For a periodic vector field $\mathbf F$, let
$\mathbf X^\epsilon$ solve the dynamical system
$$
\frac{d{\mathbf X}^\epsilon}{dt} = {\mathbf F }\left( \frac {{\mathbf X}^\epsilon }\epsilon\right) .
$$
In [1] Ennio De Giorgi enquiers whether from the existence of the limit
${\mathbf X}^0(t):=\lim\limits_{\epsilon \to 0} {\mathbf X}^\epsilon(t)$ one can conclude that $ \frac{d {\mathbf X}^0}{dt}= constant$.
Our main result settles this conjecture under fairly general assumptions on $\mathbf F$, which in some cases may also depend on $t$-variable.

Once the above problem is solved, one can apply the result to the corresponding transport equation, in a standard way. This is also touched upon in the text to follow.

[1] De Giorgi, E.: On the convergence of solutions of some evolution differential equations. Set Valued Anal. 2(1–2), 175–182 (1994)

We study the seminilinear problem $$\Delta u=\lambda_+(x) (u^+)^{q-1}-\lambda_- (x) (u^-)^{q-1} \qquad \hbox{in } \ B_1,$$ from a regularity point of view for solutions and the free boundary $\partial\{\pm u>0\}$. Here $B_1$ is the unit ball, $1< q< 2 $ and $\lambda_\pm$ are Lipschitz. Our main results concern local regularity analysis of solutions and their free boundaries. One of the main difficulties encountered in studying this equation is classification of global solutions. In dimension two we are able to present a fairly good analysis of global homogeneous solutions, and hence a better understanding of the behavior of the free boundary. In higher dimensions the problem becomes quite complicated, but we are still able to state partial results; e.g. we prove that if a solution is close to one-dimensional solution in a small ball, then in an even smaller ball the free boundary can be represented locally as two $C^1$-regular graphs $\Gamma^+=\partial\{u>0\}$ and $\Gamma^-=\partial\{u< 0\}$, tangential to each other. It is noteworthy that the above problem (in contrast to the case $q=1$) introduces interesting and quite challenging features, that are not encountered in the case $q=1$. E.g. one obtains homogeneous global solutions that are not one-dimensional. This complicates the analysis of the free boundary substantially.

We introduce a new lattice growth model, which we name boundary sandpile and which amounts to potential theoretic redistribution of the given continuous mass on the lattice $\mathbb{Z}^d$ onto the combinatorial (free) boundary of some unknown domain, which is being determined by the dynamics of the model.

Our initial motivation was triggered by an intriguing connection established by Levine and Peres, between quadrature domains, a potential theoretic concept, and divisible sandpile, a growth model on the lattice. In this paper we attempted to create a sandpile dynamics (a growth model on $\mathbb{Z}^d$) which would have a quadrature surface as its scaling limit. On the way to this quest, which is yet to be fulfilled, we introduce a new growth model, the boundary sandpile, which seems to represent a new alluring phenomenon not encountered previously.

To define boundary sandpile model, assume we have concentration of (continuous) mass $n > 0$ at the origin of $\mathbb{Z}^d$
$(d\geq 2)$. Then, at any discrete time $k\geq 0$, we say that a site $x\in \mathbb{Z}^d$ is visited, if at a certain time before (and including) $k$ it carried positive amount of mass. Next, at time $k$ we call a site $x$ unstable, if either $x$ is in the interior of the visited sites and has positive mass, or $x$ is on the (discrete) boundary of the set of visited sites, and has mass larger than $n^{1/d}$
(this threshold is due to scaling considerations). Then, any unstable site can topple, by evenly distributing all its mass among its neighboring vertices on the lattice. The process terminates, if there are no unstable sites. Clearly, this redistribution never stops in finite time, except for trivial cases, and it is not clear a priori, if the order in which one topples the unstable sites, can affect in any way the limiting growth cluster. What we showed in the paper, is that in the limit, i.e. as the discrete time goes to infinity, the growth process finds a stable configuration provided all unstable sites are toppled infinitely often; moreover, the limiting configuration is independent of the toppling sequence, and in this sense the model is abelian.
By definition, the total mass of the system is being redistributed to a combinatorial free boundary determined by the underlying dynamical system. Some limiting shapes of the sandpile are illustrated below.

Absorbing part of the topplings into linear operators acting on graphs, we show that the model reaches a stable state in finitely many steps.
We also prove that the final set of visited sites is the smallest domain (with respect to inclusions) in $\mathbb{Z}^d$ where the process can stabilize itself. The analysis is carried out in terms of a special function $u$, called the odometer, which is defined on $\mathbb{Z}^d$ and for each $x\in \mathbb{Z}^d$, $u(x)$ shows the total mass emitted from $x$ during the entire life-time of the process. We show a uniform (with respect to initial mass $n$) Lipschitz estimate on the odometer function, which (in part) implies Lipschitz regularity of the free boundary of the sandpile.

The underlying ideas of the analysis are largely motivated by potential theory, free boundary problems, and elliptic PDEs, building on top of these their combinatorial and probabilistic (random walk) counterparts. Our methods of this paper had already shown to be useful in other situations too. For instance, combined with the Least Action Principle for sandpiles proved by Fey, Levine, and Peres,
the techniques developed for boundary sandpile enabled us to show for the first time, that the boundary of the scaling limit of the classical Abelian sandpile model with a single source, is Lipschitz regular in dimensions $d\geq 2$. To prove this, we show that the Least Action Principle combined with the discrete variant (considered in our paper) of the celebrated moving plane method of A.D. Aleksandrov, implies monotonicity of the discrete odometer function of the sandpile with respect to special directions determined by mirror symmetries of the unit cube of $\mathbb{R}^d$. We then observe, that this monotonicity of the odometer is being transferred to the scaling limit of the model, the existence of which was established only recently by Pegden and Smart.
Relying on directional monotonicity of the odometer function, we show that at each point of the free boundary of the scaling limit,
there are $d$-linearly independent directions along which the odometer of the Abelian sandpile is monotone. The latter allows us to show that at each point of the free boundary there is a double-cone (of uniform size) with the vertex at that point such that the free boundary remains entirely outside the interior of the cone. That condition implies Lipschitz regularity.

We close the paper with some open problems.

ArticleBoundary behaviour for a singular perturbation problem

In this paper we study the boundary behaviour of the family of solutions $\{u^\epsilon\}$ to
singular perturbation problem $\Delta u^\epsilon=\beta_\epsilon(u^\epsilon), |u^\epsilon|\le 1$ in $B_1^+=\{x_n>0\}\cap \{|x|< 1\}$,
where a smooth boundary data $f$ is prescribed on the flat portion of $\partial B_1^+$.
Here $\beta_\epsilon(\cdot)=\frac1\epsilon\beta\left({\frac\cdot\epsilon}\right), \beta\in C_0^\infty(0, 1), \beta\ge0, \int_0^1\beta(t)=M>0$ is an approximation of identity. If $\nabla f(z)=0$ whenever $f(z)=0$ then the level sets
$\partial \{u^\epsilon > 0\} $ approach the
fixed boundary in tangential fashion with uniform speed. The methods we employ here uses delicate analysis of
local solutions, along with elaborated version of the so-called monotonicity formulas and classification of global profiles.

ArticleMulti-phase quadrature domains and a related minimization problem

Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis 45 (1), 135-155, 2016

In this paper we introduce the multi-phase version of the so-called Quadrature Domains (QD), which refers
to a generalized type of mean value property for harmonic functions. The well-established and developed theory
of one-phase QD was recently generalized to a two-phase version, by one of the current authors (in collaboration). Here we introduce the concept of
the multi-phase version of the problem, and prove existence as well as several properties of such solutions.
In particular, we discuss possibilities of multi-junction points.

Article$L^2$-estimates for singular oscillatory integrals

This article is motivated by our papers treating homogenization of Dirichlet problem,
the classical paper by Phong and Stein, and a recent interest in PDE problems involving rough boundaries, in particular
the paper by Kenig and Prange. For a real-valued function $\psi \in C^\infty(\mathbb{R}^n)$ having bounded derivatives of all orders,
consider the hypersurface $\Gamma = \{ (y, \psi(y)) \in \mathbb{R}^{n+1}: \ y \in \mathbb{R}^n \} $.
For $f\in L^2(\mathbb{\mathbb{R}^n})$, $ \lambda>0$, and $(x, x_{n+1} ) \in \mathbb{R}^n \times \mathbb{R}$ define
$$
T_\lambda f (x, x_{n+1}) = \int_\Gamma e^{i \lambda x\cdot y} \varphi_0((x,x_{n+1}),y )
K(x-y, x_{n+1}-y_{n+1}) f(y ) d\sigma(y,y_{n+1}) ,
$$
where $d\sigma$ is the surface measure on $\Gamma$, $\varphi_0$ is a real-valued
function from the class $C_0^\infty (\mathbb{R}^{n+1} \times \mathbb{R}^n )$, and $K$ is a singular
kernel satisfying $K\in C^\infty(\mathbb{R^{n+1}} \setminus \{0\} )$ and
$| \nabla^\alpha K(z) | \lesssim_\alpha \frac{|z|^m}{|z|^{n+|\alpha|}}$ with $0\leq m < n$ for all $z\in \mathbb{R}^{n+1}\setminus \{0\}$ and
any multi-index $\alpha \in \mathbb{Z}^n_+$. Here we have $n\geq 1$ and do not assume that $m$
is necessarily an integer.

For each fixed $x_{n+1}$ we study $T_\lambda$ as an operator from $L^2(\mathbb{R}^{n})$ to $L^2(\mathbb{R}^{n})$
and prove decay estimates for its operator norm as $\lambda \to \infty$.
A special attention is paid to obtaining precise bounds with respect to the smoothness norms of the hypersurface $\Gamma$,
as that estimates are being used to analyse the behavior of the operator $T_\lambda$
under small perturbations of a given fixed surface $\Gamma$. The latter problem
is motivated by PDE problems with rough boundaries, where, for instance, the boundary can be technically $C^\infty$
however oscillating rapidly (and hence having large smoothness norms).
In that setting standard techniques by partial integration toward controlling operators of the form $T_\lambda$,
become inefficient, in view of the high oscillations of the surface.
The approach we propose here handles efficiently the effects coming from surface oscillations.

In a similar spirit, when we allow the surface to oscillate,
we consider a certain maximal operator associated with operators of the form $T_\lambda$
which captures the effect of the oscillation of the surface. More precisely, for a
family of hypersurfaces $\{\Gamma_\varepsilon\}_{0< \varepsilon\leq 1}$ (having a certain structure) we analyse the boundedness of the operator
$ T_\lambda^\ast f(x, x_{n+1}) = \sup\limits_{0< \varepsilon \leq 1} | T_\lambda^\varepsilon f(x,x_{n+1} ) |$, where $ (x,x_{n+1} ) \in \mathbb{R}^{n+1} $
and $T_\lambda^\varepsilon$ is defined as above but for the surface $\Gamma_\varepsilon$.
Here the small parameter $0< \varepsilon \leq 1$ is meant to model an oscillatory behavior of the given
family of hypersurfaces.

The second part of the paper studies operators of the form $T_\lambda$ where instead of the linear phase $x\cdot y$
we consider a fractional-type nonlinearity in the form of $|x-y|^\gamma$ with real $\gamma \geq 1$.
This special choice of the phase function is partially motivated by PDE problems, in particular by the Helmholtz operator,
and requires a completely new approach. We then apply our analysis to eigenvalue problem for the Helmholtz equation
in $\mathbb{R}^3$ with some special source terms, and establish a decay estimate for solutions satisfying Sommerfeld radiation condition, as the eigenvalue tends to infinity.

The paper is essentially self-contained, and the proofs are based, among other things, on various decomposition arguments.

ArticleHopf's lemma for a class of singular/degenerate PDE-s

If you plan to formulate more than one problem all sharing the same background (e.g. they are all from the same paper) then please choose "Group", otherwise select "Single" option.

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