Related papers: On H\"ormander's solution of the dbar-equation

We prove some extension theorems for quaternionic holomorphic functions in the sense of Fueter. Starting from the existence theorem for the nonhomogeneous Cauchy-Riemann-Fueter Problem, we prove that an $\mathbb{H}$-valued function $f$ on a…

We study the long-time behavior of small solutions for a broad class of 2D Dirac-type equations with suitable nonlinearities. First, we prove that for nonlinearities with power $p\geq 5$ (massless case) and $p\geq7$ (massive case), any…

We prove that $k$-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) in the whole space, ${\mathbb R}^{n}_x$ with $n \ge 3$, converge in…

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded, Lipschitz domain. We consider bounded, weak solutions ($u\in W^{1, 2}\cap L^{\infty}(\Omega;\mathbb{R}^N)$) of the vector-valued, Euler-Lagrange system: \text{div } \big( A(x, u)Du\big)=g(x,…

We prove the H\"older continuity of the solution to complex Hessian equation with the right hand side in $L^p$, $p>\frac{n}{m}$, $1< m< n$, in a $m$-strongly pseudoconvex domain in $\mathbb{C}^n$ under some additional conditions on the…

We consider the decay of solution to fractional diffusion equation with the distributed order Caputo derivative. We assume that the elliptic operator is time-dependent and that the weight function contained in the definition of the…

We consider the Navier-Stokes Cauchy problem with an initial datum in a weighted Lebesgue space. The weight is a radial function increasing at infinity. Our study partially follows the ideas of the paper by G.P. Galdi and P. Maremonti "On…

In this note we analyze the large time behavior of solutions to an initial/boundary problem involving a damped nonlinear beam equation. We show that under physically realistic conditions on the nonlinear terms in the equation of motion the…

We prove the expansion formula for the classical Futaki invariants on the blowup of K\"ahler surfaces, which explains the balancing condition of Arezzo-Pacard. The relation with Stoppa's result is also discussed.

We consider the boundary value problem associated to the divergence operator with vanishing Dirichlet boundary conditions and we prove the existence of classical solutions under slight assumptions on the regularity of the datum.

In this note, we prove the boundary H\"{o}lder regularity for the infinity Laplace equation under a proper geometric condition. This geometric condition is quite general, and the exterior cone condition, the Reifenberg flat domains, and the…

We construct optimal Hardy weights to subcritical energy functionals $h$ associated with quasilinear Schr\"odinger operators on locally finite graphs. Here, optimality means that the weight $w$ is the largest possible with respect to a…

We prove necessary optimality conditions for problems of the calculus of variations on time scales with a Lagrangian depending on the free end-point.

In this paper we consider the Schr{\"o}dinger equation with nonlinear derivative term. Our goal is to initiate the study of this equation with non vanishing boundary conditions. We obtain the local well posedness for the Cauchy problem on…

Some classic second-order sufficient optimality conditions in the calculus of variations are shown to be equivalent, while also introducing a new equivalent second-order condition which is extremely easy to apply: simply integrate a linear…

In this paper, we study the asymptotic behavior of solutions to the wave equation with damping depending on the space variable and growing at the spatial infinity. We prove that the solution is approximated by that of the corresponding heat…

The goal of this work is to study a model of the wave equation with dynamic boundary conditions and a viscoelastic term. First, applying the Faedo-Galerkin method combined with the fixed point theorem, we show the existence and uniqueness…

We prove that the viscous Burgers equation has a globally defined smooth solution in all dimensions provided the initial condition and the forcing term are smooth and bounded together with their derivatives. Such solutions may have infinite…

We study the Cauchy problem for a multidimensional scalar conservation law on the Bohr compactification of $\R^n$. The existence and uniqueness of entropy solutions are established in the general case of merely continuous flux vector. We…

In the setting of the massive Klein-Gordon equation on de Sitter space, we discuss Vasy's asymptotic data at conformal infinity in terms of plane waves. In particular, we derive a short-hand formula for reconstructing solutions from their…