数理解析研究所講究録
第 1896 巻 2014 年 12-25

12Construction of solutions toa fourth order parabolic obstacle problem
via minimizing movements
東北大学大学院理学研究科

岡部真也

SHINYA OKABE
Mathematical Institute,
Tohoku University

1

Introduction

The paper is devoted to a fourth order parabolic obstacle problem. We shall announce a
result ([13]) which is a joint work with M. Novaga of Pisa University.
The obstacle problem for elliptic and parabolic PDE’s is a topics which attracted
a great interest in the past years, and has been widely discussed in the mathematical
literature. However, even if many studies are available for second order elliptic andparabolic equations (see for instance [6, 9] and references therein), there are relatively
few results for higher order obstacle problems, even in the linear fourth order case. Moreprecisely, the elliptic obstacle problem for the biharmonic operator has been considered in[5, 7, 8, 10, 11, 14]. To the best of our knowledge, there is no analog for the corresponding
parabolic obstacle problem. The purpose of this paper is to investigate the regularity ofa solution to the obstacle problem for the parabolic biharmonic equation.
In the sequel we letsome $\alpha\in(0,1)$ , and we let$\Omega\subset \mathbb{R}^{N}$

(1.1)

$f$

$C^{2+\alpha}$for$\Omegaarrow \mathbb{R}$$f\in C^{2}(\overline{\Omega})$

We consider an initial datum
(1.2)

:

be a bounded domain, with boundary of classbe the obstacle function, satisfying,$u_{0}:\Omegaarrow \mathbb{R}$$u_{0}\in H_{0}^{2}(\Omega)$,$f<0$ on$\partial\Omega.$such that$u_{0}\geq f$

implies $u=0$ andWe recall thatsatisfies Dirichlet boundary condition on$u\in H_{0}^{2}(\Omega)$$\partial\Omega.$

a.e. in$\nabla u\cdot\nu^{\Omega}=0$$\Omega.$on$\partial\Omega$(see [7, 8]), that is,

$u$

13
In this paper, we consider the following fourth order parabolic obstacle problem:(P)$\{\begin{array}{ll}\partial_{t}u(x, t)+\triangle^{2}u(x, t)\geq 0 in \Omega\cross \mathbb{R}_{+},\partial_{t}u(x, t)+\triangle^{2}u(x, t)=0 in \{(x, t)\in\Omega\cross \mathbb{R}_{+}:u(x, t)>f(x)\},u(x, t)=0 on \partial\Omega\cross \mathbb{R}_{+},\nabla u(x, t)\cdot\nu^{\Omega}(x)=0 on \partial\Omega\cross \mathbb{R}_{+},u(x, t)\geq f(x) in \Omega\cross \mathbb{R}_{+},u(x, 0)=u_{0}(x) in \Omega.\end{array}$

The main result of this paper is the following:
Theorem 1.1. Let $N\leq 3$ , and let be a function satisfying (1.1). Then, for any initial
data
satisfying (1.2), the problem (P) has a unique solution$f$$u_{0}$(1.3)$u\in L^{\infty}(\mathbb{R}_{+};H_{0}^{2}(\Omega))\cap H_{loc}^{1}(\mathbb{R}_{+};L^{2}(\Omega))$

Furthermore the solution

$u$

,

with

$u_{t}\in L^{2}(\mathbb{R}_{+}\cross\Omega)$.

satisfies the following properties:(i) $u\in L^{2}(0, T;W^{2,\infty}(\Omega))$ for any $T>0$ . In particular, if $N=1,$(1.4)$u\in C^{0,\beta}([0, T];C^{1,\gamma}(\Omega))$with$0< \gamma<\frac{1}{2}$and$0< \beta<\frac{1-2\gamma}{8},$if $N=2,3,$(1.5)(ii) For $a.e.$$t\in \mathbb{R}_{+}$

the quantity(1.6)$\mu_{t}$defines a Radon measure insuch that(1.7)

with $0< \gamma<\frac{4-N}{2}$ and $0< \beta<\frac{4-N-2\gamma}{8}.$$u\in C^{0,\beta}([0, T];C^{0,\gamma}(\Omega))$$\Omega$$:=u_{t}(\cdot, t)+\triangle^{2}u(\cdot, t)$, and for any $T>0$ there exists a constant $C>0$$\int_{0}^{T}\mu_{t}(\Omega)^{2}dt<C.$

Let us point out that the problem (P) corresponds to the gradient flow of a convex
functional defined on the Hilbert space, hence we can apply the general theory of
maximal monotone operators developed in [4]. Indeed, given as above, we can define
the functional $E_{f}(u):L^{2}(\Omega)arrow[0, +\infty]$ as$L^{2}(\Omega)$$f$$E_{f}(u)=\{\begin{array}{ll}\int_{\Omega}|\triangle u|^{2} if u\in H^{2}(\Omega) and u\geq f,+\infty otherwise.\end{array}$

14
Notice thatis convex and lower semicontinuous on
the gradient flow
$E_{f}(u)$(1.8)$L^{2}(\Omega)$, and (P) corresponds to$u_{t}+\partial E_{f}(u)\ni 0, u(O)=u_{0},$denotes the subdifferential ofin. In particular, given an initial datumwith, by the results in [4] it follows that the evolution problem (1.8)
has a unique solution satisfying
where$\partial E_{f}$$E_{f}$$u_{0}\in H_{0}^{2}(\Omega)$$L^{2}(\Omega)$$u_{0}\geq f$
$u$

(1.9)

$u\in L^{\infty}(\mathbb{R}_{+};H_{0}^{2}(\Omega))\cap H_{loc}^{1}(\mathbb{R}_{+};L^{2}(\Omega))$with$u_{t}\in L^{2}(\mathbb{R}_{+}\cross\Omega)$.

The purpose of this paper is to give an extra regularity of solution to (P). To this aim,
we characterize the solution by means of an implicit variational scheme, corresponding$u$to the minimizing movements introduced by De Giorgi (see e.g. [2]). This approach willallow us to extend some of the arguments in [7], concerning the regularity of the elliptic
obstacle problem for the biharmonic operator.
We point out that the method does not rely on the linear structure of the problem andcan be applied to more general fourth order parabolic equations. Indeed our motivation
for this work rise from an analysis of a motion of planar closed curves which is governedby the straightening flow with obstacle. Curve straightening flow is a
gradient flow for
the total squared curvature$L^{2}$$\mathcal{E}(\gamma):=\frac{1}{2}\int_{\gamma}\kappa^{2}ds,$

where is a closed planar curve, and denote respectively the curvature and the arc
length parameter of . Although the flow is a fourth order quasilinear parabolic equation,
we expect that the method of this paper will be available for the geometric obstacle
problem.
The paper is organized as follows: in Section 2 we introduce the implicit scheme
corresponding to problem (P), by means of an appropriate variational problem; in Section
3 we study the regularity of solutions to the variational problem; in Section 4 we pass to
the limit in the approximating scheme and prove Theorem 1.1.
$\kappa$$\gamma$$s$$\gamma$2

Preliminary

The equation in (P) is the$L^{2}$

gradient flow for the functional$E(u)= \frac{1}{2}\int_{\Omega}|\triangle u(x)|^{2}dx.$

Let $T>0,$$n\in \mathbb{N}$, and set$\tau_{n}=\frac{T}{n}.$15
Let us set $u_{0,n}=u_{0}$ . For $i=1,$
minimum problem$\cdots,$ $n$, we define inductively$u_{i,n}$as a solution of the$\min\{G_{i,n}(u):u\in K\},$$(M_{i,n})$

where
(2.1)

$G_{i,n}(u) :=E(u)+P_{i,n}(u)$

with
(2.2)

$P_{i,n}(u):= \frac{1}{2\tau_{n}}\int_{\Omega}(u-u_{i-1,n})^{2}dx,$

and $K$ is a convex set given by$K:=\{u\in H_{0}^{2}(\Omega):u(x)\geq f(x) a.e.

in \Omega\}.$In the following, we let(2.3)$V_{i,n}(x) := \frac{u_{i,n}(x)-u_{i-1,n}(x)}{\tau_{n}}.$We give a definition of a piecewise linear interpolations of$\{u_{i,n}\}$:Definition 2.1. (Piecewise linear interpolation) Let $f$ be a function satisfying (1.1).
Letwith $u_{0}(x)\geq f(x)a.e$ . in . Defineas$u_{0}\in H_{0}^{2}(\Omega)$

(2.4)

if

$u_{n}:\Omega\cross[0, T]arrow \mathbb{R}$$\Omega$$u_{n}(x, t) :=u_{i-1,n}(x)+(t-(i-1)\tau_{n})V_{i,n}(x)$$(x, t)\in\Omega\cross[(i-1)\tau_{n}, i\tau_{n}]$for $i=1,$$\cdots,$$n.$

By a technical reason, additionally we need a piecewise constant interpolations ofand$\{u_{i,n}\}$$\{V_{i,n}\}.$

Definition 2.2. (Piecewise constant interpolation) Define(2.5)$\tilde{u}_{n}(x, t):=u_{i,n}(x)$

,

(2.6)

$V_{n}(x, t) :=V_{i,n}(x)$

,

if

3

$(x, t)\in\Omega\cross[(i-1)\tau_{n}, i\tau_{n})$for $i=1,$$\cdots,$$\tilde{u}_{n}:\Omega\cross[0, T]arrow \mathbb{R}$$n.$Existence and regularity of minimizers of

We first mention a well-known compactness result in$H_{0}^{2}(\Omega)$([1]).

$(M_{i,n})$as

16Proposition 3.1. The following embedding is compact:(3.1)$H_{0}^{2}(\Omega)arrow\{\begin{array}{ll}C^{1,\gamma}(\overline{\Omega}) for 0<\gamma<\frac{1}{2} if N=1,C^{0,\gamma}(\Omega) for 0<\gamma<2-\frac{N}{2} if N=2,3,L^{q}(\Omega) for 1\leq\forall q<+\infty if N=4,L^{q}(\Omega) for 1\leq\forall q<\frac{2N}{N-4} if N\geq 5.\end{array}$

We start with the existence of minimizers of$(M_{i,n})$.

Theorem 3.1. (Existence of minimizers) Let be a function satisfying (1.1). Letwith $u_{0}(x)\geq f(x)a.e$ . in . Then the problem
possesses a uniquesolutionwith $u_{i,n}(x)\geq f(x)a.e$ . in for each $i=1,$$f$$u_{0}\in H_{0}^{2}(\Omega)$$(M_{i,n})$$\Omega$$u_{i,n}\in H_{0}^{2}(\Omega)$

Proof. Fix

solution$u$$n\in \mathbb{N},$to$(M_{i,n})$$\Omega$$\cdots,$$T>0$ , and $i=1,$$\cdots,$ $n$$n.$. From $(2.1)-(2.2)$ and the minimality of a

, we obtain that
$E(u)\leq G_{i,n}(u)\leq G_{i,n}(u_{i-1,n})=E(u_{i-1,n})$,and then$0 \leq\inf_{H_{0}^{2}(\Omega)}G_{i,n}(u)\leq G_{i,n}(u_{i-1,n})=E(u_{i-1,n})\leq\cdots\leq E(u_{0})$Thus we can take a minimizing sequence
fora.e. in for each
and $\sup_{j}G_{i,n}(u_{j})<\infty.$
Observing that the norm
is equivalent tofrom$\{u_{j}\}\subset H_{0}^{2}(\Omega)$$\Omega$$(M_{i,n})$.

such that $u_{j}(x)\geq f(x)$$j\in \mathbb{N}$$|u\Vert_{H_{0}^{2}(\Omega)}$$\Vert\triangle u\Vert_{L^{2}(\Omega)}$(e.g., see [12]), it follows$\Vert\Delta u_{j}\Vert_{L^{2}(\Omega)}=\sqrt{2E(u_{j})}\leq\sqrt{2E(u_{0})}=\Vert\triangle u_{0}\Vert_{L^{2}(\Omega)}$that$\{u_{j}\}$is uniformly bounded in(3.2)$H_{0}^{2}(\Omega)$. Thus there exists$u_{j}arrow u$in$H_{0}^{2}(\Omega)$$u\in H_{0}^{2}(\Omega)$such that,in particular,(3.3)$\triangle u_{j}arrow\triangle u$in$L^{2}(\Omega)$,

up to a subsequence. Thenks to Proposition 3.1, we obtain that$u_{j}arrow u$

in$\{\begin{array}{ll}C^{1,\gamma}(\overline{\Omega}) for 0<\gamma<\frac{1}{2} if N=1,C^{0,\gamma}(\Omega) for 0<\gamma<2-\frac{N}{2} if N=2,3,L^{q}(\Omega) for 1\leq\forall q<+\infty if N=4,L^{q}(\Omega) for 1\leq\forall q<\frac{2N}{N-4} if N\geq 5.\end{array}$

17
In particular, for the case of $N\geq 4,$(3.4)$u_{j}arrow u$a.e. inRecalling $u_{j}\geq f$ a.e. in for eachof Fatou’s Lemma, we conclude that$\Omega$$j\in \mathbb{N},$(3.5)

up to a subsequence.

$\Omega$$(3.4)$

yields that $u\geq f$ a.e. in$\Omega$. Making use$P_{i,n}(u) \leq\lim\inf P_{i,n}(u_{j})jarrow\infty.$

Furthermore (3.3) implies(3.6)$E(u)= \frac{1}{2}\Vert\triangle u\Vert_{L^{2}(\Omega)}^{2}\leq\frac{1}{2}\lim infjarrow\infty\Vert\triangle u_{j}\Vert_{L^{2}(\Omega)}^{2}=\lim\inf E(u_{j})jarrow\infty.$

Combining (3.5) with (3.6), we see thatis the minimizer ofina.e.
. The uniqueness follows from the fact that the functional
convex.
$u\in H_{0}^{2}(\Omega)$$\Omega$

Regarding the regularity of the minimizer
following:

Theorem 3.2. Let, it holds that$u_{i,n}$be the solution of$u_{i,n}$$(M_{i,n})$$(M_{i,n})$with $u\geq f$
is strictly$G_{i,n}(\cdot)$$\square$obtained in Theorem 3.1, we show theobtained by Theorem 3.1. Then, for any$n\in \mathbb{N}$

(3.7)

(3.8)

$\sup_{i}\Vert\triangle u_{i,n}\Vert_{L^{2}(\Omega)}\leq\sqrt{2E(u_{0})}.$

Proof. Fix $T>0$ andminimality of

(3.9)

,

$\int_{0}^{T}\int_{\Omega}|V_{n}(x, t)|^{2}dxdt\leq 2E(u_{0})$$u_{i,n}$$n\in \mathbb{N}$. For each $i=1,$$\cdots,$ $n$, it follows from $(2.1)-(2.2)$ and thethat$G_{i,n}(u_{i,n})\leq G_{i,n}(u_{i-1,n})=E(u_{i-1,n})$.

Hence we get$P_{i,n}(u_{i,n})\leq E(u_{i-1,n})-E(u_{i,n})$

,

i.e.,

(3.10)$\frac{1}{2\tau_{n}}\int_{\Omega}(u_{i,n}-u_{i-1,n})^{2}dx\leq E(u_{i-1,n})-E(u_{i,n})$.

Combining (3.10) with definitions (2.3) and (2.6), we obtain$\frac{1}{2}\int_{0}^{T}\int_{\Omega}|V_{n}(x, t)|^{2}dxdt=\frac{1}{2}\sum_{i=1}^{n}\int_{(i-1)\tau_{n}}^{i\tau_{n}}\int_{\Omega}|V_{i,n}(x)|^{2}dxdt$$\leq\sum_{i=1}^{n}(E(u_{i-1,n})-E(u_{i,n}))=E(u_{0})-E(u_{n,n})\leq E(u_{0})$,18i.e., (3.7).
By (3.9), we obtain that $E(u_{i,n})\leq E(u_{i-1,n})$ for each $i=1,$(3.11)$\frac{1}{2}\int_{\Omega}(\triangle u_{i,n})^{2}dx=E(u_{i,n})\leq E(u_{0})$$\cdots,$ $n$, and then.

It is clear that (3.11) is equivalent to (3.8).
By the definition of$u_{i,n}$$\square$, we see that$\int_{\Omega}|\triangle(u_{i,n}+\epsilon\zeta)|^{2}dx+\frac{1}{2\tau_{n}}\int_{\Omega}(u_{i,n}-u_{i-1,n}+\epsilon\zeta)^{2}dx$$\geq\int_{\Omega}|\Delta u_{i,n}|^{2}dx+\frac{1}{2\tau_{n}}\int_{\Omega}(u_{i,n}-u_{i-1,n})^{2}dx$

for any$\epsilon>0$and$\zeta\in H_{0}^{2}(\Omega)$with$\zeta\geq 0$. This implies$\int_{\Omega}\triangle u_{i,n}\triangle\zeta dx+\frac{1}{\tau_{n}}\int_{\Omega}(u_{i,n}-u_{i-1,n})\zeta dx\geq 0,$

so that(3.12)$\mu_{i,n} :=\triangle^{2}u_{i,n}+V_{i,n}\geq 0$

in the sense of the distribution. Henceis a measure in (e.g., see [15]).
$N\leq3$, Proposition 3.1 implies that
When we restrict dimensions to
is continuous.
Under such restriction, we define$\Omega$$\mu_{i,n}$$u_{i,n}$(3.13)$C_{i,n}:=\{x\in\Omega :(3.14)$\mathcal{N}_{i,n}:=\{x\in\Omega :It is clear thatthe sets.

$C_{i,n}\cup \mathcal{N}_{i,n}=\Omega$

u_{i,n}(x)=f(x)\},$
u_{i,n}(x)>f(x)\}.$. We can show a relation between the support of, then there exists a neighborhood $W$ of
Lemma 3.1. Let $N\leq 3$ . Ifthat $\mu_{i,n}(W)=0$ . Furthermore we have$x_{0}\in \mathcal{N}_{i,n}$(3.15)$\delta>0$

arbitrarily. Sinceof such thatand a neighborhood$x^{0}\in \mathcal{N}_{i,n}$$W$

Notice that$u_{i,n}$$\mathcal{N}_{i,n}$such

is an open set, there exist a$x^{0}$$u_{i,n}(x)-f(x)>\delta$(3.16)$x_{0}$

and$supp\mu_{i,n}\subseteq C_{i,n}.$Proof. Let $N\leq 3$ and fix

constant$\mu_{i,n}$

for all$x\in W.$satisfies$\int_{\Omega}\triangle u_{i,n}\triangle(u_{i,n}-\varphi)dx\leq-\int_{\Omega}V_{i,n}(u_{i,n}-\varphi)dx$

19for any, forthe function$\varphi\in K$$u_{i,n}$is a solution of$(M_{i,n})$. Then for any$\zeta\in C_{0}^{\infty}(W)$with$0\leq\zeta\leq\delta/2,$$\psi=u_{i,n}-\zeta$

belongs to $K$ . Taking this$\psi$

as$\varphi$in (3.16), we have$\int_{\Omega}[\triangle u_{i,n}\triangle\zeta+V_{i,n}\zeta]dx\leq 0.$

Since$\mu_{i,n}\geq 0$, this asserts that$\int_{\Omega}[\triangle u_{i,n}\triangle\zeta+V_{i,n}\zeta]dx=0,$i.e., $\mu_{i,n}=0$ in $W.$$\square$Regarding the finiteness of$\mu_{i,n}$, we have the following:

Theorem 3.3. ([13]) Let
be the solution ofobtained by Theorem 3.1. Thenin
(3.12)
is a measure in for each $i=1,$. Moreover there exists a
definedpositive constantbeing independent of such that$(M_{i,n})$$u_{i,n}$$\Omega$$\mu_{i,n}$$C$$\cdots,$ $n$$n$(3.17)$\tau_{n}\sum_{i=1}^{n}\mu_{i,n}(\Omega)^{2}<C.$

Regarding the regularity of$u_{i,n}$, we obtain the following result:

Theorem 3.4. ([13]) Let $N\leq 3$ . It holds that

(3.18)

for each$u_{i,n}\in W^{2,\infty}(\Omega)$

and $i=1,$. Moreover, for any $R>0$ withpositive constants
andbeing independent of such that$n\in \mathbb{N}$$\cdots,$ $n$$C_{1}$$C_{2}$(3.19)

4$\overline{B}_{R}\subset\Omega$, there exist$n$$\tau_{n}\sum_{i=1}^{n}\Vert D^{2}u_{i,n}\Vert_{L(\Omega)}^{2_{\infty}}\leq C_{1}+C_{2}\Vert\triangle f\Vert_{L\infty(\Omega)}^{2}.$Existence and regularity of solutions to (P)In this section, we start with a convergence result of the piecewise linear interpolation
We state several results without its proof. For the precise proof, see [13]. We first show
a convergence result which holds in any dimension $N\geq 1.$$u_{n}.$

Theorem 4.1. Letfunction$u_{n}$

be the piecewise linear interpolation of$\{u_{i,n}\}$$u\in L^{\infty}([0, +\infty);H_{0}^{2}(\Omega))\cap H_{loc}^{1}(0, +\infty;L^{2}(\Omega))$. Then there exists a20such that
(4.1)

$in$

$u_{n}arrow u$$L^{2}(0, T;H_{0}^{2}(\Omega))\cap H^{1}(0, T;L^{2}(\Omega))$as$narrow+\infty,$

up to a subsequence, for any $0<T<+\infty$ . Moreover$\int_{0}^{T}\int_{\Omega}u_{t}^{2}dxdt\leq 2E(u_{0})$$u(x, t)\geq f(x)$(4.2)

for $a.e.$$u_{n}arrow u$

Proof. Recalling that$t_{1}<t_{2}$$x\in\Omega$,and for evew $t\in[0, +\infty)$ , and for each$in$$C^{0,\alpha}([0, T];L^{2}(\Omega))$as$\alpha\in(0, \frac{1}{2})$it holds$narrow+\infty.$

is absolutely continuous on $[0, T]$ , for all, H\"older’s inequality and Fubini’s Theorem give us$u_{n}(x, \cdot)$$t_{1},$$t_{2}\in[0, T]$with$\Vert u_{n}(\cdot, t_{2})-u_{n}(\cdot, t_{1})\Vert_{L^{2}(\Omega)}=(\int_{0}^{L}(\int_{t_{1}}^{t_{2}}\frac{\partial u_{n}}{\partial t}(x, t)dt)^{2}dx)^{\frac{1}{2}}$$\leq(\int_{t_{1}}^{t_{2}}\Vert\frac{\partial u_{n}}{\partial t}(\cdot, t)\Vert_{L^{2}(\Omega)}^{2}dt)^{\frac{1}{2}}(t_{2}-t_{1})^{\frac{1}{2}}.$

Then it follows from (3.7) that
(4.3)$\int_{t_{1}}^{t_{2}}\int_{\Omega}u_{t}^{2}dxdt\leq 2E(u_{0})$

and

(4.4)

$\Vert u_{n}(\cdot, t_{2})-u_{n}(\cdot, t_{1})\Vert_{L^{2}(\Omega)}\leq\sqrt{2E(u_{0})}(t_{2}-t_{1})^{\frac{1}{2}}.$Since (3.8) yields that
(4.5)$\sup_{t\in[0,T]}\Vert\triangle u_{n}(\cdot, t)\Vert_{L^{2}(\Omega)}\leq\sup_{1\leq i\leq n}\Vert\triangle u_{i,n}\Vert_{L^{2}(\Omega)}\leq\sqrt{2E(u_{0})},$

there exists a function $u\in L^{2}(0, T;H_{0}^{2}(\Omega))$ such that $u_{n}arrow u$ in
subsequence. On the other hand, the estimate (3.7) implies that
(4.6)

$V_{n}= \frac{\partial u_{n}}{\partial t}arrow\frac{\partial u}{\partial t}$

in$L^{2}(0, T;L^{2}(\Omega))$$L^{2}(0, T;H_{0}^{2}(\Omega))$

up to a.

This means that
, i.e., $u\in H^{1}(0, T;L^{2}(\Omega))$ . Combining (4.4) with
Ascoli-Arzel\‘a’s Theorem (see e.g. [3, Proposition 3.3.1]), we conclude (4.2).
Since (4.5) means that $\{u_{n}(t)\}$ is uniformly bounded inwith respect to $t\in[0, T]$, we deduce from (4.2) that, for each $t\in[0, T]$and$\partial u/\partial t\in L^{2}(0, T;L^{2}(\Omega))$$H_{0}^{2}(\Omega)$$n\in \mathbb{N}$(4.7)$u_{n}(t)arrow u(t)$in$H_{0}^{2}(\Omega)$

21
up to a subsequence. This asserts that
implies that for each $t\in[0, T]$

(4.8)

in

$u_{n}(t)arrow u(t)$$u\in L^{\infty}([O, T];H_{0}^{2}(\Omega))$. Moreover, Proposition 3.1$\{\begin{array}{ll}C^{1,\gamma}(\Omega) for 0<\gamma<\frac{1}{2} if N=1,C^{0,\gamma}(\Omega) for 0<\gamma<2-\frac{N}{2} if N=2,3,L^{q}(\Omega) for 0<q<+\infty if N=4,L^{q}(\Omega) for 0<q<\frac{2N}{N-4} if N\geq 5.\end{array}$

In particular, if $N\geq 4,$(4.9)$u_{n}(t)arrow u(t)$a.e. in$\Omega$

up to a subsequence. Since $u_{n}(t)\geq f$ a.e. in for each
and $t\in[0, T]$ , the fact$(4.8)-(4.9)$ yields that $u(t)\geq f$ a.e. infor each $t\in[0, T]$ . This completes the proof.
$\Omega$$n\in \mathbb{N}$$\Omega$$\square$

When $N=1$ , we can improve the convergence result obtained in Theorem 4.1:
Theorem 4.2. ([13]) Let $N=1$ . Let
it holds that$u$be the function obtained by Theorem 4.1. Thenand$u\in L^{2}(0, T;W^{2,\infty}(\Omega))\cap C^{0,\beta}([0, T];C^{1,\alpha}(\Omega))$(4.10)$u_{n}arrow u$$weakly^{*}$ $in$

(4.11)$u_{n}arrow u$$in$for every

and$\alpha\in(0, \frac{1}{2})$as$C^{0,\beta}([0, T];C^{1,\alpha}(\Omega))$$\beta\in(0, \frac{1-2\alpha}{8})$as$L^{2}(0, T;W^{2,\infty}(\Omega))$$narrow\infty,$$narrow\infty$. Furthermore $u(\cdot, t)arrow u_{0}$ in$C^{1,\alpha}(\Omega)$as$t\downarrow 0.$

When $N=2,3$ , we can also improve the result obtained in Theorem 4.1:

Theorem 4.3. ([13]) Let $N=2,3$ . Let be the function obtained by Theorem 4.1. Thenit holds that $u\in L^{2}(0, T;W^{2,\infty}(\Omega))\cap C^{0,\beta}([0, T];C^{0,\gamma}(\Omega))$ and$u$(4.12)$u_{n}arrow u$$weakly^{*}$ $in$

(4.13)$u_{n}arrow u$

$in$

as

$L^{2}(0, T;W^{2,\infty}(\Omega))$$C^{0,\beta}([0, T];C^{0,\gamma}(\Omega))$as$narrow+\infty,$$narrow+\infty$

for evew$0< \beta<(\frac{1}{2}-\frac{N}{8})(1-\frac{\gamma}{2-N/2}) , 0<\gamma<2-\frac{N}{2}.$Furthermore$u(\cdot, t)arrow u_{0}$in$C^{0,\gamma}(\Omega)$as$t\downarrow 0.$

Regarding the piecewise constant interpolation for
2.2, we can verify the following:

Lemma 4.1. Let(4.14)$\tilde{u}_{n}$$\{u_{i,n}\}$, i.e.,

be the piecewise constant interpolation of$\tilde{u}_{n}arrow u$$in$$L^{\infty}([0, T];C^{1,\gamma}(\Omega))$as$\tilde{u}_{n}$defined in Definition$\{u_{i,n}\}$$narrow+\infty$. If $N=1$ , then22for evew$\gamma\in(0,1/2)$(4.15), where$\tilde{u}_{n}arrow u$

is the function obtained in Theorem 4.1. If $N=2,3$ , then

$u$

$in$

$L^{\infty}([0, T];C^{0,\gamma}(\Omega))$for every $\gamma\in(0,2-N/2)$ . Furthermore, for any(4.16)$\Delta\tilde{u}_{n}arrow\triangle u$

Let us define$\mu_{n}$$N\geq 1$$L^{2}(0, T;L^{2}(\Omega))$

$in$

as

$narrow+\infty$, it holds thatas$narrow+\infty.$as(4.17)$\mu_{n}(t)=\mu_{i,n}$if$t\in[(i-1)\tau_{n}, i\tau_{n})$.

We close the paper with an outline of proof of Theorem 1.1:

Proof of Theorem 1.1

4.1

Let be the function in Theorem 4.1. Fix $T>0$ and, let
be a unique minimizer of the functional
arbitrary. For each$\varphi\in C_{c}^{\infty}(\Omega\cross(0, T))$$u$$\epsilon>0$(4.18)$w_{\epsilon}$$G_{i,n}^{\epsilon}$$G_{i,n}^{\epsilon}(v) := \int_{\Omega}[\frac{1}{2}(\triangle v)^{2}+\frac{1}{2\tau_{n}}(v-u_{i-1,n})^{2}+\gamma_{\epsilon}(v-f)]dx,$

where(4.19)$\gamma_{\epsilon}(\lambda)=\{\begin{array}{ll}\frac{\lambda^{2}}{\epsilon} if \lambda<0,0 if \lambda>0,\end{array}$(4.20)$\beta_{\epsilon}(\lambda)=\gamma_{\epsilon}’(\lambda)$

Since$w_{e}$.

satisfies$\int_{\Omega}[\triangle w_{\epsilon}\Delta\varphi+\frac{1}{\tau_{n}}(w_{\epsilon}-u_{i-1,n})\varphi+\beta_{\epsilon}(w_{\epsilon}-f)\varphi]dx=0,$

we observe from the definition of$\beta_{\epsilon}$that$\int_{\Omega}[\triangle w_{\epsilon}\triangle\varphi+\frac{1}{\tau_{n}}(w_{\epsilon}-u_{i-1,n})\varphi]dx=-\int_{\Omega}\beta_{\epsilon}(w_{\epsilon}-f)\varphi dx\geq 0.$

Letting$\epsilonarrow 0$, the $pro$ of of Theorem 3.3 implies that$\int_{\Omega}[\triangle u_{i,n}\Delta\varphi+\frac{1}{\tau_{n}}(u_{i,n}-u_{i-1,n})\varphi]dx\geq 0.$

Integrating it over $[0, T]$ and using Definitions 2.1 and 2.2, we deduce that
(4.21)$\int_{0}^{T}\int_{\Omega}[\triangle\tilde{u}_{n}(x, t)\triangle\varphi(x, t)+V_{n}(x, t)\varphi(x, t)]dxdt\geq 0.$

with
defined by$\varphi\geq 0$

23
It follows from (4.16) that$\int_{0}^{T}\int_{\Omega}\triangle\tilde{u}_{n}(x, t)\triangle\varphi(x, t)dxdtarrow\int_{0}^{T}\int_{\Omega}\triangle u(x, t)\triangle\varphi(x, t)dxdt$

as$narrow+\infty,$

and while (4.6) gives us$\int_{0}^{T}\int_{\Omega}V_{n}(x, t)\varphi(x, t)dxdtarrow\int_{0}^{T}\int_{\Omega}u_{t}(x, t)\varphi(x, t)dxdt$

as$narrow+\infty.$

Thus, letting $narrow+\infty$ in (4.21), we observe that
(4.22)

Since$\int_{0}^{T}\int_{\Omega}[\triangle u(x, t)\triangle\varphi(x, t)+u_{t}(x, t)\varphi(x, t)]dxdt\geq 0.$$\varphi$

is arbitrary, (4.22) implies that

(4.23)

a.e. in$\triangle^{2}u(x, t)+u_{t}(x, t)\geq 0$$\Omega\cross(0, T)$

,

where
is written in the distribution sense. Moreover, the regularity ofTheorems 4.1-4.3.
We now prove (1.7). By (4.17) and Theorem 3.3, we observe that$\triangle^{2}u$(4.24)$u$follows from$\Vert\mu_{n}\Vert_{L^{2}([0,T];\mathcal{M}(\Omega))};=\int_{0}^{T}(\int_{\Omega}d\mu_{n})^{2}dt$$= \sum_{i=1}^{n}\int_{(i-1)\tau_{n}}^{i\tau_{n}}(\int_{\Omega}d\mu_{i,n})^{2}dt=\tau_{n}\sum_{i=1}^{n}\mu_{i,n}(\Omega)^{2}<C.$

This implies that$\mu_{n}arrow\overline{\mu}$weakly in$L^{2}(0, T;\mathcal{M}(\Omega))$

up to a subsequence. Setting$\mu_{t}:=\triangle^{2}u+u_{t},$we observe from (4.23) that is a measure on
uniqueness of the limit. Since
converges tofrom (4.24) that$\mu$$\mu_{n}$, and it holds thatbyweakly in, it follows$\Omega\cross(0, T)$$\mu_{t}$$\overline{\mu}=\mu_{t}$$L^{2}(0, T;\mathcal{M}(\Omega))$$\Vert\mu_{t}\Vert_{L^{2}(0,T;\mathcal{M}(\Omega))}\leq\lim_{narrow}\inf_{\infty}\Vert\mu_{n}\Vert_{L^{2}(0,T,\mathcal{M}(\Omega))}\leq C.$

This is equivalent to (1.7), and implies that$t\in(0, T)$.

$\mu_{t}$is a positive Radon measure on$\Omega$

for a.e.

Finally we prove that is a solution of the problem (P). To prove this assertion, it issufficient to show that, if $u>f$ , then $\triangle^{2}u+u_{t}=0$ holds. Let us set$u$$\mathcal{N}:=\{(x, t)\in\Omega\cross(O, T):u(x, t)>f(x)\}.$

24
by Theorems 4.2 and 4.3,
is continuous in$\delta>0$
and a neighborhood, there exist
for any
that

Since

is an open set, so that,suchof$\mathcal{N}$$\Omega\cross(0, T)$$u$$W\cross(t_{1}, t_{2})$$(x^{0}, t^{0})\in \mathcal{N}$(4.25)$u(x, t)-f(x)>\delta$in$W\cross(t_{1}, t_{2})$$(x^{0}, t^{0})$.

Lemma 4.1 implies that there exists a number $N>0$ such that$\tilde{u}_{n}(x, t)>u(x, t)-\frac{\delta}{2}$in$W\cross(t_{1}, t_{2})$for any $n>N.$

Combining this with (4.25), we have, for any $n>N,$(4.26)$\tilde{u}_{n}(x, t)>f(x)+\frac{\delta}{2}$

Let

with

$\zeta\in C_{0}^{\infty}(W\cross(t_{1}, t_{2}))$$0\leq\zeta\leq\delta/2$

in$\psi$(4.27)

as$\varphi$.

. Then (4.26) asserts that$\psi(x, t);=\tilde{u}_{n}(x, t)-\zeta(x, t)\in K$

Taking this$W\cross(t_{1}, t_{2})$for each$t\in[O, T].$in (3.16) and integrating it with respect to$t$on $(0, T)$ , we obtain$\int_{0}^{T}\int_{\Omega}\triangle u_{i,n}(x)\zeta(x, t)dxdt\leq-\int_{0}^{T}\int_{\Omega}V_{i,n}(x)\zeta(x, t)dxdt.$

From the definition (4.17), the inequality can be reduced to(4.28)

Since$\sum_{i=1}^{n}\int_{(i-1)\tau_{n}}^{i\tau_{n}}\int_{\Omega}\zeta(x, t)d\mu_{n}dt\leq 0.$$\mu_{n}\geq 0$, we see that the integral in (4.28) must be equal to , i.e.,$0$(4.29)$\mu_{n}(W\cross(t_{1}, t_{2}))=0.$

It follows from (4.24) that$\Vert\mu_{n}\Vert_{\mathcal{M}(\Omega\cross(0,T))}:=\int_{0}^{T}\int_{\Omega}d\mu_{n}dt<C.$

Thus we deduce that$\mu_{n}$converges to$\mu_{t}$weakly in$\mathcal{M}(\Omega\cross(0, T))$, i.e.,$\int_{0}^{T}\int_{\Omega}\varphi(x, t)d\mu_{n}dtarrow\int_{0}^{T}\int_{\Omega}\varphi(x, t)d\mu_{t}dt$

for any(4.30)$\varphi\in C_{0}^{\infty}(\Omega\cross(0, T))$

. This fact also yields that$\Vert\mu_{t}\Vert_{\mathcal{M}(\Omega\cross(0,T))}\leq\lim_{narrow+}\inf_{\infty}\Vert\mu_{n}\Vert_{\mathcal{M}(\Omega\cross(0,T))}.$

Combining (4.29) with (4.30), we conclude that
(4.31)

which completes the proof.

$\mu_{t}(W\cross(t_{1}, t_{2}))=0,$$\square$25

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