American Institute of Mathematical Sciences

We define a family of functionals, called \begin{document}$ p $\end{document}-oscillation functionals, that can be interpreted as discrete versions of the classical total variation functional for \begin{document}$ p = 1 $\end{document} and of the \begin{document}$ p $\end{document}-Dirichlet functionals for \begin{document}$ p>1 $\end{document}. We introduce the notion of minimizers and prove existence of solutions to the Dirichlet problem. Finally we provide a description of Class A minimizers (i.e. minimizers under compact perturbations) in dimension \begin{document}$ 1 $\end{document}.

We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle

\begin{document}$ \epsilon^2 \Delta u+ u(1-|u|^2) = 0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \ \frac{\partial u}{\partial \nu} = 0 \ \mbox{on}\ \partial \Omega $\end{document}

where \begin{document}$ \Omega $\end{document} is a smooth bounded domain in \begin{document}$ {\mathbb R}^d $\end{document} (\begin{document}$ d\geq 2 $\end{document}), which is referred as the obstacle and \begin{document}$ \epsilon>0 $\end{document} is sufficiently small. We first construct a vortex free solution of the form \begin{document}$ u = \rho_\epsilon (x) e^{i \frac{\Phi_\epsilon}{\epsilon}} $\end{document} with \begin{document}$ \rho_\epsilon (x) \to 1-|\nabla \Phi^\delta(x)|^2, \Phi_\epsilon (x) \to \Phi^\delta (x) $\end{document} where \begin{document}$ \Phi^\delta (x) $\end{document} is the unique solution for the subsonic irrotational flow equation

\begin{document}$ \nabla ( (1-|\nabla \Phi|^2)\nabla \Phi ) = 0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \frac{\partial \Phi}{\partial \nu} = 0 \ \mbox{on} \ \partial \Omega, \ \nabla \Phi (x) \to \delta \vec{e}_d \ \mbox{as} \ |x| \to +\infty $\end{document}

and \begin{document}$ |\delta | <\delta_{*} $\end{document} (the sound speed).

In dimension \begin{document}$ d = 2 $\end{document}, on the background of this vortex free solution we also construct solutions with single vortex close to the maximum or minimum points of the function \begin{document}$ |\nabla \Phi^\delta (x)|^2 $\end{document} (which are on the boundary of the obstacle). The latter verifies the vortex nucleation phenomena (for the steady states) in superfluids described by the Gross-Pitaevskii equations. Moreover, after some proper scalings, the limits of these vortex solutions are traveling wave solution of the Gross-Pitaevskii equation. These results also show rigorously the conclusions drawn from the numerical computations in [26,27].

Extensions to Dirichlet boundary conditions, which may be more consistent with the situation in the physical experiments and numerical simulations (see [1] and references therein) for the trapped Bose-Einstein condensates, are also discussed.

Many mathematical models of biological processes can be represented as free boundary problems for systems of PDEs. In the radially symmetric case, the free boundary is a function of \begin{document}$ r = R(t) $\end{document}, and one can generally prove the existence of global in-time solutions. However, the asymptotic behavior of the solution and, in particular, of \begin{document}$ R(t) $\end{document}, has not been explored except in very special cases. In the present paper we consider two such models which arise in cancer treatment by combination therapy with two drugs. We study the asymptotic behavior of the solution and its dependence on the dose levels of the two drugs.

In this paper, we give a mathematically rigorous proof of the averaging behavior of oscillatory surface integrals. Based on ergodic theory, we find a general geometric condition which we call irrational direction dense condition, abbreviated as IDDC, under which the averaging takes place. It should be stressed that IDDC does not imply any control on the curvature of the given surface. As an application, we prove homogenization for elliptic systems with Dirichlet boundary data, in \begin{document}$ C^1 $\end{document}-domains.

We study the equation \begin{document}$u_{11}u_{22} = 1$\end{document} in \begin{document}$\mathbb{R}^2$\end{document}. Our results include an interior \begin{document}$C^2$\end{document} estimate, classical solvability of the Dirichlet problem, and the existence of non-quadratic entire solutions. We also construct global singular solutions to the analogous equation in higher dimensions. At the end we state some open questions.

Motivated by the study of the equilibrium equations for a soap film hanging from a wire frame, we prove a compactness theorem for surfaces with asymptotically vanishing mean curvature and fixed or converging boundaries. In particular, we obtain sufficient geometric conditions for the minimal surfaces spanned by a given boundary to represent all the possible limits of sequences of almost-minimal surfaces. Finally, we provide some sharp quantitative estimates on the distance of an almost-minimal surface from its limit minimal surface.

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere \begin{document}$ S^2 $\end{document},

\begin{document}$ \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u & = u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) & = u_0 \quad \text{in } \Omega , \end{align*} $\end{document}

with \begin{document}$ u(x,t): \bar \Omega\times [0,T) \to S^2 $\end{document}. Here \begin{document}$ \Omega $\end{document} is a bounded, smooth axially symmetric domain in \begin{document}$ \mathbb{R}^3 $\end{document}. We prove that for any circle \begin{document}$ \Gamma \subset \Omega $\end{document} with the same axial symmetry, and any sufficiently small \begin{document}$ T>0 $\end{document} there exist initial and boundary conditions such that \begin{document}$ u(x,t) $\end{document} blows-up exactly at time \begin{document}$ T $\end{document} and precisely on the curve \begin{document}$ \Gamma $\end{document}, in fact

\begin{document}$ | {\nabla} u(\cdot ,t)|^2 \rightharpoonup | {\nabla} u_*|^2 + 8\pi \delta_\Gamma \quad\mbox{as}\quad t\to T . $\end{document}

for a regular function \begin{document}$ u_*(x) $\end{document}, where \begin{document}$ \delta_\Gamma $\end{document} denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng [5,6].

Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of \begin{document}$ \Delta u = f(u) $\end{document} in \begin{document}$ \mathbb{R}^n $\end{document}, where \begin{document}$ f $\end{document} is smooth, non-negative, with support in the interval \begin{document}$ [0,1] $\end{document}. In such setting, any "blow-down" of the solution \begin{document}$ u $\end{document} will converge to a global solution to the classical one-phase free boundary problem of Alt–Caffarelli.

In analogy to a famous theorem of Savin for the Allen–Cahn equation, we study here the 1D symmetry of solutions \begin{document}$ u $\end{document} that are energy minimizers. Our main result establishes that, in dimensions \begin{document}$ n<6 $\end{document}, if \begin{document}$ u $\end{document} is axially symmetric and stable then it is 1D.

We provide an overview of some recent results about the regularity of the solution and the free boundary for so-called two-phase free boundary problems driven by uniformly elliptic equations.

This is a survey of some recent results on the asymptotic behavior of solutions to critical nonlinear wave equations.

Combining situations originally considered in [7] - [8], a semilinear elliptic system is treated and a nondegeneracy condition leading to the existence of multibump solutions is considerably weakened.

The purpose of this paper is twofold. Firstly I present an optimal regularity result for minimizers of a \begin{document}$ 1D $\end{document} convex functional involving the BV-norm, under Neumann boundary condition. This functional is a simplified version of models occuring in Image Processing. Secondly I investigate the existence of minimizers for the same functional under Dirichlet boundary condition. Surprisingly, this turns out to be a delicate issue, which is still widely open.

In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in [12], we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of [10].

We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in [30,Section 5.11] to illustrate the need of certain definitions in the calculus of variations.

The Almgren-Federer example, besides its intended goal of illustrating subtle aspects of geometric measure theory, is also a problem in the theory of geodesics. Hence, we wrote an exposition of the beautiful ideas of Almgren and Federer from the point of view of geodesics.

In the language of geodesics, the Almgren-Federer example constructs metrics in \begin{document}$ \mathbb{S}^1\times \mathbb{S}^2 $\end{document}, with the property that none of the Tonelli geodesics (geodesics which minimize the length in a homotopy class) are Class-A minimizers in the sense of Morse (any finite length segment in the universal cover minimizes the length between the end points; this is also sometimes given other names). In other words, even if a curve is a minimizer of length among all the curves homotopic to it, by repeating it enough times, we get a closed curve which does not minimize in its homotopy class.

In that respect, the example is more dramatic than a better known example due to Hedlund of a metric in \begin{document}$ \mathbb{T}^3 $\end{document} for which only 3 Tonelli minimizers (and their multiples) are Class-A minimizers.

For dynamics, the example also illustrates different definitions of "integrable" and clarifies the relation between minimization and hyperbolicity and its interaction with topology.

This paper is devoted to several small data existence results for semi-linear wave equations on negatively curved Riemannian manifolds. We provide a simple and geometric proof of small data global existence for any power \begin{document}$ p\in (1, 1+\frac{4}{n-1}] $\end{document} for the shifted wave equation on hyperbolic space \begin{document}$ {\mathbb{H}}^n $\end{document} involving nonlinearities of the form \begin{document}$ \pm |u|^p $\end{document} or \begin{document}$ \pm|u|^{p-1}u $\end{document}. It is based on the weighted Strichartz estimates of Georgiev-Lindblad-Sogge [9] (or Tataru [29]) on Euclidean space. We also prove a small data existence theorem for variably curved backgrounds which extends earlier ones for the constant curvature case of Anker and Pierfelice [1] and Metcalfe and Taylor [22]. We also discuss the role of curvature and state a couple of open problems. Finally, in an appendix, we give an alternate proof of dispersive estimates of Tataru [29] for \begin{document}$ {\mathbb H}^3 $\end{document} and settle a dispute, in his favor, raised in [21] about his proof. Our proof is slightly more self-contained than the one in [29] since it does not make use of heavy spherical analysis on hyperbolic space such as the Harish-Chandra \begin{document}$ c $\end{document}-function; instead it relies only on simple facts about Bessel potentials.

The regularity of monotone transport maps plays an important role in several applications to PDE and geometry. Unfortunately, the classical statements on this subject are restricted to the case when the measures are compactly supported. In this note we show that, in several situations of interest, one can to ensure the regularity of monotone maps even if the measures may have unbounded supports.

We consider the steady fractional Schrödinger equation \begin{document}$ L u + V u = f $\end{document} posed on a bounded domain \begin{document}$ \Omega $\end{document}; \begin{document}$ L $\end{document} is an integro-differential operator, like the usual versions of the fractional Laplacian \begin{document}$ (-\Delta)^s $\end{document}; \begin{document}$ V\ge 0 $\end{document} is a potential with possible singularities, and the right-hand side are integrable functions or Radon measures. We reformulate the problem via the Green function of \begin{document}$ (-\Delta)^s $\end{document} and prove well-posedness for functions as data. If \begin{document}$ V $\end{document} is bounded or mildly singular a unique solution of \begin{document}$ (-\Delta)^s u + V u = \mu $\end{document} exists for every Borel measure \begin{document}$ \mu $\end{document}. On the other hand, when \begin{document}$ V $\end{document} is allowed to be more singular, but only on a finite set of points, a solution of \begin{document}$ (-\Delta)^s u + V u = \delta_x $\end{document}, where \begin{document}$ \delta_x $\end{document} is the Dirac measure at \begin{document}$ x $\end{document}, exists if and only if \begin{document}$ h(y) = V(y) |x - y|^{-(n+2s)} $\end{document} is integrable on some small ball around \begin{document}$ x $\end{document}. We prove that the set \begin{document}$ Z = \{x \in \Omega : \rm{no solution of } (-\Delta)^s u + Vu = \delta_x \rm{ exists}\} $\end{document} is relevant in the following sense: a solution of \begin{document}$ (-\Delta)^s u + V u = \mu $\end{document} exists if and only if \begin{document}$ |\mu| (Z) = 0 $\end{document}. Furthermore, \begin{document}$ Z $\end{document} is the set points where the strong maximum principle fails, in the sense that for any bounded \begin{document}$ f $\end{document} the solution of \begin{document}$ (-\Delta)^s u + Vu = f $\end{document} vanishes on \begin{document}$ Z $\end{document}.

For a stationary system representing prey and \begin{document}$ N $\end{document} groups of competing predators, we show classification results about the set of positive solutions. In particular, we show that if the number of components \begin{document}$ N $\end{document} is too large or if the competition between different groups is too small, then the system has only constant solutions, which we then completely characterize.

All \begin{document}$ (-1) $\end{document}-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus north and south poles have been classified in our earlier work as a four dimensional surface with boundary. In this paper, we establish near the no-swirl solution surface existence, non-existence and uniqueness results on \begin{document}$ (-1) $\end{document}-homogeneous axisymmetric solutions with nonzero swirl which are smooth on the unit sphere minus north and south poles.

We discuss the extent to which solutions to one-phase free boundary problems can be characterized according to their topological complexity. Our questions are motivated by fundamental work of Luis Caffarelli on free boundaries and by striking results of T. Colding and W. Minicozzi concerning finitely connected, embedded, minimal surfaces. We review our earlier work on the simplest case, one-phase free boundaries in the plane in which the positive phase is simply connected. We also prove a new, purely topological, effective removable singularities theorem for free boundaries. At the same time, we formulate some open problems concerning the multiply connected case and make connections with the theory of minimal surfaces and semilinear variational problems.

We consider the class of stable solutions to semilinear equations \begin{document}$ -\Delta u = f(u) $\end{document} in a bounded smooth domain of \begin{document}$ \mathbb{R}^n $\end{document}. Since 2010 an interior a priori \begin{document}$ L^\infty $\end{document} bound for stable solutions is known to hold in dimensions \begin{document}$ n\le 4 $\end{document} for all \begin{document}$ C^1 $\end{document} nonlinearities \begin{document}$ f $\end{document}. In the radial case, the same is true for \begin{document}$ n\leq 9 $\end{document}. Here we provide with a new, simpler, and unified proof of these results. It establishes, in addition, some new estimates in higher dimensions —for instance \begin{document}$ L^p $\end{document} bounds for every finite \begin{document}$ p $\end{document} in dimension 5.

Since the mid nineties, the existence of an \begin{document}$ L^\infty $\end{document} bound holding for all \begin{document}$ C^1 $\end{document} nonlinearities when \begin{document}$ 5\leq n\leq 9 $\end{document} was a challenging open problem. This has been recently solved by A. Figalli, X. Ros-Oton, J. Serra, and the author, for nonnegative nonlinearities, in a forthcoming paper.

We study the large time behaviour of the Fisher-KPP equation \begin{document}$ \partial_t u = \Delta u +u-u^2 $\end{document} in spatial dimension \begin{document}$ N $\end{document}, when the initial datum is compactly supported. We prove the existence of a Lipschitz function \begin{document}$ s^\infty $\end{document} of the unit sphere, such that \begin{document}$ u(t, x) $\end{document} approaches, as \begin{document}$ t $\end{document} goes to infinity, the function

\begin{document}$ U_{c_*}\bigg(|x|-c_*t + \frac{N+2}{c_*} \mathrm{ln}t + s^\infty\Big(\frac{x}{|x|}\Big)\bigg), $\end{document}

where \begin{document}$ U_{c*} $\end{document} is the 1D travelling front with minimal speed \begin{document}$ c_* = 2 $\end{document}. This extends an earlier result of Gärtner.

We show that, on a \begin{document}$ 2 $\end{document}-dimensional compact manifold, the optimal transport map in the semi-discrete random matching problem is well-approximated in the \begin{document}$ L^2 $\end{document}-norm by identity plus the gradient of the solution to the Poisson problem \begin{document}$ - {\Delta} f^{n, t} = \mu^{n, t}-1 $\end{document}, where \begin{document}$ \mu^{n, t} $\end{document} is an appropriate regularization of the empirical measure associated to the random points. This shows that the ansatz of [8] is strong enough to capture the behavior of the optimal map in addition to the value of the optimal matching cost.

As part of our strategy, we prove a new stability result for the optimal transport map on a compact manifold.