Nina Nikolaevna Uraltseva

The first three decades of Nina Uraltseva’s mathematical career were devoted to the theory of linear and quasilinear PDEs of elliptic and parabolic type. Her first round of works in the 1960s, mostly in collaboration with Olga Ladyzhenskaya, was related to Hilbert’s 19th and 20th problem on the existence and regularity of the minimizers of the energy integrals

$$I(u)=\int_{\Omega}F(x,u,\nabla u)dx,$$

where $F(x,u,p)$ is a smooth function of its arguments and $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq 2$. In her Candidate of Science thesis, based on work [49],⁵⁵5In those years, it was quite unusual to base the Candidate of Science thesis on just a single paper and some of the committee members voiced their concerns. However, Olga Ladyzhenskaya objected decisively that it depends on the quality of the paper. Nina Uraltseva has shown that under the assumption that $F$ is $C^{2,\alpha}$ and satisfies the uniform ellipticity condition

$$F_{p_{i}p_{j}}\xi_{i}\xi_{j}\geq m|\xi|^{2},\quad m>0,$$

the minimizers $u$ are $C^{2,\alpha}$ locally in $\Omega$ (i.e., on compact subdomains of $\Omega$), provided they are Lipschitz. (It has to be mentioned here that the Lipschitz regularity of the minimizers was known from the earlier works of Ladyzhenskaya under natural growth condition on $F$ and its partial derivatives.) Uraltseva has also shown that $C^{2,\alpha}$ regularity extends up to the boundary $\partial\Omega$ under the natural requirement that both $\partial\Omega$ and $u|_{\partial\Omega}$ are $C^{2,\alpha}$. This generalized the results of Morrey in dimension $n=2$ to higher dimensions.

Uraltseva’s proof was based on a deep extension of the ideas of De Giorgi for the solutions of uniformly elliptic equations in divergence form with bounded measurable coefficients, which were applicable only to the integrands of the form $F(\nabla u)$. In particular, one of the essential steps was to establish that $v=\pm u_{x_{i}}$, $i=1,\dots,n$, which are assumed to be bounded, satisfy the energy inequalities

$$\int_{A_{k,\rho}}|\nabla v|^{2}\zeta^{2}\leq C\int_{A_{k,\rho}}(v-k)^{2}|\nabla\zeta|^{2}+C|A_{k,\rho}|$$

(1)

for all $|k|\leq M$, where $A_{k,\rho}$ is intersections of $\{v>k\}$ with the ball $B_{\rho}(x^{0})\Subset\Omega$, $\zeta$ is a cutoff function, and $M$ is a bound for $\max|\nabla u|$.

Using similar ideas, Uraltseva was able to deduce the existence and regularity of solutions for the class of quasilinear uniformly elliptic equations in divergence form,

$$\partial_{x_{i}}(a_{i}(x,u,\nabla u))+a(x,u,\nabla u)=0,$$

(2)

under natural growth conditions on $a_{i}(x,u,p)$, $a$ and some of their partial derivatives, which were mainly needed for proving the bounds on $\max|\nabla u|$. These results were further refined in the joint works with Olga Ladyzhenskaya [19, 20, 21, 26, 22] as well as in [50], for the case of Neumann-type boundary conditions. The latter paper also contained similar results for certain quasilinear diagonal systems (important, e.g., for the applications in harmonic maps).

Quasilinear uniformly elliptic equations in nondivergence form,

$$a_{ij}(x,u,\nabla u)\,u_{x_{i}x_{j}}+a(x,u,\nabla u)=0,$$

(3)

were trickier to treat, but already in [49], Uraltseva found a key: quadratic growth of $a(x,u,p)$ in the $p$-variable,

$$|a(x,u,p)|\leq\mu(1+|p|)^{2},$$

along with the corresponding conditions on the partial derivatives of $a$ and $a_{ij}$ in their variables. In [51], Uraltseva proved $C^{1,\alpha}$ a priori bounds for solutions of (3), as wells as for diagonal systems of similar type.

The results in the elliptic case were further extended to the parabolic case (including systems) in a series of works of Ladyzheskaya and Uraltseva [24, 23].

This extensive research, that went far beyond the original scope of Hilbert’s 19th and 20th problems, was summarized in two monographs, Linear and Quasilinear Equations of Elliptic Type [25] (substantially enhanced in the 2nd edition [28]) and Linear and Quasilinear Equations of Parabolic Type [17], written in collaboration with Vsevolod Solonnikov; see Figure 3. The monographs became instant classics and were translated to English [30, 18] and other languages and have been extensively used for generations of mathematicians working in elliptic and parabolic PDEs and remain so to this date.

In a series of papers in 1979–1985, summarized in her talk at the International Congress of Mathematicians in Berkeley, CA, 1986 [61] and a survey paper with Ladyzhenskaya [29], Uraltseva and collaborators have studied uniformly elliptic quasilinear equations of nondivergence type (3) and their parabolic counterparts, when $a$ and the first derivatives of $a_{ij}$ are possibly unbounded. The typical conditions read

$$|a(x,u,p)|\leq\mu|p|^{2}+b(x)|p|+\Phi(x),$$

where $\mu$ is a constant and $b,\Phi\in L^{q}(\Omega)$, $q>n$. Uraltseva and collaborators were able to establish the existence and up to the boundary $C^{1,\alpha}$ regularity of $W^{2,n}$ strong solutions of the problem, vanishing on $\partial\Omega$ (provided the latter is sufficiently regular). The proofs were based on the extension of methods of Ladyzhenskaya and Uraltseva already in their books [30, 18], as well as those of Krylov and Safonov using the Aleksandrov-Bakelman-Pucci (ABP) estimate, in the elliptic case, and a parabolic version of the ABP estimate due to Nazarov and Uraltseva [36], in the parabolic case.

Most recent results of Nina Uraltseva in this direction are in the joint work with Alexander Nazarov [37] on the linear equations in divergence form,

$$\partial_{x_{i}}(a_{ij}(x)u_{x_{j}})+b_{i}(x)u_{x_{i}}=0\quad\text{in }\Omega,$$

and their parabolic counterparts. Their goal was to find conditions on the lower order coefficients $\mathbf{b}=(b_{1},\dots,b_{n})$ that guarantee the validity of classical results such as the strong maximum principle, Harnack’s inequality, and Liouville’s theorem. It was shown by Trudinger [47] that such results hold when $\mathbf{b}\in L^{q}$, $q>n$. Motivated by applications in fluid dynamics, in one of their theorems, Nazarov and Uraltseva showed that under the additional assumption,

$$\operatorname{div}\mathbf{b}=0,$$

(4)

the condition on $\mathbf{b}$ can be relaxed to being in the Morrey space

$$\sup_{B_{r}(x^{0})\subset\Omega}r^{q-n}\int_{B_{r}(x^{0})}|\mathbf{b}|^{q}<\infty$$

for some $n/2<q\leq n$. In the borderline case $q=n$, the Morrey space above is locally the same as $L^{n}$. Remarkably, in that case the divergence free condition (4) on $\mathbf{b}$ can be dropped when $n\geq 3$, i.e., $\mathbf{b}\in L^{n}_{\mathrm{loc}}$ alone is sufficient to have the classical theorems; moreover, this result is optimal. In dimension $n=2$, to drop (4) one needs a stronger condition $\mathbf{b}\ln^{1/2}(1+|\mathbf{b}|)\in L^{2}_{\mathrm{loc}}$.

Nina Uraltseva has also made a pioneering work on the regularity theory for degenerate quasilinear equations. A particular result in this direction is her 1968 proof [52] of the $C^{1,\alpha}$ regularity of $p$-harmonic functions, $p>2$, which are the weak solutions of the $p$-Laplace equation

$$\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0\quad\text{in }\Omega,$$

(5)

or, equivalently, are the minimizers of the energy functional

$$\int_{\Omega}|\nabla u|^{p}dx.$$

The difficulty here lies in the fact that the $p$-Laplace equation (5) degenerates at the points where the gradient vanishes and that the solutions are not generally twice differentiable in the Sobolev sense. As stated in her paper, this problem was posed to Nina Uraltseva by Yurii Reshetnyak in relation with the study of quasiconformal mappings in higher dimensions.

Uraltseva has obtained the $C^{1,\alpha}$ regularity of $p$-harmonic functions as an application of the Hölder regularity of the solutions of the degenerate quasilinear diagonal systems

$$\partial_{x_{i}}(a_{ij}(x,\mathbf{u})\,\mathbf{u}_{x_{j}})=\mathbf{0},$$

with scalar coefficients $a_{ij}$ satisfying the degenerate ellipticity condition

$$\nu(|\mathbf{u}|)|\xi|^{2}\leq a_{ij}(x,\mathbf{u})\xi_{i}\xi_{j}\leq\mu\nu(|\mathbf{u}|)|\xi|^{2},$$

with $\mu\geq 1$ and a nonnegative increasing function $\nu(\tau)$ satisfying $\nu(\lambda\tau)\leq\lambda^{s}\nu(\tau)$ for $\lambda\geq 1$ and $s>0$.

Unfortunately, despite the utmost importance of this result, Nina Uraltseva’s proof remained unknown outside of the Soviet Union. In 1977, nine years later, it was independently reproved by Karen Uhlenbeck [48]. Other proofs were given by Craig Evans [14], John Lewis [31], who extended the range of exponents to $1<p\leq 2$, Di Benedetto [13] and Tolksdorf [46], who both extended it to the case of general degenerate quasilinear equations in divergence form.

Another work in this area that has gained the status of classic is the paper of Nina Uraltseva and Anarkul Urdaletova [64], where they proved uniform gradient estimates for bounded solutions of anisotropic degenerate equations,

$$\partial_{x_{i}}(a_{i}(x,u_{x_{i}}))+a(x,u,\nabla u)=0\quad\text{in }\Omega,$$

under ellipticity, growth, and monotonicity conditions on the coefficients. Their results were applicable to the minimizers of the energy functional

$$\int_{\Omega}\sum_{i=1}^{n}|u_{x_{i}}|^{m_{i}}+f(x,u),$$

with the exponents $m_{1},\dots,m_{n}$ satisfying $m_{i}>3$, $2m_{i}>m_{0}$, $i=1,\dots,n$, $m_{0}=\max\{m_{i}\}$, under the monotonicity condition $f_{u}(x,u)\geq 0$. This was the very first paper to prove regularity results for degenerate quasilinear equations with nonstandard growth, which appeared first in the 1980s, motivated by applications in elasticity and material science, and continue to be the subject of extensive research today [35]. Major contributions in this direction have been made by Paolo Marcellini [32, 33, 34] and many others.

In [27], Ladyzhenskaya and Uraltseva developed a method of local a priori estimates for nonuniformly elliptic and parabolic equations, including the equations of minimal surface type,

$$\operatorname{div}\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}=a(x,u,\nabla u)\quad\text{in }\Omega.$$

A particular case with $a(x,u,\nabla u)=\kappa u$, $\kappa>0$, together with the Neumann-type condition ${\partial_{\nu}u}/{\sqrt{1+|\nabla u|^{2}}}=\varkappa$ on $\partial\Omega$, $|\varkappa|<1$, is known as the capillarity problem. The boundary estimates, as well as the existence of classical solutions for such problems were obtained in [55, 56, 58, 53]. Remarkably, the estimates in the last two papers required only the smoothness of the domain $\Omega$, but not its convexity.

In the 1990s, in a series of joint works [40, 41, 38, 39] with Vladimir Oliker, Nina Uraltseva studied the evolution of surfaces $S(t)$ given as graphs $u=u(x,t)$ over a bounded domain $\Omega\subset\mathbb{R}^{n}$ with the speed depending on the mean curvature of $S(t)$ under the condition that the boundary of the surface $S(t)$ is fixed. More precisely, they considered a parabolic PDE of the type

$$u_{t}=\sqrt{1+|\nabla u|^{2}}\,\operatorname{div}\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\quad\text{in }\Omega\times(0,\infty)$$

with the boundary condition $u(x,t)=\phi(x)$ on $\partial\Omega\times(0,\infty)$ and initial condition $u(x,0)=u_{0}(x)$. Even in the stationary case, when this problem is the Dirichlet problem for the mean curvature equation, the existence of up to the boundary classical solutions requires a geometric condition on the domain $\Omega$, namely, the nonnegativity of the mean curvature of $\partial\Omega$. For such domains, Huisken [15] has shown the existence of the classical solutions of the evolution problem and proved that the surfaces $S(t)$ converge to a classical minimal surface $S$ as $t\to\infty$. Oliker and Uraltseva have studied this problem with no geometric conditions on the domain $\Omega$. For this purpose, they introduced a notion of a generalized solution to the parabolic problem (as a limit of regularized problems). They have proved its existence and convergence $u(\cdot,t)\to\Phi$ as $t\to\infty$ to a generalized solution $\Phi$ of the stationary problem, in the sense that $\Phi$ minimizes the area functional

$$\int_{\Omega}\sqrt{1+|\nabla u|^{2}}+\int_{\partial\Omega}|u-\phi|$$

among all competitors in $W^{1,1}(\Omega)$. Such minimizer $\Phi$ is unique, but may differ from the Dirichlet data $\phi$ on the “bad” part of the boundary where the mean curvature is negative. The study of the behavior of the minimizer near the “contact points” on the boundary where $\Phi|_{\partial\Omega}$ “detaches” from $\phi$ later served as one of Uraltseva’s motivations for studying the touch between free and fixed boundaries; see Section 4.1.

Let $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$, be a bounded domain with a smooth boundary and $S$ a relatively open nonempty subset of $\partial\Omega$. Suppose we are also given two functions $\psi,g\in W^{1,2}(\Omega)$ satisfying $g\geq\psi$ on $S$ (in the sense of traces). Consider then a closed convex subset $\mathfrak{K}\subset W^{1,2}(\Omega)$ defined by

$$\mathfrak{K}\coloneq\{v\in W^{1,2}(\Omega):\text{$v\geq\psi$ on $S$, $v=g$ on $\partial\Omega\setminus S$}\}.$$

In other words, $\mathfrak{K}$ consists of functions that need to stay above $\psi$, called a boundary (or thin) obstacle, on $S$ and equal to $g$ on $\partial\Omega\setminus S$. Then, one wants to find $u\in\mathfrak{K}$ that minimizes the generalized Dirichlet energy

$$J(v)=\int_{\Omega}a_{ij}(x)v_{x_{j}}v_{x_{i}}+2f(x)u,$$

where $a_{ij}(x)$ are uniformly elliptic coefficients and $f$ is a certain function. Equivalently, the minimizer $u$ satisfies the variational inequality

$$u\in\mathfrak{K},\quad\int_{\Omega}a_{ij}(x)u_{x_{j}}(v-u)_{x_{i}}\\ +f(x)(v-u)\geq 0\quad\text{for any }v\in\mathfrak{K}.$$

In turn, it is equivalent to the following boundary value problem

	$\displaystyle\partial_{x_{i}}(a_{ij}(x)u_{x_{j}})=f(x)$	$\displaystyle\quad\text{in }\Omega,$
	$\displaystyle u=g$	$\displaystyle\quad\text{on }\partial\Omega\setminus S,$
	$\displaystyle u\geq\psi,\ \partial_{\nu}^{A}u\geq 0,\ (u-\psi)\partial_{\nu}^{A}u=0$	$\displaystyle\quad\text{on }S,$

to be understood in the appropriate weak sense, where $\partial_{\nu}^{A}u\coloneq a_{ij}(x)\nu_{j}u_{x_{j}}$ is the conormal derivative of $u$ on $\partial\Omega$, with $\nu=(\nu_{1},\dots,\nu_{n})$ being the outward unit normal. The conditions on $S$ are known as the Signorini complementarity conditions and are remarkable because they imply that

$$\text{either $u=\psi$ or $\partial_{\nu}^{A}u=0$ on $S$},$$

yet the exact sets where the first or the second equality holds are unknown. The interface $\Gamma$ between these sets in $S$ is called free boundary (see Figure 4). The study of the free boundary is one of the main objectives in such problems (see Section 4 for Uraltseva’s contributions in that direction), yet the regularity of the solutions $u$ is a challenging problem by itself and is often an important step towards the study of the free boundary.

One of the theorems of Nina Uraltseva [60, 59] states that when

	$\displaystyle a_{ij}$	$\displaystyle\in W^{1,q}(\Omega),$
	$\displaystyle\psi$	$\displaystyle\in W^{1,2}(\Omega)\cap W^{2,q}_{\mathrm{loc}}(\Omega\cup S),$
	$\displaystyle f$	$\displaystyle\in L^{q}(\Omega),$

for some $q>n$, then

$$u\in C^{1,\alpha}_{\mathrm{loc}}(\Omega\cup S),$$

with a universal exponent $\alpha\in(0,1)$. Prior to this result, similar conclusion was known only under higher regularity assumptions on the coefficients and the obstacle in the works of Caffarelli [12] and Kinderlehrer [16]. The lower regularity assumptions in Uraltseva’s result, particularly on the obstacle $\psi$, were instrumental in Schumann’s proof of the corresponding result in the vectorial case [43]. The parabolic counterpart of Uraltseva’s theorem, with similar assumptions on the coefficients and the obstacle was established later in a joint work of Arkhipova and Uraltseva [7].

The idea of Uraltseva’s proof is based on an interplay between De Giorgi-type energy inequalities and the Signorini complementarity condition. Locally, near $x^{0}\in S$, one can assume that $S=\{x_{n}=0\}$ and $\psi=0$. First, working with the regularized problem, one can establish that for any partial derivative $v=\pm u_{x_{i}}$, $i=1,\dots,n$, there holds an energy inequality (similar to (1) in the unconstrained case)

$$\int_{A_{k,\rho}}|\nabla v|^{2}\zeta^{2}\leq C\int_{A_{k,\rho}}(v-k)^{2}|\nabla\zeta|^{2}+C_{0}|A_{k,\rho}|^{1-2/q}$$

for any $k>0$, $0<\rho<\rho_{0}$, and a cutoff function $\zeta$ in $B_{\rho}(x^{0})$, where $A_{k,\rho}=\{v>k\}\cap B_{\rho}(x^{0})\cap\Omega$. Next, one observes that as a consequence of the Signorini complementarity conditions, one has

$$u_{x_{i}}u_{x_{n}}=0\quad\text{on }\{x_{n}=0\}\cap B_{\rho}(x_{0})$$

for all $i=1,\dots,n-1$ and hence either the normal derivative $v=u_{x_{n}}$ or all tangential derivatives $v=u_{x_{i}}$, $i=1,\dots,n-1$, vanish at least on half of $\{x_{n}=0\}\cap B_{\rho}(x^{0})$ (by measure). This allows to apply Poincare’s inequality in one of the steps and obtain a geometric improvement of the Dirichlet energy for $v$ going from radius $\rho$ to $\rho/2$. By iteration, this gives that either

	$\displaystyle\sum_{i=1}^{n-1}\int_{\Omega\cap B_{\rho}(x^{0})}|\nabla u_{x_{i}}|^{2}$	$\displaystyle\leq C\rho^{n-2+2\alpha}\quad\text{or}$		(6)
	$\displaystyle\int_{\Omega\cap B_{\rho}(x^{0})}|\nabla u_{x_{n}}|^{2}$	$\displaystyle\leq C\rho^{n-2+2\alpha}$		(7)

holds, with $C$ depending on the distance from $x^{0}$ to $\partial\Omega\setminus S$. However, using the PDE satisfied by $u$, it is easy to see that (6) implies (7), and hence (7) always holds. From there, the $C^{1,\alpha}$ regularity of $u$ follows by standard results for the solutions of the Neumann problem.

The results described above were extended, in joint works with Arina Arkhipova [9, 11], to the problem with two obstacles $\psi_{-}\leq\psi_{+}$ on $S$, that corresponds to the constraint set

$$\mathfrak{K}=\{v\in W^{1,2}(\Omega):\text{$\psi_{-}\leq u\leq\psi_{+}$ on $S$},\\ \text{$u=g$ on $\partial\Omega\setminus S$}\}.$$

While substantial difficulties arise near the set where $\psi_{-}=\psi_{+}$, the results are as strong as in the case of a single obstacle. In their further work, Arkhipova and Uraltseva [8, 10] studied related problems for quasilinear elliptic systems with diagonal principal part. To describe their results, let $V=W^{1,2}(\Omega;\mathbb{R}^{N})\cap L^{\infty}(\Omega;\mathbb{R}^{N})$ and

$$\mathfrak{K}=\{\mathbf{u}\in V:\text{$\mathbf{u}(x)\in K(x)$ for every $x\in\partial\Omega$}\},$$

where $K(x)$ are given convex subset of $\mathbb{R}^{N}$ for every $x\in\partial\Omega$. Then consider the variational inequality of the type

$$\mathbf{u}\in\mathfrak{K},\quad\int_{\Omega}\left(a_{ij}(x,\mathbf{u})\mathbf{u}_{x_{j}}+\mathbf{b}_{i}(x,\mathbf{u})\right)(\mathbf{v}-\mathbf{u})_{x_{i}}\\ +\mathbf{f}(x,\mathbf{u},\nabla\mathbf{u})(\mathbf{v}-\mathbf{u})\geq\mathbf{0}\\ \text{for any }\mathbf{v}\in\mathfrak{K},$$

where $a_{ij}$ are scalar uniformly elliptic coefficients, $\mathbf{b}_{i}$ and $\mathbf{f}$ are $N$-dimensional vector functions and $\mathbf{f}(x,\mathbf{u},\mathbf{p})$ grows at most quadratically in $\mathbf{p}$. We note that the problem with two obstacles $\psi_{-}\leq\psi_{+}$ on $\partial\Omega$ fits into this framework with $N=1$ and $K(x)=[\psi_{-}(x),\psi_{+}(x)]$. Assume now that the convex sets $K(x)$ are of the form

$$K(x)=T(x)K_{0}+\mathbf{g}(x),$$

where $K_{0}$ is a convex set in $\mathbb{R}^{N}$ with a nonempty interior and a smooth ($C^{2}$) boundary, $T(x)$ is an orthogonal $N\times N$ matrix, and $\mathbf{g}(x)$ is an $N$-dimensional vector. A theorem of Arkhipova and Uraltseva [10] then states that when the entries of $T$ and $\mathbf{g}$ are extended to $W^{2,q}$ functions in $\Omega$, $q>n$, $a_{ij}(\cdot,\mathbf{u})$ and $\mathbf{b}_{i}(\cdot,\mathbf{u})$ are in $W^{1,q}(\Omega)$, uniformly in $\mathbf{u}$ and have at most linear growth in $\mathbf{u}$, and $\mathbf{f}$ has at most quadratic growth in $\mathbf{p}$, then

$$\mathbf{u}\in C^{1,\alpha}_{\text{loc}}(\Omega\cup S),$$

provided $\mathbf{u}$ is Hölder continuous in $\Omega\cup S$. The Hölder continuity assumption on $\mathbf{u}$ can be replaced by a bound on the oscillation in $\Omega$ and a local uniqueness of the solutions, which is also necessary for the continuity of the solutions of the nonlinear systems of the type

$$\partial_{x_{i}}(a_{ij}(x,\mathbf{u})\mathbf{u}_{x_{j}}+\mathbf{b}_{i}(x,\mathbf{u}))+\mathbf{f}(x,\mathbf{u},\nabla\mathbf{u})=\mathbf{0}\quad\text{in }\Omega.$$

For a more complete overview of Uraltseva’s results on variational inequalities, we refer to her own survey paper [62].

In [4] (joint with one of us) and her follow-up paper [63], Uraltseva studied the obstacle problem close to a Dirichlet data, for smooth boundaries, where she proves that the free boundary touches the fixed boundary tangentially. The idea seemed to be inspired by related works with Oliker (see Section 2.2) and the Dam-problem in filtration.

During the potential theory program at Institute Mittag-Leffler (1999–2000) she started working on free boundary problems that originated in potential theory. Specifically, the harmonic continuation problem in potential theory, that was strongly tied to obstacle problem, but with the lack of having a sign for the solution function. The simplest way to formulate this problem is as follows:

		$\displaystyle\Delta u=\chi_{\Omega(u)}\quad\text{in }B_{1}^{+},$		(8)
		$\displaystyle\qquad\qquad\text{with }\Omega(u)\coloneq\{u=|\nabla u|=0\}^{c}$
		$\displaystyle u=0\quad\text{on }\Pi\cap B_{1},$

where $B_{1}^{+}=\{|x|<1,\ x_{1}>0\}$ and $\Pi=\{x_{1}=0\}$; see Figure 5. The question of interest was the behavior of the free boundary $\Gamma=\partial\Omega(u)$ close to the fixed boundary $\Pi$.

In [3], and several follow-up papers in the parabolic regime, she shows that the free boundary $\Gamma$ is a graph of a $C^{1}$-function close to points on $\Pi$, where $\Gamma\cap B_{1}^{+}$ touches $\Pi$, or comes too close to $\Pi$.

To prove this, and the related parabolic results, there was a need for developing new tools and approaches. This was possible partly due to the availability of monotonicity formulas, such as that of Alt, Caffarelli, and Friedman [1]. One version of the latter asserts that for continuous subharmonic functions $h_{1},$ $h_{2}$ in $B_{R}(x^{0})$, satisfying $h_{1}h_{2}=0$, and $h_{1}(x^{0})=h_{2}(x^{0})=0$, we have $\varphi(r)\nearrow$ for $0<r<R$, where

$$\varphi(r)=\phi(r,h_{1},x^{0})\,\phi(r,h_{2},x^{0})$$

(9)

with

$$\phi(r,h_{i},x^{0})\coloneq\frac{1}{r^{2}}\int_{B(x^{0},r)}\frac{|\nabla h_{i}|^{2}dx}{|x-x^{0}|^{n-2}}.$$

One can use the monotonicity of the function $\varphi(r)$ to prove several important properties for $u$ and the free boundary. Indeed, one first extends $u$ to be zero in $B_{1}^{-}=\{|x|<1,\ x_{1}<0\}$ and applies the monotonicity formula (9) to $h_{1}=(\partial_{e}u)^{+}$ and $h_{2}=(\partial_{e}u)^{-}$, where $e$ is any vector tangent to the plane $\{x_{1}=0\}$. Using the fact that at least one of the sets $\{\pm\partial_{e}u>0\}$ has positive volume density at $x^{0}$, we shall have

$$c_{0}|\nabla\partial_{e}u(x^{0})|^{4}=\lim_{r\to 0}\varphi(r)\leq\varphi(1)\leq C_{0}.$$

Combining this with equation (8) we obtain the bound for $u_{x_{1}x_{1}}(x^{0})$. From here, the uniform $C^{1,1}$ regularity for $u$ in $B_{1/2}^{+}$ follows.

The $C^{1,1}$ regularity is instrumental for any analysis of the properties of the free boundary. Indeed, to study the free boundary at points where it touches the fixed boundary, one needs to rescale the solution quadratically, $u_{r}(x)=u(rx+x^{0})/r^{2}$, which keeps the equation invariant. Indeed, this scaling and “blow-up”⁶⁶6Blow-up refers to $\lim_{r\to 0}u_{r}(x)$, whenever it exists. brings one to a global setting of equation (8) in $\mathbb{R}^{n}_{+}$, where solutions can be classified (in a rotated system) as one of the following:

• (i)

  $u(x)=\frac{1}{2}x_{1}^{2}+ax_{1}x_{2}+\alpha x_{1}$ $(a>0,\ \alpha\in\mathbb{R})$,

• (ii)

  $u(x)=\frac{1}{2}((x_{1}-a)^{+})^{2}$  $(a>0)$.

The proof of the classification of global solutions uses an array of geometric tools and the monotonicity function $\varphi(r)$, implying that if $\{u=0\}\cap\{x_{1}>0\}\neq\emptyset$, then $\partial_{e}u\equiv 0$, for any direction $e$ tangential to $\Pi$. The case when this set is empty is easily handled by Liouville’s theorem.

Once this classification is done, one can argue by indirect methods that the free boundary $\partial\{u>0\}\cap\{x_{1}>0\}$ approaches the fixed one, at touching points, tangentially, and that it is a $C^{1}$-graph locally, which is optimal in the sense that (in general) it cannot be $C^{1,{\rm Dini}}$.

If one considers extension of equation (8) into $B_{1}$, by an odd reflection, then one obtains a specific example of a general problem that is referred to as two-phase obstacle problem, and is formulated as

$$\Delta u=\lambda_{+}\chi_{\{u>0\}}-\lambda_{-}\chi_{\{u<0\}}\quad\text{in }B_{1}(0),$$

where $\lambda_{\pm}$ are positive bounded Lipschitz functions. Figure 6 illustrates this problem.

In [45], Nina Uraltseva (with co-authors) proves that at any branch point $x^{0}\in\partial\{u>0\}\cap\partial\{u<0\}$ with $u(x^{0})=|\nabla u(x^{0})|=0$, the free boundaries $\partial\{u>0\}\cap B_{r_{0}}(0)$ and $\partial\{u<0\}\cap B_{r_{0}}(0)$ are $C^{1}$-surfaces, that touch each other tangentially at $x^{0}$.

The proof of this and several similar results (also in parabolic setting) relies heavily on the monotonicity function $\varphi$ mentioned above as well as on the balanced energy functional

$$\Phi_{x_{0}}(r)\coloneq r^{-n-2}\int_{B_{r}(x_{0})}\left({|\nabla u|}^{2}+\lambda_{+}u^{+}+\lambda_{-}u^{-}\right)\\ -2r^{-n-3}\int_{\partial B_{r}(x_{0})}u^{2},$$

(10)

which is strictly monotone in $r$, unless $u$ is homogeneous. Using these two monotonicity functionals in combination with geometric tools bring us to the fact that any global solution $u_{0}$ to the two-phase problem is one-dimensional and, in a rotated and translated system of coordinates,

$$u_{0}=\frac{\lambda_{+}}{2}(x_{1}^{+})^{2}-\frac{\lambda_{-}}{2}(x_{1}^{-})^{2}.$$

From here one uses a revised form of the so-called directional monotonicity argument of Luis Caffarelli, that in this setting boils down to the fact that close to branch points $x^{0}$ one can show that in a suitable cone of directions $\mathcal{C}$ one has $\partial_{e}u\geq 0$, in $B_{r}(x^{0})$ for $e\in\mathcal{C}$ and $r$ universal. This in particular implies that the free boundaries $\partial\{\pm u>0\}$ are Lipschitz graphs locally close to branch points.

The approaches here generated further application of the techniques to problems with hysteresis [6, 5].

In her work with coauthors [2], Uraltseva considers the following vectorial energy minimizing functional:

$$E(\mathbf{u})\coloneq\int_{B_{1}}\left(|\nabla\mathbf{u}|^{2}+2|\mathbf{u}|\right)dx.$$

Here $B_{1}$ is the unit ball in $\mathbb{R}^{n}$ ($n\geq 1$), and we minimize over the Sobolev space $\mathbf{g}+W_{0}^{1,2}(B_{1};\mathbb{R}^{N})$ for some smooth boundary values $\mathbf{g}=(g_{1},\dots,g_{N})$. The minimizer(s) are vector-valued functions $\mathbf{u}=(u_{1},\dots,u_{N})$, with components $u_{i}$ satisfying

$$\Delta u_{i}=\frac{u_{i}}{|\mathbf{u}|},\quad i=1,\dots,N.$$

Since the set $\{|\mathbf{u}|>0\}$ competes with the Dirichlet energy, by taking the boundary values small we may obtain $\{\mathbf{u}=\mathbf{0}\}\neq\emptyset$, which is in contrast to standard variational problems. The set $\partial\{|\mathbf{u}|>0\}$ is called the free boundary. One observes that when $N=1$ (scalar case) then we fall back to the two-phase problem.

Simple examples of solutions to this problem are

• (i)

  $u_{i}=\alpha_{i}P(x)$, with $P(x)\geq 0$, $\Delta P(x)=1$, and $\sum_{i=1}^{N}\alpha_{i}^{2}=1$,

• (ii)

  $u_{i}=\frac{\alpha_{i}}{2}(x_{1}^{+})^{2}+\frac{\beta_{i}}{2}(x_{1}^{-})^{2}$  (2-phase),
  $\sum_{i=1}^{N}\alpha_{i}^{2}=1$, $\sum_{i=1}^{N}\beta_{i}^{2}=1$,

• (iii)

  $u_{i}=\frac{\alpha_{i}}{2}(x_{1}^{+})^{2}$  (1-phase),
  $\sum_{i=1}^{N}\alpha_{i}^{2}=1$.

Using the vectorial version of the monotonicity formula (10), one can show that $\mathbf{u}$ has a quadratic growth away from the free boundary.

The regularity of the free boundary follows through the homogeneity improvement approach with the so-called epiperimetric inequality, which is used to show that the functional

$$\mathcal{M}(\mathbf{v})\coloneq\int_{B_{1}(0)}(|\nabla\mathbf{v}|^{2}+2|\mathbf{v}|)-\int_{\partial B_{1}(0)}|\mathbf{v}|^{2}$$

satisfies

$$|\mathcal{M}(\mathbf{u}_{r_{1}})-\mathcal{M}(\mathbf{u}_{r_{2}})|\leq c|r_{2}-r_{1}|^{\alpha},\quad\alpha>0,$$

where $\mathbf{u}_{r}=\mathbf{u}(x^{0}+rx)/r^{2}$, and $x^{0}$ is such that $\mathbf{u}_{r}$ is close to rotated version of a half-space solution of type $\mathbf{h}=\frac{1}{2}(x_{1}^{+})^{2}\mathbf{e}$.

This, in particular, gives uniqueness of the blow-ups, and can be used to show that (in a rotated system of coordinates) there exist $\beta^{\prime}>0$, $r_{0}>0$, and $C<\infty$ such that

$$\int_{\partial B_{1}(0)}\left|\mathbf{u}_{r}-\mathbf{h}\right|\leq Cr^{\beta^{\prime}}\\ \text{for every }x^{0}\in\mathcal{R}_{\mathbf{u}}\text{ and every }r\leq r_{0},$$

where $\mathcal{R}_{\mathbf{u}}$ is the set of free boundary points whose blow-ups are half-spaces. This implies that $\mathcal{R}_{\mathbf{u}}$ is locally in $B_{1/2}$ a $C^{1,\beta}$-surface.