Henrik Shahgholian

Abstract: We introduce a new boundary Harnack principle in Lipschitz domains for equations with a right hand side. Our approach, which uses comparisons and blow-ups, seems likely to adapt to more general domains, as well as other types of operators.

We consider operators of the form $$ \mathcal{L} u = (a^{ij}u_i)_j + b^i u_i + cu , $$ with the following ellipticity conditions $$ \Lambda^{-1} |\xi|^2 \leq \langle a^{ij}(x) \xi, \xi \rangle \leq \Lambda |\xi|^2, $$ for some $\Lambda >0$ and for all nonzero $\xi \in \R^n$. Furthermore, $a^{ij}(x)$ is a real $n \times n$ matrix. For the lower order terms we assume $$|c(x)|, \sum |b^i(x)| \leq \Lambda-1$$ and that $c(x) \leq 0$. We say $u \in \mathcal{S_{\mathcal{L}}}(\Omega_{L,R})$ if $$ \begin{aligned} \mathcal{L} u(x) &=0 \text{ in } \Omega_{L,R} \cap B_R ,\\ u(x)&=0 \text{ on } \Omega_{L,R}^c\cap B_R, \end{aligned} $$ and that $u \in \mathcal{S_{\mathcal{L}}}(\Omega_{L,R},d^{\gamma})$ if $$ \begin{aligned} |\mathcal{L} u(x)| &\leq (\text{dist}(x,\partial \Omega_{L,R} \cap B_R))^{\gamma} \ \text{ in } \Omega_{L,R} , \\ u(x)&=0 \ \text{ on } \Omega_{L,R}^c\cap B_R. \end{aligned} $$ To apply the Hölder continuity estimates for elliptic operators we will require that $\gamma>-2/n $. Since the boundary is Lipschitz, this will ensure the correct integrability assumptions for the right hand side.

Theorem:

Let $0 \in \partial \Omega_L$ with $L< M$, and fix $x^0 \in \Omega_L$. Assume further that $B_1 \cap \{x_n > 1/4\} \subseteq \Omega_L$. Assume $u,v \geq 0$ and $u,v \in \mathcal{S}(\Omega_{L},d^{\gamma})$ with $\mathcal L u, \mathcal L v \leq 0$ and $u(x^0)=v(x^0)=1$, and also assume that $2-\alpha+\gamma>0$, with $\gamma>-2/n$ and $\alpha$ be such that $$ \sup_{B_{r}(x)} u \geq c_1 u(e_n/2) r^{\alpha}. $$

Then there exists a uniform constant $C>0$ (depending only on dimension $n$, Lipschitz constant $M$, $M-L$, and dist$(x^0,\partial \Omega_L)$) such that \begin{equation} \label{e:ine3} C^{-1} v(x) \leq u(x) \leq Cv(x) \end{equation} for all $x \in B_{1/2}$.

This paper studies the fully nonlinear free boundary problem $$ F(D^2u)=1 \ \text{a.e. in }B_1 \cap \Omega \qquad \hbox{and }\qquad |D^2 u| \leq K , \ \text{a.e. in }B_1\setminus\Omega, $$ where $K>0$, and $\Omega$ is an unknown open set. The main result is the optimal regularity for solutions to this problem: namely, we prove that $W^{2,n}$ solutions are locally $C^{1,1}$ inside $B_1$. Under the extra condition that $ {\Omega \supset \{D{u} \neq 0 \}}$, and a uniform thickness assumption on the coincidence set $\{Du=0\}$. It is also shown a local regularity for the free boundary $\partial \Omega \cap B_1$.

In this paper we initiate the study of maps minimising the energy $$ \int_{D} (|\nabla {\mathbf u}|^2+2| {\mathbf u}|)\ dx. $$ which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations $$ \Delta {\mathbf u}=\frac{ {\mathbf u}}{| {\mathbf u}|}\chi_{\left\lbrace | {\mathbf u}|>0\right\rbrace}, \qquad {\mathbf u} = (u_1, \cdots, u_m) \ . $$ 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.

We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, $\hbox{div}(A_s (x, u)u) = \hbox{div} { \mathbf f (x)}$, where $A_s (x, u)$ has " broken " derivatives of order $s \geq 0$, such as $A_s (x, u) = a(x) + b(x)(u +)^s$ , with $(u +)^0$ being understood as the characteristic function on $ \{u > 0\}$. The vector $\mathbf f (x)$ is assumed to be $C^\alpha$ in case $s = 0$, and $C^{ 1,\alpha}$ (or higher) in case $s > 0$. Using geometric methods, we prove almost complete results (in analogy with standard PDEs) concerning the behavior of the nodal sets. More exactly, we show that the nodal sets, where solutions have (linear) nondegeneracy, are locally smooth graphs. Degenerate points are shown to have structures that follow the lines of arguments as that of the nodal sets for harmonic functions, and general PDEs.

In this paper we consider a quasilinear elliptic PDE, $\hbox{div} (A(x,u) \nabla u) =0$, where the underlying physical problem gives rise to a jump for the conductivity $A(x,u)$, across a level surface for $u$. Our analysis concerns Lipschitz regularity for the solution $u$, and the regularity of the level surfaces, where $A(x,u)$ has a jump and the solution $u$ does not degenerate. In proving Lipschitz regularity of solutions, we introduce a new and unexpected type of ACF-monotonicity formula with two different operators, that might be of independent interest, and surely can be applied in other related situations. The proof of the monotonicity formula is done through careful computations, and (as a byproduct) a slight generalization to a specific type of variable matrix-valued conductivity is presented.

We study the minimum problem for the functional \begin{equation*} \int_{\Omega}\bigl( \vert \nabla {\mathbf{u}} \vert^{2} + Q^{2}\chi_{\{\vert {\mathbf{u}} \vert>0\}} \bigr)dx \end{equation*} with the constraint $u_i\geq 0$ for $i=1,\cdots,m$ where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain and ${\mathbf{u}}=(u_1,\cdots,u_m)\in H^{1}(\Omega;\mathbb{R}^{m})$.

Using an array of technical tools, from geometric analysis for the free boundaries, we reduce the problem to its scalar counterpart and hence conclude similar results as that of scalar problem. This can also be seen as the most novel part of the paper, that possibly can lead to further developments of free boundary regularity for systems.

The aim of this paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). Given a measure $ \mu $ in $\mathbb{R}^d$ and a domain $\Omega$ containing the support of $\mu$, we call $\partial \Omega$ a quadrature surface for $\mu $ if one has $ \int_{\partial \Omega } h(x) d\mathcal{H}^{d-1} (x) = \int h(x) d\mu (x) $ for all functions $ h $ harmonic on $ \overline{\Omega} $ [Sh1] and [Sh2] by H. Shahgholian). For example, if $\mu$ is a Dirac mass concentrated at some point $x\in \mathbb{R}^d$, then a sphere with center at $x$ and of an appropriate radius would give a quadrature surface. There is no complete description of measures $\mu$ admitting a quadrature surface identity, but in a special case of a finite sum of Dirac masses, for instance, the existence is due to Shahgholian.

In this paper we show that one can obtain a quadrature surface as a scaling limit of a certain growth model on the lattice $\mathbb{Z}^d $, where $ d\geq 2 $. Our motivation comes from an already established connection between quadrature domains (these are domains admitting a similar integral identity as above, but with integration against the Lebesgue measure over the domain instead of the surface) and Laplacian growth models by Levine and Peres (see [LP1] and [LP2] ). They introduce a model called divisible sandpile, and prove that for initial distribution of finitely many points its scaling limit is a quadrature surface ( a smash sum of Euclidean balls, precisely, a concept after Diaconis and Fulton) A shift from a domain to a lower-dimensional object, a hypersurface in our case, seems to necessitate an entirely different approach.

To define our model, start with a distribution of mass $\mu_0 $ on $ \mathbb{Z}^d $, i.e. a bounded non-negative function of finite support, let $n = \sum_{x\in \mathbb{Z}^d} \mu_0(x) $ be its total mass, and fix a threshold $ m>0 $. One by one we pick a vertex $x \in \mathbb{Z}^d$ and if it carries mass larger than $m$, distribute the excess of mass from $m$ evenly among the $2d$ lattice neighbors of $x$. Otherwise, if the mass carried by $x$ is bounded by $m$ but the cumulative emissions of mass from $x$ prior to a given epoch of time exceed $\frac 1m n^{2/d}$ distribute the entire mass of $x$ evenly among its lattice neighbors. If none of the two conditions hold for $x$ we move on to another vertex. This mass redistribution procedure is called toppling of $x$. We show that provided any vertex of $\mathbb{Z}^d$ is chosen infinitely many times, the mass redistribution process will reach a stable state after countably many topplings, and regardless of the order of the topplings the final configuration will be the same (Abelian property). Informally, one may think about the proposed model, as a deformation of a divisible sandpile model of Levine and Peres, where we force the mass to accumulate on a $\frac 1m $-sub-level sets of the odometer. This model is partially inspired by singular perturbation problems in PDEs which are known to converge to Bernoulli type free boundaries.

The key concept in our analysis is the odometer function $u:\mathbb{Z}^d \rightarrow \mathbb{R}_+$ where for each $x\in \mathbb{Z}^d$ the value $u(x)$ shows total emissions of mass from $x$ during the lifetime of the process. In terms of this function, the discrete PDE problem solved by the model reads $$\Delta u(x) + \mu_0(x) \leq m \ \ \ \text{ everywhere on } \mathbb{Z}^d $$ and $$\Delta u(x) = m \mathbb{I}_{ \{ 0 < u < \frac{1}{m} n^{2/d} \} } (x) - \mu_0(x) \ \ \ \text{ for all } x \in \{u > 0\},$$ where $\Delta$ is the discrete (normalized) Laplacian. We show that the odometer $u$ is precisely the smallest super-solution to the above problem. This characterization combined with combinatorial moving plane techniques, which we introduced in our earlier paper, allows to get a certain discrete monotonicity properties for the odometer function for a single source initial distributions, using which we show that the set of visited sites of the model in $\mathbb{Z}^d$ grows proportional to $n^{1/d}$, and the entire mass of the system is being distributed to an annular ring of thickness $\sim 1/m$.

We prove the existence of the scaling limit of the model generated by a single source, and fixed threshold $m$. As we show, this limit is a ball, with the entire mass of the system being redistributed onto a narrow annular ring of thickness $\frac 1m$ near the boundary of the ball. With compactness arguments, we conclude that there is also a scaling limit for a subsequence of the odometers, if the threshold $m$ tends to infinity along with the scale of the model. That limit is spherical, with the entire mass of the system being uniformly redistributed onto the boundary of that ball. This gives a quadrature surface for a single source mass distributions.

In this paper we study the behaviour of the free boundary close to its contact points with the fixed boundary $B\cap\{x_1=0\}$ in the obstacle type problem \begin{equation*} \left\{ \begin{aligned} & \operatorname{div}(x_1^{a}\nabla u)=\chi_{\{u>0\}} \quad \text{in } \quad B^{+}\text{,} \\ & u=0 \qquad \qquad\text{on }\quad B\cap\{x_1=0\} \end{aligned}\right. \end{equation*} where $a< 1$, $B^+=B\cap\{x_1>0\}$, $B$ is the unit ball in $\mathbb{R}^{n}$ and $n\geq 2$ is an integer.

Let $\Gamma=B^{+}\cap\partial\{u>0\}$ be the free boundary and assume that the origin is a contact point, i.e. $0\in\overline\Gamma$. We prove that the free boundary touches the fixed boundary uniformly tangentially at the origin, near to the origin it is the graph of a $C^{1}$ function and there is a uniform modulus of continuity for the derivatives of this function.

In this paper we shall initiate the study of the two- and multi-phase quadrature surfaces (QS), which amounts to a two/multi-phase free boundary problems of Bernoulli type. The problem is studied mostly from a potential theoretic point of view that (for two-phase case) relates to integral representation $$ \int_{\partial \Omega^+} g h (x) \ d\sigma_x - \int_{\partial \Omega^-} g h (x) \ d\sigma_x= \int h d\mu \ , $$ where $d\sigma_x$ is the surface measure, $\mu= \mu^+ - \mu^-$ is given measure with support in (a priori unknown domain) $\Omega=\Omega^+\cup\Omega^-$, $g$ is a given smooth positive function, and the integral holds for all functions $h$, which are harmonic on $\overline \Omega$.

Our approach is based on minimization of the corresponding two- and multi-phase functional and the use of its one-phase version as a barrier. We prove several results concerning existence, qualitative behavior, and regularity theory for solutions. A central result in our study states that three or more junction points do not appear.

We study a minimum problem for a non-differentiable functional whose reaction term does not have scaling properties. Specifically we consider the functional $$ \mathcal J(v)=\int_\Omega\left(\frac{|\nabla v|^2}{2} -v^+(\log v -1)\right)dx\to \min $$ which should be minimized in some natural admissible class of non-negative functions. Here, $v^+=\max\{0,v\}.$ The Euler--Lagrange equation associated with $\mathcal J$ is $$ -\Delta u= \chi_{\{u>0\}}\log u, $$ which becomes singular along the free boundary $\partial\{u>0\}.$ Therefore, the regularity results do not follow from classical methods. Besides, the logarithmic forcing term does not have scaling properties, which are very important in the study of free boundary theory. Despite these difficulties, we obtain optimal regularity of a minimizer and show that, close to every free boundary point, they exhibit a super-characteristic growth like $$ r^2|\log r|. $$ This estimate is crucial in the study of analytic and geometric properties of the free boundary.