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Vesa Julin

Publications and source records attributed to Vesa Julin.

At least 37 records · Page 2Linked to original sources
arXiv2021

Approximation of BV functions by neural networks: A regularity theory approach

In this paper we are concerned with the approximation of functions by single hidden layer neural networks with ReLU activation functions on the unit circle. In particular, we are interested in the case when the number of data-points exceeds the number of nodes. We first study the convergence to equilibrium of the stochastic gradient flow associated with the cost function with a quadratic penalization. Specifically, we prove a Poincaré inequality for a penalized version of the cost function with explicit constants that are independent of the data and of the number of nodes. As our penalization biases the weights to be bounded, this leads us to study how well a network with bounded weights can approximate a given function of bounded variation (BV). Our main contribution concerning approximation of BV functions, is a result which we call the localization theorem. Specifically, it states that the expected error of the constrained problem, where the length of the weights are less than $R$, is of order $R^{-1/9}$ with respect to the unconstrained problem (the global optimum). The proof is novel in this topic and is inspired by techniques from regularity theory of elliptic partial differential equations. Finally we quantify the expected value of the global optimum by proving a quantitative version of the universal approximation theorem.

math.AP↗
arXiv2020

Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow

We prove a new quantitative version of the Alexandrov theorem which states that if the mean curvature of a regular set in R^{n+1} is close to a constant in L^{n}-sense, then the set is close to a union of disjoint balls with respect to the Hausdorff distance. This result is more general than the previous quantifications of the Alexandrov theorem and using it we are able to show that in R^2 and R^3 a weak solution of the volume preserving mean curvature flow starting from a set of finite perimeter asymptotically convergences to a disjoint union of equisize balls, up to possible translations. Here by weak solution we mean a flat flow, obtained via the minimizing movements scheme.

math.AP↗
arXiv2020

Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane

We consider the flat flow solutions of the mean curvature equation with a forcing term in the plane. We prove that for every constant forcing term the stationary sets are given by a finite union of disks with equal radii and disjoint closures. On the other hand for every bounded forcing term tangent disks are never stationary. Finally in the case of an asymptotically constant forcing term we show that the only possible long time limit sets are given by disjoint unions of disks with equal radii and possibly tangent.

math.AP↗
arXiv2019

Generalized Harnack's inequality for nonhomogeneous elliptic equations

This paper is concerned with nonlinear elliptic equations in nondivergence form where the operator has a first order drift term which is not Lipschitz continuous. Under this condition the equations are nonhomogeneous and nonnegative solutions do not satisfy the classical Harnack's inequality. This paper presents a new type of generalization of the classical Harnack's inequality for such equations. As a corollary we obtain the optimal Harnack type of estimate for p(x)-harmonic functions which quantifies the strong minimum principle.

math.AP↗
arXiv2018

The surface diffusion flow with elasticity in three dimensions

We establish short-time existence of a smooth solution to the surface diffusion equation with an elastic term and without an additional curvature regularization in three space dimensions. We also prove the asymptotic stability of strictly stable stationary sets.

math.AP↗
arXiv2018

Symmetry of minimizers of a Gaussian isoperimetric problem

We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. This second term has a repulsive effect, and it is in competition with the perimeter. Because of that, in general the solution is not the half-space. We characterize all the minimizers of this functional, when the volume is close to one, by proving that the minimizer is either the half-space or the symmetric strip, depending on the strength of the repulsive term. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimetric problem among symmetric sets when the volume is close to one.

math.PR↗
arXiv2017

The surface diffusion flow with elasticity in the plane

In this paper we prove short-time existence of a smooth solution in the plane to the surface diffusion equation with an elastic term and without an additional curvature regularization. We also prove the asymptotic stability of strictly stable stationary sets.

math.AP↗
arXiv2017

Robustness of the Gaussian concentration inequality and the Brunn-Minkowski inequality

We provide a sharp quantitative version of the Gaussian concentration inequality: for every $r>0$, the difference between the measure of the $r$-enlargement of a given set and the $r$-enlargement of a half-space controls the square of the measure of the symmetric difference between the set and a suitable half-space. We also prove a similar estimate in the Euclidean setting for the enlargement with a general convex set. This is equivalent to the stability of the Brunn-Minkowski inequality for the Minkowski sum between a convex set and a generic one.

math.AP↗
arXiv2016

Remark on a nonlocal isoperimetric problem

We consider isoperimetric problem with a nonlocal repulsive term given by the Newtonian potential. We prove that regular critical sets of the functional are analytic. This optimal regularity holds also for critical sets of the Ohta-Kawasaki functional. We also prove that when the strength of the nonlocal part is small the ball is the only possible stable critical set.

math.AP↗
arXiv2016

Generalized Harnack inequality for semilinear elliptic equations

This paper is concerned with semilinear equations in divergence form \[ \diver(A(x)Du) = f(u) \] where $f :\R \to [0,\infty)$ is nondecreasing. We prove a sharp Harnack type inequality for nonnegative solutions which is closely connected to the classical Keller-Osserman condition for the existence of entire solutions.

math.AP↗
arXiv2016

Nonlinear stability results for the modified Mullins-Sekerka and the surface diffusion flow

It is shown that any three-dimensional periodic configuration that is strictly stable for the area functional is exponentially stable for the surface diffusion flow and for the Mullins-Sekerka or Hele-Shaw flow. The same result holds for three-dimensional periodic configurations that are strictly stable with respect to the sharp-interface Ohta-Kawaski energy. In this case, they are exponentially stable for the so-called modified Mullins-Sekerka flow.

math.AP↗
arXiv2015

p-harmonic coordinates for Hölder metrics and applications

We show that on any Riemannian manifold with Hölder continuous metric tensor, there exists a $p$-harmonic coordinate system near any point. When $p = n$ this leads to a useful gauge condition for regularity results in conformal geometry. As applications, we show that any conformal mapping between manifolds having $C^α$ metric tensors is $C^{1+α}$ regular, and that a manifold with $W^{1,n} \cap C^α$ metric tensor and with vanishing Weyl tensor is locally conformally flat if $n \geq 4$. The results extend the works [LS14, LS15] from the case of $C^{1+α}$ metrics to the Hölder continuous case. In an appendix, we also develop some regularity results for overdetermined elliptic systems in divergence form.

math.DG↗
arXiv2014

Lipschitz regularity for local minimizers of some widely degenerate problems

We consider local minimizers of the functional \[ \sum_{i=1}^N \int (|u_{x_i}|-δ_i)^p_+\, dx+\int f\, u\, dx, \] where $δ_1,\dots,δ_N\ge 0$ and $(\,\cdot\,)_+$ stands for the positive part. Under suitable assumptions on $f$, we prove that local minimizers are Lipschitz continuous functions if $N=2$ and $p\ge 2$, or if $N\ge 2$ and $p\ge 4$.

math.AP↗
arXiv2014

On the regularity of critical and minimal sets of a free interface problem

We study a free interface problem of finding the optimal energy configuration for mixtures of two conducting materials with an additional perimeter penalization of the interface. We employ the regularity theory of linear elliptic equations to study the possible opening angles of Taylor cones and to give a different proof of a partial regularity result by Fan Hua Lin.

math.AP↗
arXiv2013

Minimality via second variation for microphase separation of diblock copolymer melts

We consider a non local isoperimetric problem arising as the sharp interface limit of the Ohta-Kawasaki free energy introduced to model microphase separation of diblock copolymers. We perform a second order variational analysis that allows us to provide a quantitative second order minimality condition. We show that critical configurations with positive second variation are indeed strict local minimizers of the nonlocal perimeter. Moreover we provide, via a suitable quantitative inequality of isoperimetric type, an estimate of the deviation from minimality for configurations close to the minimum in the $L^1$ topology .

math.AP↗