OpenYear of origin: 2020
Posted online: 2019-12-26 21:41:25Z by Henrik Shahgholian
Cite as: P-191226.2
This is a previous version of the post. You can go to the current version.
We introduce notations for vector-functions and vector-domains as $\mathbf{u}= (u^1,u^2,\cdots ,u^m)$, and $\mathbf{D} = (D_1, D_2, \cdots , D_m)$, with (bounded) $D_j \subset \mathbb R^n$. For simplicity of presentation we shall denote by $\Omega = \{|\mathbf{u}| > 0\}$, and $\mathbf{k} = (k_1, \cdots , k_m)$, with $k_i > 0$.
Consider equations of the type \begin{equation} \tag{1} \begin{cases} \Delta \mathbf{u} = \mathbf{f} (x,\mathbf{u}) &\qquad \hbox{in } \Omega \setminus \mathbf{D} ,\\ \mathbf{u} =\mathbf{ k} &\qquad \hbox{on } \mathbf{D} ,\\ G(\nabla \mathbf{u}) = g(x) &\qquad \hbox{on } \partial \Omega \\ \mathbf{u} \geq \mathbf{0} &\qquad \hbox{in } \mathbb{R}^n . \end{cases} \end{equation} In case $ |\nabla \mathbf{u}| = 0$ on $\partial \Omega$, the last equation is taken out and the first equation is replaced by $\Delta \mathbf{u} = \mathbf{f} (x,\mathbf{u}) $ in $ \mathbf{D}^c$, i.e. \begin{equation}\tag{2} \begin{cases} \Delta \mathbf{u} = \mathbf{f} (x,\mathbf{u}) &\qquad \hbox{in } \mathbf{D}^c ,\\ \mathbf{u} =\mathbf{ k} &\qquad \hbox{on } \mathbf{D} .\\ \end{cases} \end{equation} When $\mathbf{f} = \nabla_{\mathbf{u}} F$ with $F$ "reasonably" smooth, one may find solutions to (2) using minimizers of the functional \begin{equation} J(v) = \int_{\mathbb{R}^n}|\nabla \mathbf{v}|^2+F(x,\mathbf{v}), \end{equation} over $ \{\mathbf{v} \in W_{0}^{1,2}(\mathbb R^n): \mathbf{v} = \mathbf{k} \,\, \text{on} \,\, \mathbf{D} \}$. When $F (x, \nabla \mathbf{u} ) = g^2 (x)\chi_{\{|\mathbf{u}| > 0\}}$ one obtains (1) with $G(\nabla \mathbf{u}) = |\nabla \mathbf{u}|$, which is a Bernoulli type free boundary problem for systems that has been studied in [CSY] \begin{equation}\begin{cases} \Delta \mathbf{u} = 0 & \text{in } \Omega \setminus \mathbf{D}, \\ |\nabla \mathbf{u}| = g(x) & \text{on } \partial\Omega . \\ \end{cases} \end{equation} The particular case of $F(\mathbf{u}) = |\mathbf{u}|$, recently studied in [ASUW], gives rise to the obstacle type problems for systems \begin{equation} \Delta \mathbf{u} = \frac{\mathbf{u}}{|\mathbf{u}|} \chi_{\left\{|\mathbf{u}|>0\right\}}. \end{equation}
Next is to see what kind of domains $\mathbf{D}$ can be of interest to consider. This has several possible scenarios; here are a few:
(i) $D_i =D_j , \forall i,j=1,2,..m $,
(ii) $ D_1\subset D_2\subset...\subset D_m$,
(iii) $ D_1\subset D_2\subset...\subset D_m$, and they are homethetic,
(iv) $\bigcap D_i \neq \emptyset$.
We should also remark that for minimizers $\mathbf{u}$ of the functional $J$ we always have $\{u^i > 0\} = \{u^j>0\}, \,\, \forall i,j,$ provided $D_i \bigcap D_j \neq \emptyset$. This follows from the fact that in such cases we can make variations in both directions (upward and downward), $u^i \mp\epsilon \phi^i$ and $u^j \mp\epsilon \phi^j$. Hence we have the Euler Lagrange equation for both $u^i$ and $u^j$ whenever one of them is non-zero.
Question: The question we want to raise is under what conditions on $ \mathbf{D}$, and equations above we may expect geometric inheritance for solutions. E.g. will starshapness of $ \mathbf{D}$ w.r.t. a point $z \in \mathbf{D}$ imply the same for $\Omega$? Will convexity of all components of $ \mathbf{D}$ imply convexity of $\Omega$? What about spherical symmetry? And many other geometric qualitative properties that can be inherited by the solutions.
The system case of the free boundary for general $F(x, \mathbf{p})$, even if $F$ is independent of $x$, and convex in $\mathbf{p}$-variables, is of course a non-trivial problem. Let us consider the particular case as in
\begin{equation} \Delta \mathbf{u} = \frac{\mathbf{u}}{|\mathbf{u}|} \chi_{\left\{|\mathbf{u}|>0\right\}}. \end{equation} As in the Bernoulli problem, case (i) reduces to scalar cases. In fact since $D_i = D_j$, by setting $D=D_i=D_j$ for $v^i= \frac{u^i}{\lambda_i}$, we have $\Delta \mathbf{v} = \frac{\mathbf{v}}{|\mathbf{u}|}$, and if we let $h^{ij} = v^i-v^j$ then \begin{equation*}\begin{cases} \Delta h^{ij} = \frac{h^{ij}}{|\mathbf{u}|} & \text{in } \Omega\backslash D, \\ h^{ij} = 0 & \text{on } \partial(\Omega\backslash D). \\ \end{cases} \end{equation*} Hence by the maximum principle we have $h^{ij} \equiv 0$ i.e $v^i =v^j$, $\forall i,j,$ and it follows that $u^i = \frac{\lambda_i}{\lambda_j}u^j. $
The less-simple case $D_i \subset D_{i+1}$, $i=1, \cdots , m-1$ does not seem obvious how to approach.
There are other models of system case, that we mention here. These are $$ a) \ \Delta u^i = \left( 1+ \sum_{j\neq i} u^j \right) \chi_{\{|\mathbf{u}| > 0\}} , \qquad b) \ \Delta u^i = \frac{u^i}{ \sum_{j=1}^m u^j} \chi_{\{|\mathbf{u}| > 0\}} , $$ where both of them can be reduced to scalar case by defining $U= \sum_{i=1}^m u^i$, which gives us the following equations $$ c) \ \Delta U = \left( m+ m U \right) \chi_{\{U > 0\}}, \qquad d) \ \Delta U = \chi_{\{U > 0\}}, $$ respectively, where we have used $\chi_{\{|\mathbf{u}| > 0\}} = \chi_{\{U > 0\}}$. Even such a reformulation does not seem to work by earlier methods, unless $D_i = D_j$ for all $i, j$. Indeed in this case the equation becomes scalar with a fixed $D= D_i$, with boundary values $k:=\sum k_i$ on $D$. Since the right hand side is monotone in both cases we can invoke earlier results by [GK], to conclude $U$ has convex level sets. It is however not obvious whether each component will be quasi-convex. This is left to readers to explore.
As a final simple example we mention the following system $$ \Delta u^i = \left( 1 + (m-1) u^i - \sum_{j\neq i} u^j \right) \chi_{\{|\mathbf{u}| > 0\}} , $$ and with $U = \sum u^i$ we have $\Delta U = m \chi_{\{U > 0\}}$, with $D_i =D_j $ for all $i, j$, and the corresponding boundary value. This is the obstacle problem and it is well-known that the unique solution to this problem is quasi-convex.
For the above problems, with $D_i \subset D_{i+1}$, the question of quasi-convexity of each component, or any combination of the components, or even the convexity of the support $\Omega = \{|\mathbf{u}|> 0\}$ remains a tantalizing question.
We know of no quasi-convexity results (or even problems) for systems in existing literature. Even the most simple equation such as $$ \begin{cases} \Delta u = ( 1 + 2v ) \chi_{ \{u > 0\} } , &\qquad \hbox{in } D^c, \\ \Delta v = ( 1 + u )\chi_{\{v > 0\}} &\qquad \hbox{in } D^c, \\ u= 1 &\qquad \hbox{on } \partial D ,\\ v= 2 &\qquad \hbox{on } \partial D , \end{cases} $$ does not seem to be easy to handle with existing methods for convexity. Here we have chosen the equations such that it cannot be reduced to a scalar case. A major problem is that solutions in exterior domains become quasi-convex and not convex. This means that any existing method would imply the right hand side of each equation above is a quasi-convex function and this is not enough to force through the existing methods, as they require the r.h.s. to be convex functions. This may also be an indication that what we ask for is not true, but we have neither a counterexample.
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