Narrow Results

We prove some uniqueness results for Dirichlet problems for second-order linear elliptic partial differential equations in non-divergence form with singular data in suitable weighted Sobolev spaces, on an open subset Ω of ℝⁿ, n ≥ 2, not necessarily bounded or regular.

In this paper, we present some basic uniqueness results for evolution equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field satisfies a monotonicity assumption: we prove the uniqueness of a solution for first-order systems modeling crowd motion with hard congestion effects, introduced recently by Maury et al. The monotonicity of the velocity field implies that the $2$-Wasserstein distance along two solutions is $\lambda$-contractive, which in particular implies uniqueness. In the case of diffusive models, we prove the uniqueness of a solution passing through the dual equation, where we use some well-known parabolic estimates to conclude an $L^1$-contraction property. In this case, by the regularization effect of the nondegenerate diffusion, the result follows even if the given velocity field is only $L^{\infty }$ as in the standard Fokker–Planck equation.

For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions if some blowup points coincide with the singularities of the Dirac data. If the strength of the Dirac mass at each blowup point is not a multiple of $4\pi$, we prove that bubbling solutions are unique. This paper extends previous results of Lin-Yan [C. S. Lin and S. S. Yan, On the mean field type bubbling solutions for Chern–Simons–Higgs equation, Adv. Math. 338 (2018) 1141–1188] and Bartolucci et al. [D. Bartolucci, A. Jevnikar, Y. Lee and W. Yang, Uniqueness of bubbling solutions of mean field equations, J. Math. Pures Appl. (9) 123 (2019) 78–126].

We study the uniqueness of solutions to diagonal hyperbolic systems in one spatial dimension and we present two uniqueness results. First, we establish a global existence and uniqueness theorem for continuous solutions to strictly hyperbolic systems. Second, we establish a global existence and uniqueness theorem for Lipschitz continuous solutions to hyperbolic systems that need not be strictly hyperbolic. Furthermore, an application is presented for one-dimensional flows in isentropic gas dynamics.