Inheritance of geometry in free boundary problems for systems: Unbounded domains
OpenYear of origin: 2020
Posted online: 2019-12-26 21:41:25Z by Henrik Shahgholian50
Cite as: P-191226.5
General Description View the group
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 [2] \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 [3], 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.
Problem's Description
Most of the above questions and open problems can naturally be stated for unbounded domains and half-spaces. Several authors have considered the scalar case of such problems and the literature is vast. The reader may consult paper [8], and the references therein for these problems.
It would be interesting to see if methods can be invented to treat, not only problems in bounded domains, but also in unbounded ones as developed in the literature.
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Edited: (general update ) at 2020-02-06 10:03:16Z
Created at: 2019-12-26 21:41:25Z View this version
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