Related papers: A general asymptotic decay lemma for elliptic prob…
We consider an abstract first order evolution equation in a Hilbert space in which the linear part is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the…
We partially resolve a conjecture of Meeks on the asymptotic behavior of minimal surfaces in $\mathbb{R}^3$ with quadratic area growth.
We consider nonlinear parabolic equations of the type $$ u_t - div a(x, t, Du)= f(x,t) on \Omega_T = \Omega\times (-T,0), $$ under standard growth conditions on $a$, with $f$ only assumed to be integrable. We prove general decay estimates…
We study the growth of degrees in many autonomous and non-autonomous lattice equations defined by quad rules with corner boundary values, some of which are known to be integrable by other characterisations. Subject to an enabling…
We show that any unique global solution (here we do not require any smallness condition beforehand) to 3-D axisymmetric Navier-Stokes equations in some scaling invariant spaces must eventually become a small solution. In particular, we show…
We prove that the 2d Euler equation is globally well-posed in a space of vector fields having spatial asymptotic expansion at infinity of any a priori given order. The asymptotic coefficients of the solutions are holomorphic functions of…
In this paper we generalize the Deuring theorem on a reduction of elliptic curve with complex multiplication. More precisely, for an Abelian variety $A$, arising after reduction of an Abelian variety with complex multiplication by a CM…
By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of {\it asymptotic average solutions} making pointwise solvable every Poisson equation $\mathcal{L} u(x)=-f(x)$ with continuous data $f$,…
We are concerned with minimax signal detection. In this setting, we discuss non-asymptotic and asymptotic approaches through a unified treatment. In particular, we consider a Gaussian sequence model that contains classical models as special…
We study the asymptotic behavior of nonnegative solutions of the semilinear parabolic problem {u_t=\Delta u + u^{p}, x\in\mathbb{R}^{N}, t>0 u(0)=u_{0}, x\in\mathbb{R}^{N}, t=0. It is known that the nonnegative solution $u(t)$ of this…
In this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators under strong absorption conditions. We establish improved geometric regularity along the free boundary, for a sharp value…
In this paper we are concerned with global maximal regularity estimates for elliptic equations with degenerate weights. We consider both the linear case and the non-linear case. We show that higher integrability of the gradients can be…
We introduce a new equation describing epitaxial growth processes. This equation is derived from a simple variational geometric principle and it has a straightforward interpretation in terms of continuum and microscopic physics. It is also…
We study the large-time behavior of solutions to a generalized Burgers Equation, with initial zero mass data. Our main purpose is to present a modified version of the Renormalization Group map, which is able to provide the higher order…
We determine generalised asymptotic solutions for the inflaton field, the Hubble parameter, and the equation-of-state parameter valid during the oscillatory phase of reheating for potentials that close to their global minima behave as even…
We introduce in this paper a technique for the reduced order approximation of parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension…
Reformulated uniform asymptotic expansions are derived for ordinary differential equations having a large parameter and a simple turning point. These involve Airy functions, but not their derivatives, unlike traditional asymptotic…
Asymptotic solutions are derived for inhomogeneous differential equations having a large real or complex parameter and a simple turning point. They involve Scorer functions and three slowly varying analytic coefficient functions. The…
In the present paper, we propose the investigation of variable-exponent, degenerate/singular elliptic equations in non-divergence form. This current endeavor parallels the by now well established theory of functionals satisfying nonstandard…
Current theoretical results on optimization trajectories of neural networks trained by gradient descent typically have the form of rigorous but potentially loose bounds on the loss values. In the present work we take a different approach…