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.

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.

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.

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.

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.

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.