Consider the elastic scattering of a plane or point incident wave by an unbounded and rigid rough surface. The angular spectrum representation (ASR) for the time-harmonic Navier equation is derived in three dimensions. The ASR is utilized as a radiation condition to the elastic rough surface scattering problem. The uniqueness is proved through a Rellich-type identity for surfaces given by uniformly Lipschitz functions. In the case of flat surfaces with local perturbations, an equivalent variational formulation is deduced in a truncated bounded domain and the existence of solutions are shown for general incoming waves. The main ingredient of the proof is the radiating behavior of the Green tensor to the first boundary value problem of the Navier equation in a half-space.
more »
« less
This content will become publicly available on April 19, 2025
GREEN FUNCTIONS IN METRIC MEASURE SPACES
Abstract. We study existence and uniqueness of Green functions for the Cheeger Q- Laplacian in metric measure spaces that are Ahlfors Q-regular and support a Q-Poincar ́e inequality with Q > 1. We prove uniqueness of Green functions both in the case of relatively compact domains, and in the global (unbounded) case. We also prove existence of global Green functions in unbounded spaces, complementing the existing results in relatively compact domains proved recently in [BBL20].
more »
« less
- Award ID(s):
- 2141297
- PAR ID:
- 10519885
- Publisher / Repository:
- arxiv
- Date Published:
- Format(s):
- Medium: X
- Institution:
- Arxiv
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Following ideas of Caffarelli and Silvestre in [20], and using recent progress in hyperbolic fillings, we define fractional p-Laplacians (−∆p)θ with 0 < θ < 1 on any compact, doubling metric measure space (Z, d, ν), and prove existence, regularity and stability for the non- homogenous non-local equation (−∆p)θu = f. These results, in turn, rest on the new existence, global Hölder regularity and stability theorems that we prove for the Neumann problem for p-Laplacians ∆p, 1 < p < ∞, in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincaré inequality. Our work also includes as special cases much of the previous results by other authors in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure spaces context, our work does not rely on the assumption that (Z, d, ν) supports a Poincaré inequality.more » « less
-
more » « less
Abstract This paper examines the existence and uniqueness of weak solutions to the
d ‐dimensional magnetohydrodynamic (MHD) equations with fractional dissipation and fractional magnetic diffusion . The aim is at the uniqueness of weak solutions in the weakest possible inhomogeneous Besov spaces. We establish the local existence and uniqueness in the functional setting and when , and . The case when with and has previously been studied in [7, 19]. However, their approaches can not be directly extended to the fractional case when due to the breakdown of a bilinear estimate. By decomposing the bilinear term into different frequencies, we are able to obtain a suitable upper bound on the bilinear term for , which allows us to close the estimates in the aforementioned Besov spaces. -
more » « less
We establish existence of finite energy weak solutions to the kinetic Fokker-Planck equation and the linear Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the
\begin{document}$ S_p $\end{document} 7 ], we prove regularity in the kinetic Sobolev spaces\begin{document}$ S_p $\end{document} \begin{document}$ S_p $\end{document} -
We study the focusing NLS equation in $R\mathbb{R}^N$ in the mass-supercritical and energy-subcritical (or intercritical ) regime, with $H^1$ data at the mass-energy threshold $\mathcal{ME}(u_0)=\mathcal{ME}(Q)$, where Q is the ground state. Previously, Duyckaerts–Merle studied the behavior of threshold solutions in the $H^1$-critical case, in dimensions $N = 3, 4, 5$, later generalized by Li–Zhang for higher dimensions. In the intercritical case, Duyckaerts–Roudenko studied the threshold problem for the 3d cubic NLS equation. In this paper, we generalize the results of Duyckaerts–Roudenko for any dimension and any power of the nonlinearity for the entire intercritical range. We show the existence of special solutions, $Q^\pm$, besides the standing wave $e^{it}Q$, which exponentially approach the standing wave in the positive time direction, but differ in its behavior for negative time. We classify solutions at the threshold level, showing either blow-up occurs in finite (positive and negative) time, or scattering in both time directions, or the solution is equal to one of the three special solutions above, up to symmetries. Our proof extends to the $H^1$-critical case, thus, giving an alternative proof of the Li–Zhang result and unifying the critical and intercritical cases. These results are obtained by studying the linearized equation around the standing wave and some tailored approximate solutions to the NLS equation. We establish important decay properties of functions associated to the spectrum of the linearized Schrödinger operator, which, in combination with modulational stability and coercivity for the linearized operator on special subspaces, allows us to use a fixed-point argument to show the existence of special solutions. Finally, we prove the uniqueness by studying exponentially decaying solutions to a sequence of linearized equations.more » « less
