This paper concerns positive solutions to the boundary value problems of the scalar field equation in the half space with a Sobolev supercritical nonlinearity and an inhomogeneous Dirichlet...

We give an explicit representation of the fundamental solution to the heat equation on a half-space of ${\mathbb R}^N$ with the homogeneous dynamical boundary condition, and obtain upper and lower...

In this paper we introduce uniformly local weak Zygmund type spaces, and obtain an optimal sufficient condition for the existence of solutions to the critical fractional semilinear heat equation.

The nonlinear Hartree equation (NLH) in the Heisenberg picture admits steady states of the form $γ_f=f(-Δ)$ representing quantum states of infinitely many particles. In this article, we...

We prove the Riemannian version of a classical Euclidean result: every level set of the capacitary potential of a starshaped ring is starshaped. In the Riemannian setting, we restrict ourselves to...

The free elastic flow is the $L^2$-gradient flow for Euler's elastic energy, or equivalently the Willmore flow with translation invariant initial data. In contrast to elastic flows under length...

This paper presents a mathematical analysis of ODE-Net, a continuum model of deep neural networks (DNNs). In recent years, Machine Learning researchers have introduced ideas of replacing the deep...

We prove that Sobolev norms of solutions to time dependent Schrödinger equations for $d$-dimensional $N$-partcles interacting via time dependent two body potentials are bounded in time if...

In this paper, we consider the Strichartz estimates for orthonormal systems in the context of randomization. Frank, Lewin, Lieb, and Seiringer first proved the orthonormal Strichartz estimates....

This is an expository note to give a brief review of classical elastica theory, mainly prepared for giving a more detailed proof of the author's Li--Yau type inequality for self-intersecting...

This paper is concerned with nonlinear Schrödinger equations with a time-decaying harmonic potential. The nonlinearity is gauge-invariant of the long-range critical order. In [24] and [22], it...

In this paper we prove boundedness of propagators for Dirac equations with unbounded time-dependent potentials on Wiener amalgam spaces. In particular we deal with class of potentials such as...

The Cauchy problem is studied for very general systems of evolution equations, where the time derivative of solution is written by Fourier multipliers in space and analytic nonlinearity, with no...

This paper is concerned with a parabolic evolution equation of the form $A(u_t) + B(u) = f$, settled in a smooth bounded domain of ${\bf R}^d$, $d \geq 1$, and complemented with the initial...

In this paper, we consider the nonlinear Schrödinger equation with a general homogeneous nonlinearity in dimensions up to three. We assume that the degree (i.e., power) of the nonlinearity is...

In this thesis we investigate how the nonlocalities affect the study of different PDEs coming from physics, and we analyze these equations under almost optimal assumptions of the nonlinearity. In...

In this paper we prove the existence of at least one positive solution for the nonlocal semipositone problem \[ \displaystyle \left\{\begin{array}{rcll} (-Δ)_p^s(u) &=& λf(u)...

We present weighted estimates and higher order asymptotic expansions of global solutions to the complex Ginzburg--Landau (CGL) type equation in the super Fujita-critical case. Our approach is...

We study the equations in divergence form with piecewise $C^{\ga}$ coefficients. The domains contain corners and the discontinuity surfaces are attached to edges of the corners. We obtain...

We prove a smooth compactness theorem for the space of elasticae, unless the limit curve is a straight segment. As an application, we obtain smooth stability results for minimizers with respect to...

We develop a general theory for the existence, uniqueness, and higher regularity of solutions to wave-type equations on Lorentzian manifolds with timelike curves of cone-type singularities. These...

We adapt the classical theory of local well-posedness of evolution problems to cases in which the nonlinearity can be accurately quantified by two different norms. For ordinary differential...

We study Hadamard's variational formula for simple eigenvalues under dynamical and conformal deformations. Particularly, harmonic convexity of the first eigenvalue of the Laplacian under the mixed...

Hardy's uncertainty principle is a classical result in harmonic analysis, stating that a function in $L^2(\mathbb{R}^d)$ and its Fourier transform cannot both decay arbitrarily fast at infinity....

In this paper, we mainly consider nonnegative weak solution to the $D^{1,p}(\R^{N})$-critical quasi-linear static Schrödinger-Hartree equation with $p$-Laplacian $-Δ_{p}$ and nonlocal...

We study the final state problem for the nonlinear Schrödinger equation with a critical long-range nonlinearity and a long-range linear potential. Given a prescribed asymptotic profile which...

In this paper we give an estimate for the solution to the Schrödinger equation with sub-quadratic potentials in modulation spaces by the norm of the initial functions in Wiener-Amalgum spaces.

Total variation gradient flows are important in several applied fields, including image analysis and materials science. In this paper, we review a few basic topics including definition of a...

This paper is devoted to the asymptotic expansions of global solutions to the convection-diffusion equation in the Fujita-subcritical case. We improve the result by Zuazua (1993) and establish...

We study an inverse resonance problem on a radially and compactly perturbed infinite hyperbolic cylinder. Using the symmetries of this kind of geometry, we are led to study a stationary...

We establish Krylov-Safonov type Hölder regularity theory for solutions to quite general discrete dynamic programming equations or equivalently discrete stochastic processes on random geometric...

We prove an interpolation theorem for nonlinear functionals defined on scales of Banach spaces that generalize Besov spaces. It applies to functionals defined only locally, requiring only some...

We give a simpler proof for the local well-posedness of the modified Korteweg-de Vries equations and modified Benjamin-Ono equation in $H^{\frac{1}{4}}(\mathbb{R})$ and...

In recent work the authors proposed a broad global well-posedness conjecture for cubic defocusing dispersive equations in one space dimension, and then proved this conjecture in two cases, namely...

We prove sufficient conditions in order to obtain a sharp Gårding inequality for pseudo-differential operators acting on vector-valued functions on compact Lie groups. As a consequence, we...

This paper is concerned with the structure of solutions to the elliptic problem for an Henon type equation with a forcing term. Under suitable assumptions on the forcing term, we give a complete...

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: \[ λ_F(β,Ω)=λ_{F}(p,β,Ω)=...

We prove a Hopf bifurcation theorem in general Banach spaces, which improves a classical result by Crandall and Rabinowitz. Actually, our theorem does not need any compactness conditions, which...

In this paper we prove the orthonormal Strichartz estimates for the higher order and fractional Schrödinger, wave, Klein-Gordon and Dirac equations with potentials. As in the case of the...

A broad conjecture, formulated by the authors in earlier work, reads as follows: "Cubic defocusing dispersive one dimensional flows with small initial data have global dispersive solutions"....

We prove resolvent estimates in Schatten spaces for Laplace-Beltrami operators on compact manifolds at the critical exponent. Our proof only uses known bounds for the Hadamard parametrix.

Inspired by Frehse's [1] 1973 work, we show that his elliptic system $Δu = F(u, \nabla u)$ in the plane has bounded weak solutions $u$ with arbitrarily prescribed singular sets.

We examine the large-time behaviour of solutions to the compressible Navier-Stokes equations under the assumption of radial symmetry. In particular, we calculate a fast time-decay estimate of the...

We show that a mean curvature flow obtained as the limit of the Allen--Cahn equation is not only a Brakke flow but also a generalized BV flow proposed by Stuvard and Tonegawa.

We consider a wave equation with a time-dependent propagation speed, whose potential oscillations are controlled through bounds on its first and second derivatives and by limiting the integral of...

In this paper, we present counterexamples to maximal $L^p$-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It...

We give a short and self-contained proof of the interior $\mathcal C^{1,1}$ regularity of solutions $φ:Ω\to \mathbb{R}$ to the eikonal equation $|\nabla φ|=1$ in an open set...

We consider the Gelfand problem with general supercritical nonlinearities in the two-dimensional unit ball. In this paper, we prove the non-existence of an unstable solution for any positive small...

We show that, after a renormalisation, one can define the square of the modulus of the solution of the fractional Schrödinger equations on the circle with data in Sobolev spaces of arbitrary...

In this paper we establish almost-optimal stability estimates in quantum optimal transport pseudometrics for the semiclassical limit of the Hartree dynamics to the Vlasov-Poisson equation, in the...

We perform a phase space analysis of evolution equations associated with the Weyl quantization $q^{\mathrm{w}}$ of a complex quadratic form $q$ on $\mathbb{R}^{2d}$ with non-positive real part. In...

We study a class of planar Schrödinger-Poisson systems $$ -Δu+λV(x)u+ϕu=f(u) , \quad x\in{\mathbb R}^2,\qquad Δϕ=u^2, \quad x\in{\mathbb R}^2, $$ where...

We capture optimal decay for the Mullins-Sekerka evolution, a nonlocal, parabolic free boundary problem from materials science. Our main result establishes convergence of BV solutions to the...

In the regularity theory of solutions to elliptic partial differential equations often the concept of stability plays the role of a sufficient condition for smoothness. It is a natural question to...

This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger...