The Matching Problem in PDE

Problem's Description

In the book [2] there are discussions and few results on overdetermined problems with matching Cauchy data. However, there the free boundary is a priori assumed to be $C^1$ and the authors derive higher order regularity.

A similar type of problem is the one of matching Cauchy data from one side of a domain, which can be formulated as follows:

\begin{equation} \tag{1} \left\{ \begin{array}{ll} F_i(D^2u_i,\nabla u_i,u_i,x)=0 & \text{ in }B_1 \cap \Omega , \qquad (i=1,2)\\ u_1=u_2, & \text{ in }B_1 \setminus \Omega ,\\ u_1, u_2 \in W^{2,n}(B_1),\\ \end{array} \right. \end{equation} with $F_i$ ($i=1,2$) are nice elliptic operators, $F_1(M,p,z,x) \neq F_2(M,p,z,x)$ for all $(M,p,z,x)$ (so that in particular $u_1\not\equiv u_2$ in $\Omega$), and one may ask both about the regularity of solutions and that of $\partial \Omega$.

Notice that if $u_1,u_2 \in C^1(B_1)$ and $u_1=u_2$ in the complement of $\Omega$, then their gradients agree there and we are saying that $u_1$ and $u_2$ are functions solving some nice equation inside $\Omega$ and satisfy both $u_1=u_2$ and $\nabla u_1=\nabla u_2$ on $\partial\Omega\cap B_1$.

It should be remarked that if the operators are the same we will not be able to conclude any results.

A simple example of such a problem in the linear case is the following: $$ F_1(D^2u_1,\nabla u_1,u_1,x)=\Delta u_1 , \qquad F_2(D^2u_2,\nabla u_2,u_2,x)=\Delta u_2 - 1, $$ in $B_1\cap \Omega$. Then $u=u_2 - u_1$ satisfies $\Delta u= 1$ in $\Omega \cap B_1$, and $u =0$ in $B_1 \setminus \Omega $. In this case $u \in C^{1,1}$ (see [3]) and starting from there one can obtain regularity for the free boundary (under some thickness assumptions of the complement of $\Omega$) as in [3]. However, for the nonlinear problem one needs a different approach as one cannot subtract $u_1$ from $u_2$.

It should be further remarked that the regularity of $u_1, u_2$ are still unknown for the simple case above, and only the regularity for $u= u_2 - u_1$ is shown in [3].

Question: Suppose $\Omega$ admits solutions $u_1, u_2$ satisfying (1), where $F_i$ ($i=1,2$) are Lipschitz in all variables, smooth in the lower order terms, uniformly elliptic, and concave in the hessian. We assume further $F_1 \neq F_2$.

What is the optimal regularity of the solutions, and that of the free boundary $\partial \Omega$?

Probably, as a starting case, one can try with $F_1=F_1(D^2u_1)$ and $F_2 =\Delta u_2 - 1 $.

1. ArticleIs an originAn overview of unconstrained free boundary problems

   • Alessio Figalli,
   • Henrik Shahgholian

   Philos. Trans. Roy. Soc. A 373 (2015), no. 2050, 20140281, 11 pp. A373, 2015

2. BookIs an originAn introduction to variational inequalities and their applications.

   • David Kinderlehrer,
   • Guido Stampacchia

   Reprint of the 1980 original. Classics in Applied Mathematics, 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. xx+313 pp.

3. Article Regularity of a free boundary problem with application to the Pompeiu problem

   • L.A. Caffarelli,
   • L. Karp,
   • and H. Shahgholian

   Ann. Math. 151, 269-292, 2000