OpenYear of origin: 2020
Posted online: 2019-12-26 21:41:25Z by Henrik Shahgholian
Cite as: P-191226.3
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.
Contrary to convexity problems, the symmetry methods, such as moving plane technique works very well for free boundary value problems for systems, when there is a symmetry in the given equation, the bulk domain, and boundary values. Here, we present the most simple example, leaving several obvious generalizations towards other problems to the reader. The original Saint Venant problem, with an overdetermination of the boundary gradient condition (here expressed as system) is to show that whenever there is a solution vector $\mathbf{u}$ to the following problem \begin{equation}\tag{1} \begin{cases} \Delta \mathbf{u} = - \mathbf{k} & \text{in } \Omega , \\ \mathbf{u} = \mathbf{0} & \text{on } \partial\Omega , \\ |\nabla \mathbf{u}| = 1 & \text{on } \partial\Omega , \\ \end{cases} \end{equation} the domain $\Omega $ has to be a ball. For scalar case, James Serrin [Serrin] gave a very nice proof of this based on moving plane technique (of A. D. Alexandrov).
For system case one may think of solutions being $$ \mathbf{u} = \mathbf{k} (1-\frac{|x|^2}{2n}), \qquad \hbox{with } \Omega = \{|x| < \frac{n}{|\mathbf{k}|}\} . $$ Same technique carries over to the system case above, mutatis mutandis, by reflection principle as long as the boundary is $C^{2,\beta}$ to make Serrin's argument to work. Indeed, using the moving plane technique one makes comparison component-wise along with Hopf's lemma and then adds up the gradients to achieve a contradiction. As this being a graduate exercise we leave out the details to the readers.
It is however more interesting to consider the semi-linear case of Serrin's problem for systems, which is to prove spherical symmetry for the same system as in Saint-Venant problem (1) where one replaces $-\mathbf{k} $ with $\mathbf{f}(\mathbf{u})$, with mild assumptions on $\mathbf{f}$ such as Lipschitz regularity.
Q1) How far can one stretch the conditions on $\mathbf{f}$. Can we allow each component of $\mathbf{f}$ to change sign, or be in $L^1$. This question is yet not answered in the scalar case.
Q2) Can we derive stability theory for system case, as it has been done for scalar case [BNST].
Q3) Try to prove symmetry results for Discrete Bernoulli problem in system case.
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