Discrete & Continuous Dynamical Systems
December 2019 , Volume 39 , Issue 12
Dedicated to Luis A. Caffarelli on the occasion of his 70th birthday
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We define a family of functionals, called
We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle
Extensions to Dirichlet boundary conditions, which may be more consistent with the situation in the physical experiments and numerical simulations (see [
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
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
We study the equation
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
for a regular function
Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of
In analogy to a famous theorem of Savin for the Allen–Cahn equation, we study here the 1D symmetry of solutions
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 [
The purpose of this paper is twofold. Firstly I present an optimal regularity result for minimizers of a
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 [
We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in [
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
In that respect, the example is more dramatic than a better known example due to Hedlund of a metric in
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
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
For a stationary system representing prey and
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
Since the mid nineties, the existence of an
We study the large time behaviour of the Fisher-KPP equation
We show that, on a
As part of our strategy, we prove a new stability result for the optimal transport map on a compact manifold.
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