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John Andersson

Publications and source records attributed to John Andersson.

13 recordsLinked to original sources
arXiv2020

The asymptotics of the curvature of the free discontinuity set near the cracktip for the minimizers of the Mumford-Shah functional in the plain

We consider in 2D the following special case of the Mumford-Shah functional $$ J(u, Γ)=\int_{B_1\backslashΓ} |\nabla u|^2 dx + λ^2 \fracπ{2} \mathcal{H}^1(Γ). $$ It is known that if the minimizer has a crack-tip in the ball $B_1$ (assume at the origin), then $u\approx λ\Im \sqrt{z}$ at this point. We calculate higher order terms in the asymptotic expansion, where the homogeneity orders of those terms appear to be solutions to a certain trigonometric relation.

math.AP↗
arXiv2019

Regularity up to the Crack-Tip for the Mumford-Shah problem

We prove that if $(u,Γ)$ is a minimizer of the functional $$ J(u,Γ)=\int_{B_1(0)\setminus Γ}|\nabla u|^2dx +\H^1(Γ) $$ and $Γ$ connects $\partial B_1(0)$ to a point in the interior, then $Γ$ satisfies a point-wise $C^{2,α}$-estimate at the crack-tip. This means that the Mumford-Shah functional satisfies an additional, and previously unknown, Euler-Lagrange condition. ******* The previous version of the paper contained some mistakes, which has been fixed. More explanations/details has been added in Section 6.

math.AP↗
arXiv2016

Almost Everywhere Regularity for the Free Boundary of the Normalized p-harmonic Obstacle problem $p>2$

Let $u$ be a solution to the normalized p-harmonic obstacle problem with $p>2$. That is, $u\in W^{1,p}(B_1(0))$, $2 0\}}\textrm{ in }B_1(0) $$ where $u(x)\ge 0$ and $χ_A$ is the characteristic function of the set $A$. Our main result is that for almost every free boundary point, with respect to the $(n-1)-$Hausdorff measure, there is a neighborhood where the free boundary is a $C^{1,β}-$graph. That is, for $\H^{n-1}-$a.e. point $x^0\in \partial \{u>0\}\cap B_1(0)$ there is an $r>0$ such that $B_r(x^0)\cap \partial \{u>0\}\in C^{1,β}$.

math.AP↗
arXiv2015

Optimal regularity for the obstacle problem for the $p$-Laplacian

In this paper we discuss the obstacle problem for the $p$-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising result we prove is the one for the $p$-obstacle problem: Find the smallest $u$ such that $$ \hbox{div} (|\nabla u|^{p-2}\nabla u) \leq 0, \qquad u\geq ϕ, \qquad \hbox{in } B_1, $$ with $ϕ\in C^{1,1}(B_1)$ and given boundary datum on $\partial B_1$. We prove that the solution is uniformly $C^{1,1}$ at free boundary points. Similar results are obtained in the case of an inhomogeneity belonging to $L^\infty$. When applied to the corresponding parabolic problem, these results imply that any solution which is Lipschitz in time is $C^{1,\frac{1}{p-1}}$ in the spatial variables.

math.AP↗
arXiv2013

Equilibrium points of a singular cooperative system with free boundary

In this paper we initiate the study of maps minimising the energy $$ \int_{D} (|\nabla \u|^2+2|\u|)\ dx. $$ which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations $$ Δ\u=\frac{\u}{|\u|}χ_{\left\lbrace |\u|>0\right\rbrace}, \qquad \u = (u_1, \cdots, u_m) \ . $$ Our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps. Our results here concern regularity of the solution as well as that of the free boundary. They are achieved by using monotonicity formulas and epiperimetric inequalities, in combination with geometric analysis.

math.AP↗
arXiv2013

Optimal Regularity for The Signorini Problem and its Free Boundary

We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if $\u=(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\R^3)$ minimizes $$ J(\u)=\int_{B_1^+}|\nabla \u+\nabla^\bot \u|^2+λ÷(\u)^2 $$ in the convex set $$ K=\big\{\u=(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\R^3);\; u^3\ge 0 \textrm{on}Π, $$ $$ \u=f\in C^\infty(\partial B_1) \textrm{on}(\partial B_1)^+ \big\}, $$ where $λ\ge 0$ say. Then $\u\in C^{1,1/2}(B_{1/2}^+)$. Moreover the free boundary, given by $$ Γ_\u=\partial \{x;\;u^3(x)=0,\; x_3=0\}\cap B_{1}, $$ will be a $C^{1,α}$ graph close to points where $\u$ is not degenerate. Similar results have been know before for scalar partial differential equations (see for instance \cite{AC} and \cite{ACS}). The novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations.

math.AP↗
arXiv2012

Optimal Regularity for the parabolic No-Sign Obstacle Problem

We study the parabolic free boundary problem of obstacle type $$ \lap u-\frac{\partial u}{\partial t}= fχ_{u\ne 0}. $$ Under the condition that $f=Hv$ for some function $v$ with bounded second order spatial derivatives and bounded first order time derivative, we establish the same regularity for the solution $u$. Both the regularity and the assumptions are optimal. Using this result and assuming that $f$ is Dini continuous, we prove that the free boundary is, near so called low energy points, a $C^1$ graph. Our result completes the theory for this type of problems for the heat operator.

math.AP↗
arXiv2012

Double Obstacle Problems with obstacles given by non-$C^2$ Hamilton-Jacobi equations

We prove optimal regularity for the double obstacle problem when obstacles are given by solutions to Hamilton-Jacobi equations that are not $C^2$. When the Hamilton-Jacobi equation is not $C^2$ then the standard Bernstein technique fails and we loose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speed in different directions) we develop a new pointwise regularity theory for Hamilton-Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that $C^1$-solutions to the Hamilton-Jacobi equation $$ \pm |\nabla h-a(x)|^2=\pm 1 \textrm{in} B_1, \qquad h=f \textrm{on} \partial B_1, $$ are in fact $C^{1,α/2}$ provided that $a \in C^α$. This result is optimal and to the authors' best knowledge new.

math.AP↗
arXiv2011

Optimal regularity for the no-sign obstacle problem

In this paper we prove the optimal $C^{1,1}(B_\frac12)$-regularity for a general obstacle type problem $$ \lap u = fχ_{\{u\neq 0\}}\textup{in $B_1$}, $$ under the assumption that $f*N$ is $C^{1,1}(B_1)$, where $N$ is the Newtonian potential. This is the weakest assumption for which one can hope to get $C^{1,1}$-regularity. As a by-product of the $C^{1,1}$-regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point $x^0$, the free boundary is locally a $C^1$-graph close to $x^0$, provided $f$ is Dini. This completely settles the question of the optimal regularity of this problem, that has been under much attention during the last two decades.

math.AP↗
arXiv2010

On the singularities of a free boundary through Fourier expansion

In this paper we are concerned with singular points of solutions to the {\it unstable} free boundary problem $$ Δu = - χ_{\{u>0\}} \qquad \hbox{in} B_1. $$ The problem arises in applications such as solid combustion, composite membranes, climatology and fluid dynamics. It is known that solutions to the above problem may exhibit singularities - that is points at which the second derivatives of the solution are unbounded - as well as degenerate points. This causes breakdown of by-now classical techniques. Here we introduce new ideas based on Fourier expansion of the nonlinearity $χ_{\{u>0\}} $. The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in ${\mathbb R}^3$. A surprising fact in ${\mathbb R}^3$ is that although $$\frac{u(r\x)}{\sup_{B_1}|u(r\x)|}$$ can converge at singularities to each of the harmonic polynomials $$ xy, {x^2+y^2\over 2}-z^2 \textrm{and} z^2-{x^2+y^2\over 2},$$ it may {\em not} converge to any of the non-axially-symmetric harmonic polynomials $α((1+ δ)x^2 +(1- δ)y^2 - 2z^2)$ with $δ\ne 1/2$. We also prove the existence of stable singularities in ${\mathbb R}^3$.

math.AP↗
arXiv2009

Uniform Regularity close to Cross Singularities in an Unstable Free Boundary Problem

We introduce a new method for the analysis of singularities in the unstable problem $$Δu = -χ_{\{u>0\}},$$ which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of "supercharacteristic" growth of the solution, i.e. points at which the solution grows faster than the characteristic/invariant scaling of the equation would suggest. At such points the classical theory is doomed to fail, due to incompatibility of the invariant scaling of the equation and the scaling of the solution. In the case of two dimensions our result shows that in a neighborhood of the set at which the second derivatives of $u$ are unbounded, the level set $\{u=0\}$ consists of two $C^1$-curves meeting at right angles. It is important that our result is not confined to the minimal solution of the equation but holds for all solutions.

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