Limit Behaviour of a Singular Perturbation Problem for the Biharmonic Operator

Abstract

We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in Dipierro et al. (arXiv:1808.07696, 2018), and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem. We also investigate the quadratic behaviour of solutions near the zero level set, at least for small values of the perturbation parameter. Some counterexamples to the uniform regularity are also provided if one does not impose some structural assumptions on the forcing term.

Appendix A: Decay Estimates for the Gradient and the Hessian

Here, we present some decay estimates for the gradient and the Hessian of solutions to (1.1).

Proposition A.1

Suppose that $u^\varepsilon$ is a solution of (1.1) such that $|u^\varepsilon |\le 1$ in $\Omega$. Let $D\Subset \Omega$ and $R_0\in \left( 0,{ \mathrm dist}(\overline{D}, \partial \Omega )\right)$. Suppose that

(A.1)

Then, we have

$$\begin{aligned}&\frac{1}{R^{n+2}}\int _{B_{R}(x_0)}|\nabla u^\varepsilon |^2+\frac{1}{R^n}\int _{B_{R}(x_0)}|D^2 u^\varepsilon |^2 \nonumber \\&\quad \le \frac{C}{R^{n+4}}\int _{B_{4R}(x_0)}(u^\varepsilon -m)^2+ \frac{\widehat{C}}{R^{n+2}}\int _{B_{4R}(x_0)}(u^\varepsilon -m), \end{aligned}$$

(A.2)

for any $x_0\in D$ and any $R\in (0,R_0/4)$, where

$$\begin{aligned} m=m^\varepsilon := \min _{B_{4R}(x_0)} u^\varepsilon , \end{aligned}$$

(A.3)

and $C>0$ depends only on n.

Proof

The proof follows from the argument used to prove Lemma A.1 in [8]. We briefly sketch the argument here. Up to a translation, we suppose $x_0:=0$. From the super biharmonicity of $u^\varepsilon$ we get

$$\begin{aligned} 0\ge \int _\Omega \Delta u^\varepsilon \Delta \phi =\sum _{i,j=1}^n\int _\Omega u_{ij}^\varepsilon \phi _{ij} \end{aligned}$$

for every $\phi \in C_0^\infty (B_{4R},[0,+\,\infty ))$, where two integration by parts are performed in the latter step. Choosing $\phi :=(u^\varepsilon -m^\varepsilon )\eta ^2$, where $m^\varepsilon$ is as in (A.3), and $\eta$ is a standard cut-off function supported in $B_{2R}\Subset \Omega$, such that $\eta =1$ in $B_{R}$ and $\eta =0$ outside $B_{2R}$ we get

$$\begin{aligned} \begin{aligned} \sum _{i,j=1}^n\int _\Omega (u_{ij}^\varepsilon )^2\eta ^2 \le \;&\frac{C}{R^2}\int _{B_{2R}}|\nabla u^\varepsilon |^2 +\frac{C}{R^4}\int _{B_{2R}}(u^\varepsilon -m)^2, \end{aligned} \end{aligned}$$

(A.4)

for some universal constant $C>0$ (compare, e.g. with formula (A.6) in [8]).

On the other hand, using the mean value property and the lower bound in (A.1), we obtain the Caccioppoli-type inequality

$$\begin{aligned} \int _{B_{2R}}|\nabla u^\varepsilon |^2 \le \frac{C}{R^2}\,\int _{B_{4R}} (u^\varepsilon - m^\varepsilon )^2+C\,\int _{B_{4R}} (u^\varepsilon -m^\varepsilon ), \end{aligned}$$

(A.5)

see e.g. formula (7.7) in [8]. Combining (A.4) and (A.5), we finish the proof. $\square$