0000000000004990
AUTHOR
Vesa Julin
showing 16 related works from this author
Robustness of the Gaussian concentration inequality and the Brunn–Minkowski inequality
2016
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.
p-harmonic coordinates for H\"older metrics and applications
2015
We show that on any Riemannian manifold with H\"older 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 $C^\alpha$ metric tensors is $C^{1+\alpha}$ regular, and that a manifold with $W^{1,n} \cap C^\alpha$ 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 $C^{1+\alpha}$ metrics to the H\"older continuous case. In an appendix, we also develop some regularity results for overdetermined el…
p-harmonic coordinates for Hölder metrics and applications
2017
We show that on any Riemannian manifold with H¨older 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 W1,n ∩ C α metric tensor and with vanishing Weyl tensor is locally conformally flat if n ≥ 4. The results extend the works [LS14, LS15] from the case of C 1+α metrics to the H¨older continuous case. In an appendix, we also develop some regularity results for overdetermined elliptic systems in divergence form. peerReviewed
Remark on a nonlocal isoperimetric problem
2017
Abstract 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.
A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation
2011
In this paper, we give a new proof for the fact that the distributional weak solutions and the viscosity solutions of the $p$-Laplace equation $-\diver(\abs{Du}^{p-2}Du)=0$ coincide. Our proof is more direct and transparent than the original one by Juutinen, Lindqvist and Manfredi \cite{jlm}, which relied on the full uniqueness machinery of the theory of viscosity solutions. We establish a similar result also for the solutions of the non-homogeneous version of the $p$-Laplace equation.
On the regularity of critical and minimal sets of a free interface problem
2015
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 [Calc Var. Partial Differential Equations, 1993].
Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality
2017
We provide a full quantitative version of the Gaussian isoperimetric inequality: the difference between the Gaussian perimeter of a given set and a half-space with the same mass controls the gap between the norms of the corresponding barycenters. In particular, it controls the Gaussian measure of the symmetric difference between the set and the half-space oriented so to have the barycenter in the same direction of the set. Our estimate is independent of the dimension, sharp on the decay rate with respect to the gap and with optimal dependence on the mass.
Generalized Harnack inequality for semilinear elliptic equations
2015
Abstract This paper is concerned with semilinear equations in divergence form div ( A ( x ) D u ) = f ( u ) , where f : R → [ 0 , ∞ ) is nondecreasing. We introduce a sharp Harnack type inequality for nonnegative solutions which is a quantified version of the condition for strong maximum principle found by Vazquez and Pucci–Serrin in [30] , [24] and is closely related to the classical Keller–Osserman condition [15] , [22] for the existence of entire solutions.
Quantitative lower bounds to the Euclidean and the Gaussian Cheeger constants
2020
We provide a quantitative lower bound to the Cheeger constant of a set $\Omega$ in both the Euclidean and the Gaussian settings in terms of suitable asymmetry indexes. We provide examples which show that these quantitative estimates are sharp.
The asymptotics of the area-preserving mean curvature and the Mullins–Sekerka flow in two dimensions
2022
AbstractWe provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions, starting from any bounded set of finite perimeter, converge with exponential rate to a finite union of equally sized disjoint disks. A similar result is established also for the periodic two-phase Mullins–Sekerka flow.
Stationary sets of the mean curvature flow with a forcing term
2020
We consider the flat flow approach for the mean curvature equation with forcing in an Euclidean space $\mathbb R^n$ of dimension at least 2. Our main results states that tangential balls in $\mathbb R^n$ under any flat flow with a bounded forcing term will experience fattening, which generalizes the result by Fusco, Julin and Morini from the planar case to higher dimensions. Then, as in the planar case, we are able to characterize stationary sets in $\mathbb R^n$ for a constant forcing term as finite unions of equisized balls with mutually positive distance.
A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term
2017
This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in (Julin, ARMA -15), to prove a generalized Carleson estimate. We also prove boundary H\"older continuity and a boundary Harnack type inequality.
Short time existence of the classical solution to the fractional mean curvature flow
2019
Abstract We establish short-time existence of the smooth solution to the fractional mean curvature flow when the initial set is bounded and C 1 , 1 -regular. We provide the same result also for the volume preserving fractional mean curvature flow.
Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane
2020
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. peerReviewed
Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow
2020
We prove a new quantitative version of the Alexandrov theorem which states that if the mean curvature of a regular set in Rn+1 is close to a constant in the Ln 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 R2 and R3 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 a weak solution we mean a flat flow, obtained via the minimizing movements scheme. peerReviewed
Minimality via second variation for microphase separation of diblock copolymer melts
2017
Abstract 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 problem. 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 {L^{1}} -topology.