Analysis of PDEs (math.AP)

Abstract: Metaplectic Wigner distributions generalize the most popular time-frequency representations, such as the short-time Fourier transform (STFT) and $\tau$-Wigner distributions, using metaplectic operators. However, in order for a metaplectic Wigner distribution to measure local time-frequency concentration of signals, the additional property of shift-invertibility is fundamental. In addition, metaplectic atoms provide different ways to model signals. Namely, signals can be written as discrete superpositions of these operators, providing original ways to represent signals, with applications to machine learning, signal analysis, theory of pseudodifferential operators, to mention a few. Among all shift-invertible distributions, Wigner-decomposable metaplectic Wigner distributions provide the most straightforward generalization of the STFT. In this work, we focus on metaplectic atoms of Wigner-decomposable shift-invertible metaplectic distributions and characterize the associated metaplectic Gabor frames.

Abstract: We consider the Kato square root problem for non-divergence second order elliptic operators $L =- a_{ij} D_iD_j$, and, especially, the normalized adjoints of such operators. In particular, our results are applicable to the case of real coefficients having sufficiently small BMO norm. We assume that the coefficients of the operator are smooth, but our estimates do not depend on the assumption of smoothness.

Abstract: We adapt Glimm's approximation method to the framework of convex integration to show density of wild data for the (complete) Euler system of gas dynamics. The desired infinite family of entropy admissible solutions emanating from the same initial data is obtained via convex integration of suitable Riemann problems pasted with local smooth solutions. In addition, the wild data belong to BV class.

Abstract: We consider a second-order nonlinear wave equation with a linear convolution term. When the convolution operator is taken as the identity operator, our equation reduces to the classical elasticity equation which can be written as a $p$-system of first-order differential equations. We first establish the local well-posedness of the Cauchy problem. We then investigate the behavior of solutions to the Cauchy problem in the limit as the kernel function of the convolution integral approaches to the Dirac delta function, that is, in the vanishing dispersion limit. We consider two different types of the vanishing dispersion limit behaviors for the convolution operator depending on the form of the kernel function. In both cases, we show that the solutions converge strongly to the corresponding solutions of the classical elasticity equation.

Abstract: The generator of the semigroup associated with linear age-structured population models including spatial diffusion is shown to have compact resolvent.

Abstract: We construct a blowing-up solution for the energy critical focusing biharmonic nonlinear Schr\"odinger equation in infinite time in dimension $N\geq 13$. Our solution is radially symmetric and converges asymptotically to the sum of two bubbles. The scale of one of the bubble is of order $1$ whereas the other one is of order $|t|^{-\frac{2}{N-12}}$. Moreover, the phase between the two bubbles form a right angle.

Abstract: The generalised Hopf equation is the first order nonlinear equation with data $\Phi$ a holomorphic functions and $\eta\geq 1$ a positive weight, \[ h_w\,\overline{h_\wbar}\,\eta(w) = \Phi.\] The Hopf equation is the special case $\eta(w)=\tilde{\eta}(h(w))$ and reflects that $h$ is harmonic with respect to the conformal metric $\sqrt{\tilde{\eta}(z)}|dz|$. This article obtains conditions on the data to ensure that a solution is open and discrete. We also prove a strong uniqueness result.

Abstract: We propose a variational approach to approximate measures with measures uniformly distributed over a 1 dimentional set. The problem consists in minimizing a Wasserstein distance as a data term with a regularization given by the length of the support. As it is challenging to prove existence of solutions to this problem, we propose a relaxed formulation, which always admits a solution. In the sequel we show that if the ambient space is $\mathbb{R}^2$ , under techinical assumptions, any solution to the relaxed problem is a solution to the original one. Finally we manage to prove that any optimal solution to the relaxed problem, and hence also to the original, is Ahlfors regular.

Abstract: We study local and global higher integrability properties for quasiminimizers of a class of double-phase integrals characterized by nonstandard growth conditions. We work purely on a variational level in the setting of a metric measure space with a doubling measure and a Poincar\'e inequality. The main novelty is an intrinsic approach to double-phase Sobolev-Poincar\'e inequalities.

Abstract: In this paper we continue the analysis of the dispersive properties of the 2D and 3D massless Dirac-Coulomb equations that has been started in arXiv:1503.00945 and arXiv:2101.07185. We prove a priori estimates of the solution of the mentioned systems, in particular Strichartz estimates with an additional angular regularity, exploiting the tools developed in the previous works. As an application, we show local well-posedness results for a Dirac-Coulomb equation perturbed with Hartree-type nonlinearities.

Abstract: In this paper we address some questions about symmetry, radial monotonicity, and uniqueness for a semilinear fourth-order boundary value problem in the ball of $\mathbb R^2$ deriving from the Kirchhoff-Love model of deformations of thin plates. We first show the radial monotonicity for a wide class of biharmonic problems. The proof of uniqueness is based on ODE techniques and applies to the whole range of the boundary parameter. For an unbounded subset of this range we also prove symmetry of the ground states by means of a rearrangement argument which makes use of Talenti's comparison principle. This paper complements the analysis in [G. Romani, Anal. PDE 10 (2017), no. 4, 943-982], where existence and positivity issues have been investigated.

Abstract: We show that the solution of the Cauchy problem for the classical ode $m \mathbf y''=\mathbf f$ can be obtained as limit of minimizers of exponentially weighted convex variational integrals. This complements the known results about weighted inertia-energy approach to Lagrangian mechanics and hyperbolic equations.