Calculus of Variations and Partial Differential Equations最新文献

The notion of quasi-metric space arises by revoking the symmetry from the definition of a distance. Semi-Lipschitz functions appear naturally as morphisms associated with the new structure. In this work, under suitable assumptions on the quasi-metric space (analogous to standard ones in the metric case), we establish existence of optimal (that is, absolutely minimal) extensions of real-valued semi-Lipschitz functions from a subset of the space to the whole space. This is done in two different ways: first, by adapting the Perron method from the classical setting to this asymmetric case, and second, by means of an iteration scheme for (an unbalanced version of) the tug-of-war game, initiating the algorithm from a McShane extension. This new iteration scheme provides, even in the symmetric case of a metric space, a constructive way of establishing existence of absolutely minimal Lipschitz extensions of real-valued Lipschitz functions.

We show global in time existence and uniqueness on any finite time interval of strong solutions to a Navier-Stokes/Cahn-Hilliard type system on a given two-dimensional evolving surface in the case of different densities and a singular (logarithmic) potential. The system describes a diffuse interface model for a two-phase flow of viscous incompressible fluids on an evolving surface. We also establish the validity of the instantaneous strict separation property from the pure phases. To show these results we use our previous achievements on local well-posedness together with suitable novel regularity results for the convective Cahn-Hilliard equation. The latter allows to obtain higher-order energy estimates to extend the local solution globally in time. To this aim the time evolution of energy type quantities has to be calculated and estimated carefully.

We propose a possible nonlocal approximation of the Willmore functional, in the sense of Gamma-convergence, based on the first variation of the fractional Allen-Cahn energies, and we prove the corresponding $\Gamma$ -limsup estimate. Our analysis is based on the expansion of the fractional Laplacian in Fermi coordinates and fine estimates on the decay of higher order derivatives of the one-dimensional nonlocal optimal profile. This result is the nonlocal counterpart of that obtained by Bellettini and Paolini, where they proposed a phase-field approximation of the Willmore functional based on the first variation of the (local) Allen-Cahn energies.

In this article we analyze the spectral properties of the curl operator on closed Riemannian 3-manifolds. Specifically, we study metrics that are optimal in the sense that they minimize the first curl eigenvalue among any other metric of the same volume in the same conformal class. We establish a connection between optimal metrics and the existence of minimizers for the $L^{\frac{3}{2}}$ -norm in a fixed helicity class, which is exploited to obtain necessary and sufficient conditions for a metric to be locally optimal. As a consequence, our main result is that we prove that $S^3$ and $R{P}^3$ endowed with the round metric are $C^1$ -local minimizers for the first curl eigenvalue (in its conformal and volume class). The connection between the curl operator and the Hodge Laplacian allows us to infer that the canonical metrics of $S^3$ and $R{P}^3$ are locally optimal for the first eigenvalue of the Hodge Laplacian on coexact 1-forms. This is in strong contrast to what happens in four dimensions.

In this paper, we revisit asymptotic stability for the two-dimensional incompressible porous media equation and the Stokes transport system in a periodic channel. It is well-known that a stratified density, which strictly decreases in the vertical direction, is asymptotically stable under sufficiently small and smooth perturbations. We provide improvements in the regularity assumptions on the perturbation and in the convergence rate. Unlike the standard approach for stability analysis relying on linearized equations, we directly address the nonlinear problem by exploiting the energy structure of each system. While it is widely known that the potential energy is a Lyapunov functional in both systems, our key observation is that the second derivative of the potential energy reveals a (degenerate) coercive structure, which arises from the fact that the solution converges to the minimizer of the energy.

The main purpose of this article is to establish the Runge-type approximation in $L^2(0,T;{\tilde{H}}^s\left(\Omega \right))$ for solutions of linear nonlocal wave equations. To achieve this, we extend the theory of very weak solutions for classical wave equations to our nonlocal framework. This strengthened Runge approximation property allows us to extend the existing uniqueness results for Calderón problems of linear and nonlinear nonlocal wave equations in our earlier works. Furthermore, we prove unique determination results for the Calderón problem of nonlocal wave equations with polyhomogeneous nonlinearities.

In this paper, we study the rigidity problem for compact minimal Legendrian submanifolds in the unit Euclidean spheres via eigenvalues of fundamental matrices, which measure the squared norms of the second fundamental form on all normal directions. By using Lu’s inequality (Lu in J Funct Anal 261:1284–1308, 2011) on the upper bound of the squared norm of Lie brackets of symmetric matrices, we establish an optimal pinching theorem for such submanifolds of all dimensions, giving a new characterization for the Calabi tori. This pinching condition can also be described by the eigenvalues of the Ricci curvature tensor. Moreover, when the third large eigenvalue of the fundamental matrix vanishes everywhere, we get an optimal rigidity theorem under a weaker pinching condition.

We prove a new existence theorem for proper solutions of Huisken and Ilmanen’s weak inverse mean curvature flow, assuming certain non-degeneracy conditions on the isoperimetric profile. In particular, no curvature assumption is imposed in our existence theorem.

The (L^{p}) dual curvature measure was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The associated Minkowski problem, known as the (L^{p}) dual Minkowski problem, asks about existence of a convex body with prescribed (L^{p}) dual curvature measure. This question unifies the previously disjoint (L^{p}) Minkowski problem with the dual Minkowski problem, two open questions in convex geometry. In this paper, we prove the existence of a solution to the (L^{p}) dual Minkowski problem for the case of (q<p+1), (-1<p<0), and (pne q) for even measures.

We study a singular limit of the classical parabolic-elliptic Patlak-Keller-Segel (PKS) model for chemotaxis with non linear diffusion. The main result is the (Gamma ) convergence of the corresponding energy functional toward the perimeter functional. Following recent work on this topic, we then prove that under an energy convergence assumption, the solution of the PKS model converges to a solution of the Hele-Shaw free boundary problem with surface tension, which describes the evolution of the interface separating regions with high density from those with low density. This result complements a recent work by the author with I. Kim and Y. Wu, in which the same free boundary problem is derived from the congested PKS model (which includes a density constraint (rho le 1) and a pressure term): It shows that the congestion constraint is not necessary to observe phase separation and surface tension phenomena.