Abstract The paper is devoted to the classification of entire solutions to the Cahn–Hilliard equation \(-\Delta u=u-u^3-\delta \) in \(\mathbb {R}^N\), with particular interest in those solutions whose nodal set is either bounded or contained in a cylinder. The aim is to prove either radial or cylindrical symmetry, under suitable hypothesis.

Abstract We study the question whether Lipschitz minimizers of \(\int F(\nabla u)\,dx\) in \(\mathbb {R}^n\) are \(C^1\) when F is strictly convex. Building on work of De Silva–Savin, we confirm the \(C^1\) regularity when \(D^2F\) is positive and bounded away from finitely many points that lie in a 2-plane. We then construct a counterexample in \(\mathbb {R}^4\), where F is strictly convex but \(D^2F\)

Abstract We study measures \(\mu \) on the plane with two independent Alberti representations. It is known, due to Alberti, Csörnyei, and Preiss, that such measures are absolutely continuous with respect to Lebesgue measure. The purpose of this paper is to quantify the result of A–C–P. Assuming that the representations of \(\mu \) are bounded from above, in a natural way to be defined in the introduction

Abstract We consider a point cloud \(X_n := \{ {\mathbf {x}}_1, \ldots , {\mathbf {x}}_n \}\) uniformly distributed on the flat torus \({\mathbb {T}}^d : = \mathbb {R}^d / \mathbb {Z}^d \), and construct a geometric graph on the cloud by connecting points that are within distance \(\varepsilon \) of each other. We let \({\mathcal {P}}(X_n)\) be the space of probability measures on \(X_n\) and endow

Abstract We consider the Robin boundary value problem \({\mathrm {div}}\,(A\nabla u) = {\mathrm {div}}\,\varvec{f}+F\) in \(\Omega \), a \(C^1\) domain, with \((A\nabla u - \varvec{f})\cdot {\varvec{n}}+ \alpha u = g\) on \(\Gamma \), where the matrix A belongs to \(VMO ({\mathbb {R}}^3) \), and discover the uniform estimates on \(\Vert u\Vert _{W^{1,p}(\Omega )}\), with \(1< p < \infty \), independent

Abstract We consider the boundary value problem $$\begin{aligned} \begin{array}{rcl} -\Delta _p u &{} = &{} \alpha |u|^{p-2}u^+-\beta |u|^{p-2}u^- \text{ in } \Omega ,\\ u &{} = &{} 0 \text{ on } \partial \Omega , \end{array} \end{aligned}$$where \(\Delta _p u:=\nabla \cdot (|\nabla u|^{p-2}\nabla u)\) for \(1

Abstract Given \(\alpha >0\), we establish the following two supercritical Moser–Trudinger inequalities $$\begin{aligned} \mathop {\sup }\limits _{ u \in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( (\alpha _n + |x|^\alpha ) |u|^{\frac{n}{n-1}} \big ) dx < +\infty \end{aligned}$$and $$\begin{aligned} \mathop {\sup }\limits _{ u\in W^{1,n}_{0,\mathrm{rad}}(B): \int

Abstract In this paper, we justify the low Mach number limit of the steady irrotational Euler flows for the airfoil problem, which is the first result for the low Mach number limit of the steady Euler flows in an exterior domain. The uniform estimates on the compressibility parameter \(\varepsilon \), which is singular for the flows, are established via a variational approach based on the compressible–incompressible

Abstract We consider nonlocal variational problems in \(L^p\), like those that appear in peridynamics, where the functional object of the study is given by a double integral. It is known that convexity of the integrand implies the lower semicontinuity of the functional in the weak topology of \(L^p\). If the integrand is not convex, a usual approach is to compute the relaxation, which is the lower

Abstract Nonexistence of nontrivial nonnegative classical solutions is obtained for the problems: 0.1$$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2 u=u^p \;\;\; &{}\text{ in } {\mathbb {R}}^N \backslash {\overline{B}},\\ u=\Delta u=0 \;\;\; &{}\text{ on } \partial B \end{array} \right. \end{aligned}$$with \(1

Abstract We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak \((1,1)\)-Poincaré inequality. The two main results we obtain are a decomposition theorem into indecomposable sets and a characterisation of extreme points in the space of BV functions. In both cases, the proof we propose requires an additional

Abstract We consider the inverse multiphase Stefan problem with homogeneous Dirichlet boundary condition on a bounded Lipschitz domain, where the density of the heat source is unknown in addition to the temperature and the phase transition boundaries. The variational formulation is pursued in the optimal control framework, where the density of the heat source is a control parameter, and the criteria

Abstract Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa–Holm equations are well-studied examples. A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description. Geometrically it corresponds to a geodesic initial value problem on the infinite-dimensional group of

Abstract We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependence of fractional Laplacians on the underlying Riemannian metric. It extends several previous results and applies to a wide range of variational

Abstract We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals–Allen–Cahn–Hilliard type, satisfying a linear volume constraint, on a bounded Lipschitz domain \(\Omega \subset \mathbb {R}^N\). The number of solutions is estimated in terms of topological and homological invariants of the underlying domain \(\Omega \)

Abstract In this paper we study the quasilinear equation \(- \varepsilon ^2 \varDelta u-\varDelta _p u=f(u)\) in a smooth bounded domain \(\varOmega \subset {\mathbb {R}}^N\) with Dirichlet boundary condition, where \(p>2\) and f is a suitable subcritical and p-superlinear function at \(\infty \). First, for \(\epsilon \ne 0\) we prove that Morse index is two for every least energy nodal solution.

Abstract We are concerned with the following Schrödinger–Newton problem $$\begin{aligned} -\varepsilon ^2\Delta u+V(x)u=\frac{1}{8\pi \varepsilon ^2} \left( \int _{\mathbb {R}^3}\frac{u^2(\xi )}{|x-\xi |}d\xi \right) u,~x\in {\mathbb {R}}^3. \end{aligned}$$For \(\varepsilon \) small enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points of V(x). The

Abstract This paper concerns a class of optimal control problems, where a central planner aims to control a multi-agent system in \({\mathbb {R}}^d\) in order to minimize a certain cost of Bolza type. At every time and for each agent, the set of admissible velocities, describing his/her underlying microscopic dynamics, depends both on his/her position, and on the configuration of all the other agents

Abstract Motivated by Ball and Majumdar’s modification of Landau-de Gennes model for nematic liquid crystals, we study energy-minimizer Q of a tensor-valued variational obstacle problem in a bounded 3-D domain with prescribed boundary data. The energy functional is designed to blow up as Q approaches the obstacle. Under certain assumptions, especially on blow-up profile of the singular bulk potential

Abstract We study an old variational problem formulated by Euler as Proposition 53 of his Scientia Navalis by means of the direct method of the calculus of variations. Precisely, through relaxation arguments, we prove the existence of minimizers. We fully investigate the analytical structure of the minimizers in dependence of the geometric parameters and we identify the ranges of uniqueness and non-uniqueness

Abstract We study the free boundary problem describing the micro phase separation of diblock copolymer melts in the regime that one component has small volume fraction \(\rho \) such that the micro phase separation results in an ensemble of small disks of one component. We consider the two dimensional case in this paper, whereas the three dimensional case was already considered in Niethammer and Oshita

Abstract In this article, we improve the partial regularity theory for minimizing 1/2-harmonic maps of Millot and Sire (Arch Ration Mech Anal 215:125–210, 2015), Moser( J Geom Anal 21:588–598, 2011) in the case where the target manifold is the \((m-1)\)-dimensional sphere. For \(m\geqslant 3\), we show that minimizing 1/2-harmonic maps are smooth in dimension 2, and have a singular set of codimension

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller-Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the

We prove the global existence of Dirac-wave maps with curvature term with small initial data on globally hyperbolic manifolds of arbitrary dimension which satisfy a suitable growth condition. In addition, we also prove a global existence result for wave maps under similar assumptions.

A classification of [Formula: see text] contravariant Minkowski valuations on convex functions and a characterization of the projection body operator are established. The associated LYZ measure is characterized. In addition, a new [Formula: see text] covariant Minkowski valuation on convex functions is defined and characterized.

In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining

We continue our work (Y. Li, C. Zhao in J Differ Equ 212:208-233, 2005) to study the structure of positive solutions to the equation epsilon(m) Delta(m)u - u(m-1) + f(u) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of RN (N >/= 2). First, we study subcritical case for 2 < m < N and show that after passing by a sequence positive solutions go to a constant in C(1, alpha)

The convergence of spectra via two-scale convergence for double-porosity models is well known. A crucial assumption in these works is that the stiff component of the body forms a connected set. We show that under a relaxation of this assumption the (periodic) two-scale limit of the operator is insufficient to capture the full asymptotic spectral properties of high-contrast periodic media. Asymptotically

We analyse the fine convergence properties of one parameter families of hyperbolic metrics that move always in a horizontal direction, i.e. orthogonal to the action of diffeomorphisms. Such families arise naturally in the study of general curves of metrics on surfaces, and in one of the gradients flows for the harmonic map energy.

We consider the space of probability measures on a discrete set X , endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset Y ⊆ X , it is natural to ask whether they can be connected by a constant speed geodesic with support in Y at all times. Our main result answers this question affirmatively, under a suitable geometric condition on Y introduced in