Related papers: Regularity of Milne Problem with Geometric Correct…
This paper investigates the Newton's problem of minimal resistance for a body moving through a fluid whose density decreases exponentially with altitude. We prove the local existence and regularity of radial solutions $u(r)$ satisfying the…
For the thin obstacle problem in $\mathbb{R}^n$, $n\geq 2$, we prove that at all free boundary points, with the exception of a $(n-3)$-dimensional set, the solution differs from its blow-up by higher order corrections. This expansion…
We use convex relaxation techniques to provide a sequence of solutions to the matrix completion problem. Using the nuclear norm as a regularizer, we provide simple and very efficient algorithms for minimizing the reconstruction error…
In the paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in [5]. Our result shows that if a geometric hexagonal triangulation of the…
As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of numerical results is increasingly important. This presentation concerns the development of another benchmark for…
We examine the conditional regularity of the solutions of Navier-Stokes equations in the entire three-dimensional space under the assumption that the data are axially symmetric. We show that if positive part of the radial component of…
A mathematical model for the discrete nonlinear fragmentation (collision-induced breakage) equation with diffusion is studied. The existence of global weak solutions is established in arbitrary spatial dimensions without assuming a strictly…
We study stability and sample complexity properties of divergence regularized optimal transport (DOT). First, we obtain quantitative stability results for optimizers of DOT measured in Wasserstein distance, which are applicable to a wide…
In the classic work of Beale-Kato-Majda ({[}2{]}) for the Euler equations in $\mathbb{R^{\mathrm{3}}}$, regularity of a solution throughout a given interval $[0,T_{*}]$ is obtained provided that the curl $\omega$ satisfies $\omega\in…
We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales characterized by the magnitude of a…
We develop the regularity theory of viscosity solutions to transmission problems for fully nonlinear second order uniformly elliptic equations. Our results give a complete theory of existence, uniqueness, comparison principle, and…
Consider reconstructing a signal $x$ by minimizing a weighted sum of a convex differentiable negative log-likelihood (NLL) (data-fidelity) term and a convex regularization term that imposes a convex-set constraint on $x$ and enforces its…
We investigate the well-posedness and approximation of mild solutions to a class of linear transport equations on the unit interval $[0,1]$ endowed with a linear discontinuous production term, formulated in the space $\mathcal{M}([0,1])$ of…
We study the nonlinear steady Boltzmann equation in the half space, with phase transition and Dirichlet boundary condition. In particular, we study the regularity of the solution to the half-space problem in the situation that the gas is in…
Understanding the orientation of geological structures is crucial for analyzing the complexity of the Earths' subsurface. For instance, information about geological structure orientation can be incorporated into local anisotropic…
We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we establish…
A convex polyhedron, that is, a compact convex subset of $\mathbb{R}^3$ which is the intersection of finitely many closed half-spaces, can be rectified by taking the convex hull of the midpoints of the edges of the polyhedron. We derive…
It is found the exact solution of the Poisson equation for the multidimensional space with topology $M_{3+d}=\mathbb{R}^3\times T^d$. This solution describes smooth transition from the newtonian behavior $1/r_3$ for distances bigger than…
Recent result [Wu and Guo, Comm. Math. Phys., 2015] has shown that over the 2D unit disk, the classical half-space equation (CHS) for the neutron transport does not capture the correct boundary layer behaviour as long believed. In this…
Optimal transport has emerged as a fundamental methodology with applications spanning multiple research areas in recent years. However, the convergence rate of the empirical estimator to its population counterpart suffers from the curse of…