Porosity of the free boundary for quasilinear parabolic variational problems
 Jun Zheng^{1}Email author,
 Binhua Feng^{2} and
 Peihao Zhao^{3}
Received: 22 April 2015
Accepted: 24 October 2015
Published: 2 November 2015
Abstract
In this paper we consider a certain quasilinear parabolic variational problem with identically zero constraint. By using intrinsic scaling, the exact growth of the solution near the free boundary is established. A consequence of this is that the time level of the free boundary is porous (in N dimensions) and therefore its Hausdorff dimension is less than N. In particular, the Ndimensional Lebesgue measure of the free boundary is zero for each time level.
Keywords
MSC
1 Introduction and main theorem
Under certain conditions on f and θ, we will show that the free boundary of the solution to the variational problems (1) is porous for each tlevel cut, which implies that the tcuts of the free boundary has Lebesgue measure zero.
As is well known, in the obstacle problems associated with elliptic operators, to obtain the porosity of the free boundary one needs to prove that every solution has a certain growth rate near the free boundary; see [2–4] for instance. When focusing on pparabolic variational problem (\(1< p<\infty\)), we remark that due to the lack of the strong minimum principle or the Harnack inequality one cannot inherit each technique from the elliptic obstacle problems, and we need further arguments to establish the growth rate of solutions near the free boundary. In pparabolic variational problems (\(p\geq 2\)), Shahgholian overcame this difficulty by using Hölder’s estimates for solutions of parabolic equations. As a byproduct, the author obtained the porosity of the free boundary for \(p\geq 2\); see [5]. A fact that should be noticed is: although neither the technique of Hölder’s estimates nor Harnack inequality can be applied to get the growth of solutions in the case of \(1< p<2\) in [5], a ‘minimum principle (in spatial variables)’ for singular parabolic equations given in [6] (Lemma 2.3) may be used in our problem as a substitute tool at this step. Thus in this paper, using the main idea of [5] and techniques of compactness, we are interested in studying the porosity of the free boundary in a large class of variational problems governed by quasilinear parabolic operators. Our result contains not only the case of \(p\geq 2\), but the singular case of \(1< p<2\) as well, which is naturally an extension of [5].
 (a_{1}):

\(a_{i}(x,0)=0\),
 (a_{2}):

\(\sum^{N}_{i,j=1}\frac{\partial a_{i}}{\partial \eta_{j}}(x,\eta)\xi _{i}\xi_{j}\geq\gamma _{0}\eta^{p2}\xi^{2}\),
 (a_{3}):

\(\sum^{N}_{i,j=1}\frac{\partial a_{i}}{\partial \eta _{j}}(x,\eta)\leq \gamma_{1}\eta^{p2}\),
 (a_{4}):

\(\sum^{N}_{i,j=1}\frac{\partial a_{i}}{\partial x _{j}}(x,\eta)\leq\gamma_{1}\eta^{p1}\),
Remark 1.1
 (f):

\(0<\lambda_{0}\leq f\leq\Lambda\) in \(\Omega_{T}\), \(f(x,t)\) is monotone nonincreasing in t;
 (θ):

\(\theta(x,0)=0\), \(\theta(x,t)\) is monotone nondecreasing in t.
Classical theorem
We recall the concept of porosity; see [2, 5].
Porosity
A set E in \(\mathbb{R}^{N}\) is called porous with porosity constant δ if there is an \(r_{0}>0\) such that for each \(x\in E\) and \(0< r< r_{0}\) there is a point y such that \(B_{\delta r}(y)\subset B_{r}(x)\setminus E\).
According to [9], a porous set has Hausdorff dimension not exceeding \(NC\delta^{N}\); thus, it is of Lebesgue measure zero.
Now we state the main theorem in this paper.
Theorem 1.1
2 A class of functions on the unit cylinder
In this section, we discuss the behavior of solutions to problem (1) and functions in \(\mathscr{G}_{a}\) near the free boundary.
2.1 Nondegeneracy of the solution near the free boundary
The following result gives a description of the solution u to problem (1) showing that it cannot grow too slowly near the free boundary. This property and the growth rate of the elements in \(\mathscr{G}_{a}\) will pave the way to establish the porosity of the free boundary.
Lemma 2.1
Proof
2.2 Growth rate of the function u in \(\mathscr{G}_{a}\)
In this subsection we prove that every function u in \(\mathscr{G}_{a}\) cannot grow too fast near the free boundary but has a growth rate of order \(q=\frac{p}{p1}\) (Theorem 2.1).
It should be noticed that \(\mathbb{M}_{a}(u)\neq\emptyset\) for all \(u\in \mathscr{G}_{a}\) since \(0\in\mathbb{M}_{a}(u)\). Indeed, it follows from Lemma 2.1 that \(S(1,u) \leq 1=(\frac{1}{c_{0}2^{q}})c_{0}2^{q}\leq(\frac {1}{c_{0}2^{q}})S(2^{1},u)=AS(2^{1},u)\).
Now we state the growth property of the elements in the class \(\mathscr{G}_{a}\).
Theorem 2.1
To prove this theorem we need the following lemma.
Lemma 2.2
Proof
Case 1 (\(1< p\leq2\)). In this case, we need the following lemma originating from [6], where the authors stated it for pparabolic equations (\(1< p<2\)). One should pay attention to the fact that the proof of the following lemma can be repeated as in [6] with slight modifications. Moreover, the result is valid for \(p=2\) since the process is ‘stable’ as \(p\nearrow2\) so that one may recover the regularity results by letting \(p\nearrow2\) (see the proofs of Theorems 1 and 2, or the remarks in 1(iii) on p.323 of [6]).
Lemma 2.3
(Theorem 2 [6])
Now notice that \(\sup_{B_{\frac{1}{2}}}u(x,0)\geq1\) by \(\partial_{t} u \geq0\) and (6). One may find \(x_{0}\in B_{\frac{1}{2}}(0)\) such that \(u(x_{0},0)\geq\frac{1}{2}\). On the other hand, since, for any \(\overline{Q'}\subset Q_{1}\), \(u\in C^{1,\alpha}(Q')\) for some \(\alpha\in(0,1)\) (see Chapter IX of [1]), and then Lemma 2.3 gives \(u(0,0)\geq\frac{1}{2}\), which is a contradiction. Indeed, in Lemma 2.3, one may let \(\rho=\frac{1}{2}\in(x_{0},1x_{0})\) and \(\Omega=B_{r}\subset B_{1}\) with \(r=\frac{\frac{1}{2}+x_{0}+1}{2}\). Therefore \(B_{\rho}(x_{0})\subset\Omega\) and \(0\in B_{\rho}(x_{0})\).
Proof of Theorem 2.1
3 Proof of the main theorem
Having the estimates from below and above for the function u, one can prove our main result as in [5]. For completeness we carry out the minor changes in the proof of [5].
Proof of the main theorem
Without loss of generality, we assume that the compact set K in the main theorem is the closed unit cylinder \(\overline{Q}_{1}\), and, moreover, that \(\overline{Q}_{2}\subset\Omega_{T}\).
Declarations
Acknowledgements
The authors would like to thank the referee for his/her careful reading and valuable suggestions, which made this article more readable. This work is supported by the Fundamental Research Funds for the Central Universities: 10801B10096018 and 10801X10096022.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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