenerate trapping potential with non-isolated critical points, we obtain the existence

S.-Y. A. Chang, J. Qing, and P. Yang proved an important gap theorem for Bach-flat metrics with round sphere as model case in 2007. In this article, we generalize this result by establishing conformally invariant gap theorems for Bach-flat 4-manifolds with \((\mathbb {CP}^2, g_{FS})\) and \((S^2 \times S^2,g_{prod})\) as model cases. An iteration argument plays an important role in the case of \((\mathbb

In this paper we study the Nirenberg problem on standard half spheres \((\mathbb {S}^n_+,g), \, n \ge 5\), which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: $$\begin{aligned} (\mathcal {P}) \quad {\left\{ \begin{array}{ll}

In this paper, we study quasilinear parabolic equations with the nonlinearity structure modeled after the p(x, t)-Laplacian on nonsmooth domains. The main goal is to obtain end point Calderón-Zygmund type estimates in the variable exponent setting. In a recent work [1], the estimates obtained were strictly above the natural exponent p(x, t) and hence there was a gap between the natural energy estimates

We study the discrete-to-continuum variational limit of the antiferromagnetic XY model on the two-dimensional triangular lattice. The system is fully frustrated and displays two families of ground states distinguished by the chirality of the spin field. We compute the \(\Gamma \)-limit of the energy in a regime which detects chirality transitions on one-dimensional interfaces between the two admissible

Let \((\phi _\alpha , \psi _\alpha )\) be a sequence of \(\alpha \)-Dirac-harmonic maps from a Riemann surface M to a compact Riemannian manifold N with uniformly bounded energy. If the target N is a sphere \(S^{K-1}\), we show that the energy identity and necklessness hold during the interior blow-up process as \(\alpha \searrow 1\) for such a sequence .

Given two-dimensional Riemannian manifolds \(\mathcal {M},\mathcal {N}\), we prove a lower bound on the distortion of embeddings \(\mathcal {M}\rightarrow \mathcal {N}\), in terms of the areas’ discrepancy \(V_{\mathcal {N}}/V_{\mathcal {M}}\), for a certain class of distortion functionals. For \(V_{\mathcal {N}}/V_{\mathcal {M}} \ge 1/4\), homotheties, provided they exist, are the unique energy minimizing

Let \(\Omega \subset \mathbb {R}^3\) be an open and bounded set with Lipschitz boundary and outward unit normal \(\nu \). For \(11 \quad \text {if }p = \frac{3}{2}. \end{aligned}$$ Specifically, there exists a constant \(c=c(p,\Omega ,r)>0\) such that the inequality $$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{3\times 3})}\le c\,\left( \Vert {{\,\mathrm{sym}\,}}P \Vert _{L^p(\Omega ,\mathbb

In this paper, we study the nonrelativistic limit and some properties of solutions for the following nonlinear Dirac equation $$\begin{aligned} -ic\sum _{k=1}^3 \alpha _k \partial _k \psi + m c^2 \beta \psi - \omega \psi =g(|\psi |)\psi , ~~\text {in}\ \mathbb {R}^3 \end{aligned}$$ where c denotes the speed of light, \(m>0\) is the mass of the electron. We show that solutions of nonlinear Dirac equation

We study the chemotaxis-(Navier–)Stokes system modeling coral fertilization: \(n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nS(x,n,c)\nabla c)-nm\), \(c_t+u\cdot \nabla c=\Delta c-c+m\), \(m_t+u\cdot \nabla m=\Delta m-nm\), \(u_t+\kappa (u\cdot \nabla )u+\nabla P=\Delta u+(n+m)\nabla \phi \) and \(\nabla \cdot u=0\) in a bounded and smooth domain \(\Omega \subset \mathbb {R}^2\), where \(\kappa \in \mathbb

A vectorial Modica–Mortola functional is considered and the convergence to a sharp interface model is studied. The novelty of the paper is that the wells of the potential are not constant, but depend on the spatial position in the domain \(\Omega \). The mass constrained minimization problem and the case of Dirichlet boundary conditions are also treated. The proofs rely on the precise understanding

We prove that the combination of the norm of gradient and the Gaussian curvature for the convex level sets of harmonic function is superharmonic.

We consider the natural time-dependent fractional p-Laplacian equation posed in the whole Euclidean space, with parameter \(12\) has been dealt with in a recent paper. We study here the “fast” regime \(1

We study the boundary behaviors of a complete conformal metric which solves the \(\sigma _k\)-Ricci problem on the interior of a manifold with boundary. We establish asymptotic expansions and also \(C^1\) and \(C^2\) estimates for this metric multiplied by the square of the distance in a small neighborhood of the boundary.

In the cases where there is no Sobolev-type or Gagliardo–Nirenberg-type fractional estimate involving \({|u |}_{W^{s,p}}\), we establish alternative estimates where the strong \(L^p\) norms are replaced by Lorentz norms.

We prove new existence results for a nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$\begin{aligned} - \Delta u - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}\left( {\mathbb {R}}^{N}\right) \end{aligned}$$ with \(k>0,\) \(N \ge 3\), \(p \in \left[ \left. \frac{2(N+1)}{N-1},\frac{2N}{N-2}\right) \right. \) and \(Q \in L^{\infty }({\mathbb {R}}^{N})\). Due to the sign-changes

In this paper we establish some symmetry results for positive solutions of semilinear elliptic equations with mixed boundary conditions. In particular, we show that the positive solution in a super-spherical sector must be symmetric. The monotonicity property is also proved. Our proof is based on the well-known moving plane methods.

Following Rivière’s study of conservation laws for second order quasilinear systems with critical nonlinearity and Lamm/Rivière’s generalization to fourth order, we consider similar systems of order 2m. Typical examples are m-polyharmonic maps. Under natural conditions, we find a conservation law for weak solutions on 2m-dimensional domains. This implies continuity of weak solutions.

We prove a Liouville type theorem for minimizing extrinsic biharmonic maps from \({\mathbb {R}}^m\) to the Euclidean unit sphere \({\mathbb {S}}^n\) : if \(u \in W^{2,2} _{loc}( {\mathbb {R}}^m, {\mathbb {S}}^{n})\) is a minimizing extrinsic biharmonic map and \(m\le 4\) then u is constant. In the case that \(m \ge 5\), if we suppose that \( \int _{{\mathbb {R}}^m}\left| \Delta u\right| ^{2} dx\) is

Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with the Dirichlet boundary condition are considered. The principal eigen map of such elliptic operators is introduced and some basic properties of this map, including continuity, concavity, and differentiability with respect to the parameter in the diffusibility matrix, are established

Given \(n_1\ge n_2\ge 2\), let \(\Sigma \) be an orientable, minimal hypersurface of \(\mathbb {S}^{n_1}(1)\times \mathbb {S}^{n_2}(a)\) with index one. Assume \(\Sigma \) is closed and either smooth or singular with a singular set \({{\,\mathrm{sing}\,}}(\Sigma )\) satisfying \(\mathcal {H}^{n-2}({{\,\mathrm{sing}\,}}(\Sigma ))=0\). By using the almost product structure, we prove that, when \(a^{2}\ge

We develop a functional analytic approach for the study of nonlocal minimal graphs. Through this, we establish existence and uniqueness results, a priori estimates, comparison principles, rearrangement inequalities, and the equivalence of several notions of minimizers and solutions.

We consider Riesz transforms of any order associated to an Ornstein–Uhlenbeck operator with covariance given by a real, symmetric and positive definite matrix, and with drift given by a real matrix whose eigenvalues have negative real parts. In this general Gaussian context, we prove that a Riesz transform is of weak type (1, 1) with respect to the invariant measure if and only if its order is at most

In dimension n isolated singularities—at a finite point or at infinity—for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in Esposito (Anal Non Linéaire 35(3):781–801, 2018) and establish a quantization result for entire solutions of the singular n-Liouville equation.

We construct certain non-polynomial entire solutions to \(\sigma _k\) equations on \({\mathbb {R}}^n\).

We consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their \(C^{1,\beta }\) regularity on the either side of the thin manifold, the optimal growth away from the free boundary, the \(C^{1,\gamma }\) regularity of the regular part of the free boundary, as well as a structural theorem for the singular set. The analysis of the

Klein et al. (J Fluid Mech 288:201–248, 1995) have formally derived a simplified asymptotic motion law for the evolution of nearly parallel vortex filaments in the context of the three dimensional Euler equation for incompressible fluids. In the present work, we rigorously derive the corresponding asymptotic motion law in the context of the Gross–Pitaevskii equation.

For a class of semilinear elliptic systems, the existence of a broader class of multibump solutions is established under considerably weaker conditions than in earlier works. The key tool is the use of variational methods.

Given a complete non-compact Riemannian manifold (M, g) with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries G of (M, g) that characterizes the coerciveness of G in the sense of Skrzypczak and Tintarev (Arch Math 101(3): 259–268, 2013). Furthermore, under these conditions, compact Sobolev-type embeddings à la Berestycki-Lions are proved for the full

We develop a general method to compute the Morse index of branched Willmore spheres and show that the Morse index is equal to the index of certain matrix whose dimension is equal to the number of ends of the dual minimal surface (when the latter exists). As a corollary, we find that for all immersed Willmore spheres \(\vec {\Phi }:S^2\rightarrow \mathbb {R}^3\) such that \(W(\vec {\Phi })=4\pi n\)

We consider elliptic systems of order 2m in dimension 2m which are generalizations of extrinsic and intrinsic polyharmonic maps. We show the existence of a conservation law for these systems by using a small perturbation of Uhlenbeck’s gauge fixing matrix.

This paper utilizes the p-capacity-flux and the p-capacity-radius to make an intrinsic three-fold exploration of the p-capacity for the subconformal case \(1

This article deals with the study of the following singular quasilinear equation: $$\begin{aligned} (P) \left\{ \ -\Delta _{p}u -\Delta _{q}u = f(x) u^{-\delta },\; u>0 \text { in }\; \Omega ; \; u=0 \text { on } \partial \Omega , \right. \end{aligned}$$ where \(\Omega \) is a bounded domain in \({\mathbb {R}}^N\) with \(C^2\) boundary \(\partial \Omega \), \(1< q< p<\infty \), \(\delta >0\) and \(f\in

Let (M, g) be a compact Riemannian n-dimensional manifold with umbilic boundary It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have \(\partial M\) as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set in the case of low dimensional manifolds, that is \(n=6,7,8\), provided that the Weyl tensor

Let \(\mathcal {B}\) be a homogeneous differential operator of order \(l=1\) or \(l=2\). We show that a sequence of functions of the form \((\mathcal {B}u_j)_j\) converging in the \(L^1\)-sense to a compact, convex set K can be modified into a sequence converging uniformly to this set provided that the derivatives of order l are uniformly bounded. We prove versions of our result on the whole space

In this paper, we study the behavior of a sequence of harmonic maps from surfaces with uniformly bounded energy on the generalized neck domain. The generalized neck domain is a union of ghost bubbles and annular neck domains, which connects non-trivial bubbles. An upper bound of the energy density is proved and we use it to study the limit of the nullity and index of the sequence.

The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq system helps gain insight into certain weather phenomena. The 2D Boussinesq system focused here is anisotropic and involves only horizontal dissipation and horizontal thermal diffusion. Due to the lack of the vertical dissipation

We study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is

In this paper, we study \(C^{1, 1}\) regularity for solutions to the degenerate \(L_p\) Dual Minkowski problem. Our proof is motivated by the idea of Guan and Li’s work on \(C^{1,1}\) estimates for solutions to the Aleksandrov problem.

This note studies the asymptotic behavior of global solutions to the fourth-order Schrödinger equation $$\begin{aligned} i\dot{u}+\Delta ^2 u+F(x,u)=0 . \end{aligned}$$ Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing mass super-critical and energy sub-critical regimes, with radial setting. This work uses a new approach due to Dodson and Murphy (Proc

In this paper we study the geometry and the topology of unbounded domains in the Hyperbolic Space \(\mathbb {H}^n\) supporting a bounded positive solution to an overdetermined elliptic problem. Under suitable conditions on the elliptic problem and the behaviour of the bounded solution at infinity, we are able to show that symmetries of the boundary at infinity imply symmetries on the domain itself

In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies \(I_\alpha \) defined on probability measures in \({\mathbb {R}}^n\), with \(n\ge 3\). The energy \(I_\alpha \) consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for \(\alpha =0\) and is anisotropic otherwise, and a quadratic confinement

We study some non-linear systems of PDEs that turn out to be related to the classical inverse problem in conductivity. They have a variational structure in the sense that, at least formally, they are the Euler–Lagrange systems of some explicit vector variational problems. The underlying, non-negative integrands are, however, non-quasiconvex in such a way that the existence of weak solutions for the

\(\alpha \)-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to \(\alpha \)-harmonic maps that were introduced by Sacks–Uhlenbeck to attack the existence problem for harmonic maps from closed surfaces. For \(\alpha >1\), the latter are known to satisfy a Palais–Smale condition, and so, the technique of Sacks–Uhlenbeck consists in constructing \(\alpha \)-harmonic maps for \(\alpha

In this paper, we will study harmonic functions on the complete and incomplete spaces with nonnegative Ricci curvature which exhibit inhomogeneous collapsing behaviors at infinity. The main result states that any nonconstant harmonic function on such spaces yields a definite exponential growth rate which depends explicitly on the geometric data at infinity.

In this paper, we establish uniform a priori estimates for positive solutions to higher critical order fractional Lame-Emden equations for all large exponents by adopting a new idea of using the combination of Green’s representations with suitable cut-off functions. We obtain two key ingredients needed in the estimates. One is the monotonicity of solutions in the inward normal direction near the boundary

This paper is concerned with the following Maxwell–Dirac system $$\begin{aligned}&-i\sum ^3_{k=1}\alpha _{k}\partial _{k}u + (V(x)+a)\beta u + \omega u-K(x)\phi u =F_u(x,u),\\&\quad -\Delta \phi =4\pi K(x)|u|^2, \end{aligned}$$ in \(\mathbb {R}^3\), where V(x) is a potential function and F(x, u) is a nonlinear function modeling various types of interaction and K(t, x) is the varying pointwise charge

We consider Hamilton–Jacobi equations in one space dimension with Hamiltonians of the form \(H(p,x,\omega ) = G(p) +\beta V(x,\omega )\), where \(V(\cdot ,\omega )\) is a stationary and ergodic potential of unit amplitude. The homogenization of such equations is established in a 2016 paper of Armstrong, Tran and Yu for all continuous and coercive G. Under the extra condition that G is a double-well

We prove optimal gradient estimates for distributional solutions to non-uniformly elliptic equations of multi-phase type in divergence form by investigating sharp conditions on such nonlinear operators for the Calderón-Zygmund theory.

Mean Field Games with state constraints are differential games with infinitely many agents, each agent facing a constraint on his state. The aim of this paper is to provide a meaning of the PDE system associated with these games, the so-called Mean Field Game system with state constraints. For this, we show a global semiconvavity property of the value function associated with optimal control problems

In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a cell motility model introduced in some previous work by the first author. We establish several facts about global minimizers with a volume constraint. In particular

We study rays and co-rays in the Wasserstein space \(P_p({\mathcal {X}})\) (\(p > 1\)) whose ambient space \({\mathcal {X}}\) is a complete, separable, non-compact, locally compact length space. We show that rays in the Wasserstein space can be represented as probability measures concentrated on the set of rays in the ambient space. We show the existence of co-rays for any prescribed initial probability

We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge–Ampère equation on a compact Hermitian manifold for a very general measure on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh–Nguyen–Sibony. As a consequence, we give a characterization of measures admitting Hölder continuous quasi-plurisubharmonic

In the present paper, we develop a direct approach to find nontrivial solutions and ground state solutions for the following planar Schrödinger equation: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u=f(x,u), \;\;&{} x\in {{\mathbb {R}}}^{2},\\ u\in H^1({\mathbb {R}}^2), \end{array}\right. } \end{aligned}$$ where V(x) is an 1-periodic function with respect to \(x_1\) and \(x_2\), 0 lies

In this article, we extend the mean curvature flow with surgery to mean convex hypersurfaces with entropy less than \(\varLambda _{n-2}\). In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity assumption \(H - \frac{\langle x, \nu \rangle }{2} > 0\) is possible and as an application we use the surgery flow to show that smooth n-dimensional closed self