Related papers: The Haj{\l}asz capacity density condition is self-…

We establish the local Hadamard well-posedness of a certain third-order nonlinear Schr\"odinger equation with a multi-term linear part and a general power nonlinearity known as the higher-order nonlinear Schr\"odinger equation, formulated…

We investigate a Hele-Shaw type free boundary problem in one spatial dimension, where heterogeneities appear both on the free boundary and within the interior of the positivity set. Our contributions are twofold. First, we establish…

We study the incompressible stationary Navier-Stokes equations in the upper-half plane with homogeneous Dirichlet boundary condition and non-zero external forcing terms. Existence of weak solutions is proved under a suitable condition on…

Potential functions in highly pertinent applications, such as deep learning in over-parameterized regime, are empirically observed to admit non-isolated minima. To understand the convergence behavior of stochastic dynamics in such…

We consider a composite optimization problem where the sum of a continuously differentiable and a merely lower semicontinuous function has to be minimized. The proximal gradient algorithm is the classical method for solving such a problem…

In the setting of spaces of homogeneous type, we give a direct proof of the local Tb theorem for singular integral operators. Motivated by questions of S. Hofmann, we extend it to the case when the integrability conditions are lower than 2,…

The study is motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory of interior capacities of condensers in a locally compact space is in general unsolvable, and this occurs even under very natural…

In this paper, we prove a discrete version of the generalized Riesz inequality on $\mathbb{Z}^d$. As a consequence, we will derive the extended Hardy-Littlewood and P\'olya-Szeg\"o inequalities. We will also establish cases of equality in…

We prove interior boundedness and H\"{o}lder continuity for the weak solutions of nonlocal double phase equations in the Heisenberg group $\mathbb{H}^n$. This solves a problem raised by Palatucci and Piccinini et. al. in 2022 and 2023 for…

In this paper we will focus on understanding the relation between Sobolev embedding theorems for Haj{\l}asz-Besov spaces defined on a doubling metric measure space $(\Omega,d,\mu)$ and the non-collapsing condition of the measure, i.e. \[…

Inspired by recent work of Mourgoglou and the second named author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincar\'e…

We consider the derivative nonlinear Schr\"odinger equation on the real line, with a background function $\psi(t,x)\in L^\infty(\mathbb{R}^2)$ that satisfies suitable conditions. Such a function may, for example, be a non-decaying solution…

Let $\psi$ be an $L^2$-normalized Hecke-Maass form with a large spectral parameter $\lambda>0$ on a compact arithmetic congruence hyperbolic 3-manifold $X=\Gamma\backslash\mathrm{SL}(2,\mathbb{C})/\mathrm{SU}(2)$, and let $Y$ be a totally…

We consider one-level additive Schwarz preconditioners for a family of Helmholtz problems with absorption and increasing wavenumber $k$. These problems are discretized using the Galerkin method with nodal conforming finite elements of any…

Inequalities for Riesz potentials are well-known to be equivalent to Sobolev inequalities of the same order for domain norms ``far" from $L^1$, but to be weaker otherwise. Recent contributions by Van Schaftingen, by Hernandez, Rai\c{t}\u{a}…

We are concerned with the well-posedness of the density-dependent incompressible viscoelastic fluid system. By Schauder-Tychonoff fixed point argument, when $\|{1}/{\rho_0}-1\|_{\dot{B}_{p,1}^{{N}/{p}}}$ is small, local well-posedness is…

We study the Cauchy problem of the Schr\"odinger-Korteweg-de Vries system. First, we establish the local well-posedness results, which improve the results of Corcho, Linares (2007). Moreover, we obtain some ill-posedness results, which show…

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case: \mu=n. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: \mu=n and p=q, by limiting the inequality…

We study the recently introduced Busemann subgradient method due to Goodwin, Lewis, Nicolae and L\'opez-Acedo, extending it to minimize the mean of a stochastic function over general Hadamard spaces. We prove a strong convergence theorem…

We study local regularity properties for solutions of linear, non-uniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability…