In this paper we consider the so-called double-phase problem where the phase transition takes place across the zero level "surface" of the minimizer of the functional $$ J_{p,q}(v,\Omega) = \int_\Omega \left( |\nabla v^+|^p + |\nabla v^-|^q \right) dx. $$ We prove that the minimizer exists, and is H\"older regular. From here, using an intrinsic variation, one can prove a weak formulation of the free boundary condition across the zero level surface, that formally can be represented as $$ (q-1)|\nabla u^-|^q = (p-1) |\nabla u^+|^p, \quad \hbox{on } \partial \{u > 0\}. $$ We prove that the free boundary is $C^{1,\alpha}$ a.e. with respect to the measure $\Delta_p u^+$, whose support is of $\sigma$-finite $(n-1)$-dimensional Hausdorff measure.

We prove a boundary Harnack principle in Lipschitz domains with small constant for fully nonlinear and $p$-Laplace type equations with a right hand side, as well as for the Laplace equation on nontangentially accessible domains under extra conditions. The approach is completely new and gives a systematic approach for proving similar results for a variety of equations and geometries.

Nina Uraltseva has made lasting contributions to mathematics with her pioneering work in various directions in analysis and PDEs and the development of elegant and sophisticated analytical techniques. She is most renowned for her early work on linear and quasilinear equations of elliptic and parabolic type in collaboration with Olga Ladyzhenskaya, which is the category of classics, but her contributions to the other areas such as degenerate and geometric equations, variational inequalities, and free boundaries are equally deep and significant. In this article, we give an overview of Nina Uraltseva's work with some details on selected results.

Given a global 1-homogeneous minimizer $U_0$ to the Alt-Caffarelli energy functional, with $sing(F(U_0)) = \{0\}$, we provide a foliation of the half-space ${\mathbb R}^{n} \times [0,+\infty)$ with dilations of graphs of global minimizers $\underline U \leq U_0 \leq \bar U$ with analytic free boundaries at distance 1 from the origin.

In this paper we study the following parabolic system \begin{equation*} \Delta {\bf u} -\partial_t {\bf u} =|{\bf u}|^{q-1}{\bf u}\,\chi_{\{ |{\bf u}|>0 \}}, \qquad {\bf u} = (u^1, \cdots , u^m) \ , \end{equation*} with free boundary $\partial \{|{\bf u} | >0\}$. For $0\leq q< 1$, we prove optimal growth rate for solutions ${\bf u} $ to the above system near free boundary points, and show that in a uniform neighbourhood of any a priori well-behaved free boundary point the free boundary is $C^{1, \alpha}$ in space directions and half-Lipschitz in the time direction.

We consider vector-valued solutions to a linear transmission problem, and we prove that Lipschitz-regularity on one phase is transmitted to the next phase. More exactly, given a solution $u:B_1\subset {\mathbb R}^n \to {\mathbb R}^m$ to the elliptic system \begin{equation*} \hbox{div} ((A + (B-A)\chi_D )\nabla u) = 0 \quad \text{in }B_1, \end{equation*} where $A$ and $B$ are Dini continuous, uniformly elliptic matrices, we prove that if $\nabla u \in L^{\infty} (D)$ then $u$ is Lipschitz in $B_{1/2}$. A similar result is also derived for the parabolic counterpart of this problem.

We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from the theory of free boundary problems.

In this paper we prove symmetry for solutions to the semi-linear elliptic equation $$ \Delta u = f(u) \quad \hbox{ in } B_1, \qquad 0 \leq u < M, \quad \hbox{ in } B_1, \qquad u = M, \quad \hbox{ on } \partial B_1, $$ where $M>0$ is a constant, and $B_1$ is the unit ball. Under certain assumptions on the r.h.s. $f (u)$, the $C^1$-regularity of the free boundary $\partial \{u>0\}$ and a second order asymptotic expansion for $u$ at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin's celebrated boundary point lemma, which is not available in our case due to lack of $C^2$-regularity of solutions.