A convexity problem for a semi-linear PDE

ABSTRACT

In this paper, we prove convexity of super-level sets of a semi-linear PDE with a non-monotone right-hand side, and with a free boundary (1) $\begin{array}{r}\begin{array}{ll}\mathrm{\Delta }u={\chi }_{\{0<u<1\}}& \mathrm{i}\mathrm{n}{\mathbb{R}}^n\mathrm{\setminus }D,\\ u=2& \mathrm{o}\mathrm{n}\mathrm{\partial }D.\end{array}\end{array}$(1) Here D is assumed to be convex, and $n\ge 2$. The main difficulty of this problem is that the right-hand side is non-monotone and no a-priori regularity is known about the boundary $\mathrm{\partial }\{u>0\}$.

KEYWORDS:

• Convexity
• star-shapedness
• uniqueness
• free boundary

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 35R35

Acknowledgments

I would like to thank Professor Henrik Shahgholian for his careful reading of the first draft of this paper and for making several valuable comments especially concerning the regularity theory of free boundaries and the relevant references. I would also like to thank the referees for carefully reading this manuscript and for their constructive comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Layan El Hajj  http://orcid.org/0000-0003-1865-2843

Notes

1 In the sets $\{0<u<1\}$ and $\{1<u<2\}$, we can make variations in both directions (upwards and downwards) and obtain the Euler equations as stated in (2(2) $\begin{array}{ll}\mathrm{\Delta }u={\chi }_{\{0<u<1\}}& \mathrm{i}\mathrm{n}{\mathbb{R}}^n\mathrm{\setminus }D,\\ u=2& \mathrm{o}\mathrm{n}\mathrm{\partial }D,\end{array}$(2) ) in these sets. Hence $\mathrm{\Delta }u={\chi }_{\{0<u<1\}}+{\mu }_0+{\mu }_1$, where ${\mu }_0$, ${\mu }_1$ are measures supported on $\mathrm{\partial }\{u>0\}$ respectively $\mathrm{\partial }\{u>1\}$. To obtain the Euler equation in (2(2) $\begin{array}{ll}\mathrm{\Delta }u={\chi }_{\{0<u<1\}}& \mathrm{i}\mathrm{n}{\mathbb{R}}^n\mathrm{\setminus }D,\\ u=2& \mathrm{o}\mathrm{n}\mathrm{\partial }D,\end{array}$(2) ), it suffices to prove these measures are zero. This can be shown by proving that the measures are bounded functions. A similar analysis as in [12], Section 3 gives the bound of these measures.

2 The energy is strictly decreased under spherical rearrangement, unless the solution is already spherical.

3 For n = 2, we should take $u_j=\left(|x{|}^2/4\right)+c+b\log |x|.$

4 In [8], the authors assume the r.h.s. to be continuous whereas in [7], no such assumption is made. We remark that the continuity of the r.h.s. is not needed in the proof since the viscosity theory applies to equations with measurable coefficients, see e.g. [13].

5 It should be remarked that the argument in Theorem 2.4 to show $|\mathrm{\nabla }u|>0$ does not extend in an obvious manner to nonlinear operators, specially to p-Laplacian, whereas the method of [7] does.

6 This follows from the fact that maximum of the largest sub-solution is also a solution.