OpenYear of origin: 2020
Posted online: 2019-12-26 21:41:25Z by Henrik Shahgholian
Cite as: P-191226.1
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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.
Convexity results for the Bernoulli type problems, represented in the system of equations \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} seems for now out of reach, at least with the methods we know of. It is also not straightforward what conditions we should impose on the domains $\mathbf{D}$ (besides being convex and maybe $D_i$'s being homothetic).
For $F(x,\mathbf{v}) = \chi _{\{|\mathbf{v}|>0\}}$ in case (i), i.e., when $D=D_1=D_2=...$, one may reduce the problem to the scalar case by letting $v^i= \frac{u^i}{\lambda_i}$ and $\Omega = \{|\mathbf{v}| >0\}$, to arrive at \begin{equation} \begin{cases} \Delta \mathbf{v} = 0 & \text{in } \Omega\backslash \mathbf{D}, \\ \mathbf{v} = 0 & \text{on } \partial\Omega , \\ \mathbf{v} = 1 & \text{on } \partial \mathbf{D} ,\\ \sum \lambda_i^2 |\nabla v^i |^2 = g^2 & \text{on } \partial\Omega . \\ \end{cases} \end{equation} where equations and boundary values are component-wise. From the first three equations it follows that $v^i =v^j$, $\forall i,j,$ and hence the last condition turns to $( \sum \lambda_i^2 ) |\nabla v^1|^2 = g^2$. Therefore this case reduces to the scalar case.
The main question concerning the Bernoulli problem is the following: Let $\mathbf{D} = (D_1, \cdots , D_m)$ with $D_i \subset D_{i+1}$, and each $D_i$ convex. Is there a unique solution $\mathbf{u}$, with $\Omega = \{ |\mathbf{u}| > 0\}$, to the Bernoulli free boundary problem \begin{equation} \begin{cases} \Delta \mathbf{u} = 0 & \text{in } \Omega\backslash \mathbf{D}, \\ \mathbf{u} = 0 & \text{on } \partial\Omega , \\ \mathbf{u} = 1 & \text{on } \partial \mathbf{D} , \\ |\nabla \mathbf{u}| = 1 & \text{on } \partial\Omega . \\ \end{cases} \end{equation} with the set $\{|\mathbf{u}|>0\}$ convex? Uniqueness for this problem follows as in scalar case, using Lavrentiev's principle. Observe that the boundary condition $ |\nabla \mathbf{u}| = 1$ tacitly assumes that the gradient is continuous up to the boundary.
Question: Both Bernoulli and obstacle type problems for systems seems to be very hard to treat with existing methods and tools. All problems discussed for both these are open and new methods are needed to treat them. The quasi-convexification as used in the above does not apply, since each right hand side of the equation contains other components of the solution and the requirement is that these have to be convex functions in order to apply the method of quasi-convex envelop.
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