arXiv subjects

Henrik Shahgholian

Publications and source records attributed to Henrik Shahgholian.

81 records · Page 5Linked to original sources

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↗

arXiv2011

Harmonic balls and two-phase Schwarz function

Here we shall introduce the concept of harmonic balls/spheres in sub-domains of $\R^n$, through a mean value property for a sub-class of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept fails to exist due to the fact that analytic functions can not have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exists, and gives rise to the generalization of the well-known Schwarz function to the case of two-phase Schwarz function. Our primary goal is to derive simple properties for these problems, and tease the appetites of experts working on Schwarz function and related topics. Hopefully these two concepts will provoke further study of the topic.

math.AP↗

arXiv2010

Convexity of the free boundary for an exterior free boundary problem involving the perimeter

We prove that if the given compact set $K$ is convex then a minimizer of the functional $$ I(v)=\int_{B_R} |\nabla v|^p dx+\text{Per}(\{v>0\}),\,1<p<\infty, $$ over the set $\{v\in H^1_0(B_R)|\,\, v\equiv 1\,\,\text{on}\,\, K\subset B_R\}$ has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.

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.

math.AP↗

arXiv2007

The Parabolic Two-Phase Membrane Problem: Regularity in Higher Dimensions

For the parabolic obstacle-problem-like equation $$Δu - \partial_t u = λ_+ χ_{\{u>0\}} - λ_- χ_{\{u<0\}} ,$$ where $λ_+$ and $λ_-$ are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary $\partial\{u>0\} \cup\partial\{u<0\}$ is in a neighborhood of each ``branch point'' the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper \cite{imrn} to the parabolic case. The result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example.

math.AP↗

arXiv2005

The Two-Phase Membrane Problem -- an Intersection-Comparison Approach to the Regularity at Branch Points

For the two-phase membrane problem $ Δu = {λ_+\over 2} χ_{\{u>0\}} - {λ_-\over 2} χ_{\{u<0\}} ,$ where $λ_+> 0$ and $λ_->0 ,$ we prove in two dimensions that the free boundary is in a neighborhood of each ``branch point'' the union of two $C^1$-graphs. We also obtain a stability result with respect to perturbations of the boundary data. Our analysis uses an intersection-comparison approach based on the Aleksandrov reflection. In higher dimensions we show that the free boundary has finite $(n-1)$-dimensional Hausdorff measure.

math.AP↗

arXiv2000

Regularity of a free boundary with application to the Pompeiu problem

In the unit ball B(0,1), let $u$ and $Ω$ (a domain in $\R$) solve the following overdetermined problem: $$Δu =χ_Ω\quad \hbox{in} B(0,1), \qquad 0 \in \partial Ω, \qquad u=|\nabla u |=0 \quad \hbox{in} B(0,1)\setminus Ω,$$ where $χ_Ω$ denotes the characteristic function, and the equation is satisfied in the sense of distributions. If the complement of $Ω$ does not develop cusp singularities at the origin then we prove $\partial Ω$ is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.

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