Abstract
Two kinds of machinery to show regularity of solutions of bilateral/unilateral obstacle problems are presented. Some generalizations of known results in the lit- erature are included. Several important open problems in the topics are given.
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Keywords
- Bilateral/Unilateral obstacle problem
- Regularity of solutions
- Bernstein method
- Bellman-Isaacs equation
- Penalization
- Fully nonlinear elliptic equation
- Weak Harnack inequality
- \(L^{p}\) viscosity solution
Mathematics Subject Classification
1 Introduction
In this survey, we overview regularity of solutions of obstacle problems associated with second-order uniformly elliptic partial differential equations (PDE for short). Particularly, we show two different arguments to obtain estimates on solutions of obstacle problems due to maximum principles. On the other hand, there have appeared a huge amount of results concerning on regularity of solutions of variational inequalities, whose typical example is the obstacle problem. However, our methods here do not rely on integration by parts.
One of techniques here is the so-called Bernstein method, which is relatively old, while the other is quite a new one. Inspired by an idea in [20], we have found an interesting argument in [42], which can be applied to fully nonlinear PDE with unbounded coefficients and inhomogeneous terms.
According to [52], it seems that Fichera [24, 25] first studied the Signorini problem as a variational inequality, where a free boundary arises on the boundary of domains. Stampacchia in [54] announced variational inequalities in Hilbert spaces as a modification of Lax-Milgram theorem. Later, Lions-Stampacchia in [46] introduced unilateral obstacle problems in the whole domain as an example of minimization problems associated with energy functionals over closed convex sets.
Afterwards, several regularity results on solutions of variational inequalities appeared in [6, 7, 27, 44].
We shall first consider a minimizing problem of given energies under restrictions. Fix a bounded domain \(\Omega \subset {\mathbb R}^n\) with smooth boundary \(\partial \Omega \). For a given \(\psi \in C(\overline{\Omega })\), which is called an upper obstacle, we set a closed convex set
where \(H^1_0(\Omega )\) is the closure of \(C_0^\infty (\Omega )\) with respect to \(H^1(\Omega )\) norm.
For any fixed \(f\in L^2(\Omega )\), by setting our energy
for \(u\in K^\psi \), it is known that there is a unique \(u\in K^\psi \) such that
Formally, we observe that
Hence, we may write down this problem as a Bellman equation
under the Dirichlet condition \(u=0\) on \(\partial \Omega \).
Obstacle problems arise in various settings both from purely mathematical interests and from their rich applications. For later topics, we only refer to some text books [3, 26, 34, 45, 53, 56] because it is too wide for this article to mention these issues. We will concentrate on regularity of solutions of obstacle problems but not on regularity of the free boundary, which may be more interesting subject. See [11, 14, 28] and references therein for this topics.
It is worth mentioning that for (1.1), we can only expect solutions to belong to \(W^{2,\infty }(\Omega )\) in general even if \(\psi \) and f are smooth enough. The first example is a simple one.
Example 1.1
Let \(\Omega :=(-\frac{5}{4},\frac{5}{4})\) for \(n=1\), and \(\psi (x)=x^2-1\). We easily see that
satisfies
under the Dirichlet condition \(u(\pm \frac{5}{4})=0\). We notice that this u is not twice differentiable at \(x=\pm \frac{1}{2}\).
We next show the other example when there is a 0th order term of unknown functions.
Example 1.2
Let \(\Omega \) and \(\psi \) be the same ones as in Example 1.1. For the inhomogeneous term \(f\in C^2(\overline{\Omega })\), we choose
It is easy to verify that the same function u in Example 1.1 satisfies
We next consider a minimizing problem under the other kind of restriction. Given two obstacles \(\varphi ,\psi \in C(\overline{\Omega })\) satisfying the compatibility condition
we introduce the closed convex set
Again, it is known that there is a unique \(u\in K^\psi _\varphi \) such that
We observe that u satisfies at least formally
This is a bilateral obstacle problem, which is an Isaacs equation while (1.1) is called a Bellman equation for unilateral obstacle problems.
Because of (1.2), it is easy to see formally that (1.3) is equivalent to the following PDE:
Using the standard Euclidean inner product \(\langle \cdot ,\cdot \rangle \), we consider the energy
where \(A:=(a_{ij}):\Omega \rightarrow S^n\) is positively definite; \(\exists \theta >0\) such that
Here and later \(S^n\) denotes the set of real-valued symmetric matrices of order n.
When \(a_{ij}\in C^1(\overline{\Omega })\) for simplicity, the minimizer of \(E[\cdot ]\) over \(H_0^1(\Omega )\) formally satisfies
where
Here, we set
Hence, as before, we derive the Bellman equation associated with the minimization of \(E[\cdot ]\) over \(K^\psi \):
Throughout this paper, we shall suppose that there is \(M_c>0\) such that
If we suppose that c is positive in \(\overline{\Omega }\), then particularly, \(L^\infty \) estimates become easier to prove. In fact, under (1.5), we need a perturbation function such as w in Proposition 2.1. We choose \(R_0>0\) such that
Here and later, we set \(B_r:=\{ y\in {\mathbb R}^n \ | \ |x|<r\}\), and \(B_r(x):=x+B_r\) for \(x\in {\mathbb R}^n\).
In this survey, we are concerned with regularity of solutions for obstacle problems, where the PDE part is given by the above linear second-order uniformly elliptic operator L or Bellman-Isaacs ones. We will always assume that the existence of (approximate) solutions of each obstacle problem. In Sects. 2 and 3, using Bernstein method, we obtain (local) \(W^{2,\infty }(\Omega )\) estimates on solutions of approximate equations via penalization. We consider the case when the PDE part is linear with bilateral obstacles in Sect. 2 while we deal with Bellman equations with bi- and unilateral obstacles in Sect. 3. In Sect. 4, to show the Hölder continuity of the first derivative, we apply the weak Harnack inequality to solutions of bilateral obstacle problems, where the main PDE part can be of Isaacs type, and moreover, coefficients and inhomogeneous terms can be unbounded. Since fully nonlinear PDE contain 0th order terms in Sect. 4, we need to modify basic tools such as the Aleksandrov-Bakelman-Pucci (ABP for short) maximum principle, weak Harnack inequality and local maximum principle to PDE with 0th order terms. In Appendix, we present those for the reader’s convenience.
2 A Linear Operator Case
Although some results in this section will be generalized in Sect. 3, we will present those to clarify our basic argument.
In this section, for coefficients in the linear operator L, and obstacles, we impose that
To introduce penalty equations, we need \(\beta \in C^2({\mathbb R})\) such that
For instance, it is easy to verify that \(\beta \in C^2({\mathbb R})\) defined by
satisfies all the properties in (2.2).
For \(\varepsilon \in (0,1)\), we will use \(\beta _\varepsilon (t):=\beta (t/\varepsilon )\) for \(t\in {\mathbb R}\). Furthermore, we easily observe that
We shall consider approximate equations with penalized terms:
under the Dirichlet condition
Hereafter, we will use the notations: for \(t,s\in {\mathbb R}\),
For simplicity, we will write
We also use the summation convention for repeated indices, e.g..
Proposition 2.1
\((L^{\infty }\) estimates) Assume (1.2), (1.4), (1.5) and (2.1). Let \(u^\varepsilon \in C(\overline{\Omega })\cap C^2(\Omega )\) be solutions of (2.4) satisfying (2.5). Then, there is \(\hat{C}>0\) such that
Proof
We shall only prove the second inequality since the first one can be shown similarly. We shall write u for \(u^\varepsilon \) for simplicity.
Setting \(C_0:=\max _{\overline{\Omega }}\varphi ^+\ge 0\) and \(C_1:=\max _{\overline{\Omega }}f^+\), we shall suppose
Here \(\mu >0\), \(\delta \in (0,1)\) and \(w(x):=e^{2\gamma R_0}-e^{\gamma (x_1+R_0)}> 0\) for \(x=(x_1,\ldots ,x_n)\in \Omega \), where \(\gamma \ge 1\), and \(R_0>0\) is from (1.6).
By letting \(\hat{x}\in \overline{\Omega }\) satisfy \(\Theta =u(\hat{x})-C_0-\mu (C_1+\delta )w(\hat{x})\), (2.5) yields \(\hat{x}\in \Omega \). Hence, at \(\hat{x}=(\hat{x}_1,\ldots ,\hat{x}_n)\in \Omega \), the weak maximum principle implies
Here and later, to distinguish composite functions \(\beta _\varepsilon (u-\psi )\) and \(\beta _\varepsilon (\varphi -u)\), we use the following notation:
Thus, for a fixed \(\gamma := (\max _{\overline{\Omega }}|b_1|+\theta )/\theta \), (2.6) together with (1.5) implies
Since \(\varphi -u\le \varphi -C_0-\mu (C_1+\delta )w\le \varphi -C_0\le 0\) at \(\hat{x}\), this inequality yields
which is a contradiction for \(\mu >1/(\theta \gamma )\). Therefore, for fixed \(\mu ,\gamma >0\) in the above, we have \(\Theta \le 0\), which concludes the proof. \(\square \)
We notice that in the above proof, we do not need the whole of (2.1) but we do not present “minimal” hypotheses on regularity of given functions for the sake of presentations.
Proposition 2.2
\((W^{2,p}\) estimates) Assume (1.2), (1.4), (1.5) and (2.1). Let \(u^\varepsilon \in C^2(\overline{\Omega })\) be solutions of (2.4) satisfying (2.5). Then, there is \(\tilde{C}>0\) such that for \(\varepsilon \in (0,1)\),
In particular, for each \(p\in (1,\infty )\), there is \(\tilde{C}_p>0\) such that
Proof
We shall only show the bound for \(\overline{\beta _\varepsilon }\) since we can prove the other one similarly. We shall simply write u for \(u^\varepsilon \) again.
Suppose that \(\Theta :=\max _{\overline{\Omega }}\overline{\beta _\varepsilon }>0\). In view of the second inequality of (1.2), we can choose \(\hat{x}\in \Omega \) such that \(\Theta =\beta _\varepsilon (u(\hat{x})-\psi (\hat{x}))\). Since \(\overline{\beta _\varepsilon }\) is nondecreasing, we see that \(u-\psi \) attains its maximum at \(\hat{x}\in \Omega \). Hence, we have at \(\hat{x}\),
Here and later, C denotes the various positive constant depending only on known quantities.
Note that the first inequality of (1.2) yields
Therefore, we have \( 0\le \overline{\beta _\varepsilon }\le \overline{\beta _\varepsilon }(\hat{x})\le \max _{\overline{\Omega }}f^++M_c\max _{\overline{\Omega }}\psi ^-+C\Vert D\psi \Vert _{W^{1,\infty }(\Omega )}\) in \(\overline{\Omega }\), where \(M_c>0\) is the constant in (1.5) \(\square \)
Remark 2.3
When we consider Bellman operators in Sect. 3, the \(L^\infty \) estimate on the penalty terms for obstacles does not imply (2.8) because we will have one more penalty term, which cannot be evaluated by the above argument.
Now, we show local \(W^{2,\infty }\) estimates on solutions of (2.4). Our argument is more or less standard though we do not know if the next proposition has appeared somewhere to our knowledge.
Proposition 2.4
(Local \(W^{2,\infty }\) estimates) Assume (1.2), (1.4), (1.5) and (2.1). Let \(u^\varepsilon \in C^4(\Omega )\cap C^1(\overline{\Omega })\) be solutions of (2.4). Then, for each compact set \(K\Subset \Omega \), there is \(\tilde{C}_K>0\) independent of \(\varepsilon \in (0,1)\) such that
Proof
Choose \(\zeta \in C_0^\infty (\Omega )\) such that
Putting \(M:=\max _{\Omega }\zeta |D^2u^\varepsilon |\), we may suppose \(M\ge 1\).
Writing u and \(\beta \) for \(u^\varepsilon \) and \(\beta _\varepsilon \), respectively, we set
We shall write \(\overline{\beta }:=\beta (u-\psi )\) and \(\underline{\beta }:=\beta (\varphi -u)\) again for simplicity. In the proceeding calculations, we shall more simply write \(u_{ij}\), \(u_{ijk}\), \(a_{ij,k}\) etc. for \(u_{x_ix_j}\), \(u_{x_ix_jx_k}\), \((a_{ij})_{x_k}\) etc., respectively.
We may suppose that \(\max _{\overline{\Omega }}V=V(\hat{x})>0\) for some \(\hat{x}\in \Omega \). By setting \(L_0 \xi :=-a_{ij}\xi _{ij}+b_i\xi _i\), since \(L_0V(\hat{x})\ge 0\) by the weak maximum principle, at \(\hat{x}\), we have
By Young’s inequality, at \(\hat{x}\), we have
for large \(\gamma >1\).
Since (2.8) for \(p>n\) implies the \(W^{1,\infty }\) estimates on u, we will not mention the dependence on \(\Vert u\Vert _{W^{1,\infty }(\Omega )}\) in the calculations below. In order to estimate \(I_3\), we differentiate (2.4) with respect to \(x_k\) to obtain
Thus, we have
To estimate \(I_2\), we differentiate (2.4) with respect to \(x_k\) and \(x_\ell \) to obtain
Hence, we have
Thus, inserting this in (2.9) with \(\gamma \ge 2/\theta \), we have
\(\underline{\text {Case 1}: J_2\le 0\, \text {and}\, J_3\le 0}\): In this case, for a largely fixed \(\gamma>>2/\theta \), we immediately have
which together with Propositions 2.1 and 2.2 implies
\(\underline{\text {Case 2}:\, J_2>0\, \text {or}\, J_3>0}\): We shall only consider the case of \(J_2>0\) since the other one can be shown similarly. In view of (2.7), we see that
which yields
Therefore, M is bounded independently from \(\varepsilon \in (0,1)\). \(\square \)
Remark 2.5
We note that our choice of auxiliary functions V does not work for Bellman operators in Sect. 3. Instead, we will barrow a different one from [23], which can be applied only to unilateral obstacle problems.
As mentioned in Sect. 1, Jensen in [32] showed \(W^{2,\infty }(\Omega )\) estimates under additional assumptions on the coefficients. Here, in order to simplify the argument, we shall obtain the \(W^{2,\infty }\) bound near the flat boundary under additional assumptions. Setting \(x'=(x_1,\ldots ,x_{n-1})\in {\mathbb R}^{n-1}\), we suppose that \(\Omega \) satisfies
To show \(W^{2,\infty }\) estimates near \(\partial \Omega \) for bilateral obstacle problems, we follow the argument in [31].
Theorem 2.6
Assume (1.2), (1.4), (1.5), (2.1) and (2.10). Assume also that
Let \(u^\varepsilon \in C^4(\overline{\Omega })\) be solutions of (2.4). Then, there is \(\hat{C}>0\) such that
Remark 2.7
Under hypothesis (2.11), we note that
since \(u^\varepsilon _i=u^\varepsilon _{ij}=0\) for \(1\le i,j\le n-1\) on \(\partial \Omega \cap B_1\) by (2.5).
Proof
As before, we shall write u for \(u^\varepsilon \), and use other simplified notations.
We choose \(\eta \in C^\infty _0(B_1)\) such that
Setting
where \(\hat{b}_n=b_n/a_{nn}\) and \(\hat{f}=f/a_{nn}\), we define
Consider W defined by
where \(M:=\max _{\overline{\Omega }}\eta |D^2u|\), and \(A, \gamma >1\) will be fixed. We may suppose \(M\ge 1\).
Let \(\hat{x}=(\hat{x}_{1},\hat{x}_{2},\ldots ,\hat{x}_{n})\in \overline{\Omega }\cap \overline{B}_1\) be a point such that \(\max _{\overline{\Omega }\cap \overline{B}_1}W=W(\hat{x})>0\). Because of \(W(\hat{x})>0\), we may also assume that \(\hat{x}\in \overline{\Omega }\cap B_1\).
Since the argument in the proof of Proposition 2.4 can be applied to the case when \(\hat{x}\in \Omega \cap B_1\) with some minor changes, we may suppose \(\hat{x}\in \partial \Omega \cap B_1\), and we will obtain a contradiction. Since \(|D^2v|^2=2\sum _{i=1}^{n-1}u_{in}^2\) at \(\hat{x}\), (2.5) implies
By noting \(u_{inn}=(\hat{b}_nu_n-\hat{f})_i\) at \(\hat{x}\), this equality implies
for a fixed \(A>1\). If the right hand side of the above is non-positive, then we have
which implies the uniform bound of M independent of \(\varepsilon \in (0,1)\). Therefore, we have \(W_n(\hat{x})>0\) but this implies that \(\hat{x}\) is not the maximum of W, which is a contradiction. \(\square \)
Following [31], we give a sufficient condition to derive (2.11). We use the following notation:
Although \(B_1^+\) is not a smooth domain, considering an appropriate smooth domain \(\Omega \supset B_1^+\), we may assume \(\partial B_1^+\) is smooth. The next proposition yields (2.11).
Proposition 2.8
Suppose that there is \(\alpha \in (0,1)\) such that
There is a \(C^4\)-diffeomorphism \(T=(T_1,\ldots , T_n):\overline{B}_1^+\rightarrow T(\overline{B}_1^+)\) such that \(T_k\in C^{4,\alpha }(\overline{B}_1^+)\) such that
and
Proof. We begin with considering the following PDE
such that \(u(x)=0\) for \(x=(x_1,\ldots ,x_{n-1},0)\in \overline{B}^+_1\). Consider the change of variable \(\hat{T}=(\hat{T}^1,\ldots ,\hat{T}^n):\overline{B}^1_+\rightarrow {\mathbb R}^n\) defined by
Here, \(T=(T^1,\ldots , T^n)\in C^{4,\alpha }(\overline{B}^+_1;{\mathbb R}^n)\) is the solution of
where \(\nu \) is the outward unit normal of \(\partial B^+_1\).
It is easy to rewrite the equation for \(v(y):=u(x)\) with this new variable \(y=\hat{T}(x)\):
where \(\hat{c}(y)=c(x)\), \(\hat{f}(y)=f(x)\), \(\hat{\psi }(y)=\psi (x)\), \(\hat{\varphi }(y)=\varphi (x)\),
and
In view of the boundary condition of (2.5), it is immediate to verify that for \(1\le i\le n-1\),
Open question 1: Is it possible to obtain \(W^{2,\infty }(\Omega )\) estimates with no extra assumption (2.11) on \(a_{ij}\)?
3 A Bellman Type Operator Case
In this section, we obtain \(W^{1,\infty }\) bounds for solutions of bilateral obstacle problems when the PDE part is of Bellman type. However, we do not know if we can show further estimates on the second derivative of solutions of penalized systems below for bilateral obstacle problems. Thus, following [43], we will discuss local \(W^{2,\infty }\) estimates on solutions of unilateral obstacle problems for Bellman equations.
3.1 Bilateral Obstacles
We first consider the following bilateral obstacle problems
where \(F:\overline{\Omega }\times {\mathbb R}\times {\mathbb R}^n\times S^n\rightarrow {\mathbb R}\) is defined by
Here, by letting \(N\ge 2\) be a fixed integer, for \(k\in \mathcal{N}:=\{ 1,2,\ldots ,N\}\), functions \(A^k=(a_{ij}^k):\overline{\Omega }\rightarrow S^n\), \(b^k=(b^k_i):\overline{\Omega }\rightarrow {\mathbb R}^n\), \(c^k:\overline{\Omega }\rightarrow {\mathbb R}\) and \(f^k:\Omega \rightarrow {\mathbb R}\) are given. We will use linear operators
As in Sect. 2, we suppose that there is \(\theta >0\) such that
and there is \(M_c>0\) such that
Following [22], we introduce a system of PDE via penalization: for \(k\in \mathcal{N}\),
where \(u^{N+1}:=u^1\) and \(\beta _\varepsilon \) is given in Sect. 2. In order to distinguish three \(\beta _\varepsilon \) in (3.5), we will simply write
For given functions, we suppose that
Setting
we have the \(L^\infty \) estimates on \(u^{k,\varepsilon }\) independent of \((\varepsilon ,k)\in (0,1)\times \mathcal{N}\).
Proposition 3.1
\((L^{\infty }\) estimates) Assume (1.2), (3.3) and (3.6). Let \(u^\varepsilon :=(u^{k,\varepsilon })\in C^2(\overline{\Omega };{\mathbb R}^N)\) be solutions of (3.5). Then, there is \(\hat{C}>0\) such that
Proof
Setting \(C_0:=\max _{\overline{\Omega }}\psi ^-\) and \(C_1:=\max _{\overline{\Omega }}\underline{f}\), we suppose
Here, \(\delta >0\) will be sent to 0 in the end, and w is the function in the proof of Proposition 2.1; \(w(x):=e^{2\gamma R_0}-e^{\gamma (x_1+R_0)}> 0\) in \(\Omega \subset B_{R_0}\), where \(\gamma >0\) will be fixed later. Dropping \(\varepsilon >0\) from \(u^{k,\varepsilon }\) and \(\beta _\varepsilon \), we may assume that there is \(\hat{x}\in \Omega \) such that
By setting \(\gamma :=\left(\max _{k\in \mathcal{N},x\in \overline{\Omega }}|b^k_1|+\theta \right)/\theta \), the weak maximum principle implies that at \(\hat{x}\in \Omega \),
Since \(u^1\le u^2\) and \(u^1-\psi \le 0\) at \(\hat{x}\), these observation yield
which gives a contradiction when \(\mu >1/(\theta \gamma )\). Therefore, we conclude the proof of the first inequality.
The second inequality can be shown more easily since we may avoid the penalty term \(\beta _\varepsilon (u^k-u^{k+1})\) in the opposite inequalities. \(\square \)
Next, we show \(L^\infty \) estimates on \(\beta _\varepsilon (u^k-\psi )\) and \(\beta _\varepsilon (\varphi -u^k)\) independent of \((\varepsilon ,k)\in (0,1)\times \mathcal{N}\).
Proposition 3.2
\((L^\infty \) estimates on penalty terms) Assume (1.2), (3.3) and (3.6). Let \(u^\varepsilon :=(u^{k,\varepsilon })\in C^2(\overline{\Omega };{\mathbb R}^N)\) be solutions of (3.5). Then, there exists \(\tilde{C}_1>0\) such that for \(\varepsilon \in (0,1)\) and \(k\in \mathcal{N}\),
Proof
We shall write \(u^k\) for \(u^{k,\varepsilon }\) as before. By the same reason in the proof of Proposition 3.1, we shall only show the estimates on \(\beta _\varepsilon (\varphi -u^k)\).
Suppose \(\max _{\overline{\Omega },\mathcal{N}}\underline{\beta }^k=\underline{\beta }^1(x_0)>0\) for some \(x_0\in \Omega \). Thus, we may assume \(\max _{\overline{\Omega },\mathcal{N}}(\varphi -u^k)=(\varphi -u^1)(x_0)>0\). Hence, at \(x_0\in \Omega \), we have
Since \(u^1-u^2\le 0\), and \(\varphi -u^1>0\) at \(x_0\), we have
which concludes the assertion as in the proof of Proposition 2.2. \(\square \)
Remark 3.3
Notice that we cannot apply the above argument to obtain \(L^\infty \)-estimates on \(\beta _\varepsilon (u^{k,\varepsilon }-u^{k+1,\varepsilon })\). Therefore, unlike Proposition 2.2, we cannot obtain \(W^{2,p}\) estimates on \(u^{k,\varepsilon }\).
For further regularity, we first obtain the estimate of first derivatives on \(\partial \Omega \) in Proposition 3.4 below. To this end, we shall use \(W^{1,\infty }\) estimates on approximate solutions of the associated unilateral obstacle problems via penalization.
Proposition 3.4
(Gradient estimates on \(\partial \Omega )\) Assume (1.2), (3.3) and (3.6). Let \(u^\varepsilon :=(u^{k,\varepsilon })\in C^1(\overline{\Omega };{\mathbb R}^n)\cap C^2(\Omega ;{\mathbb R}^n)\) be solutions of (3.5). Then, there exists \(\tilde{C}_2>0\) such that for \(\varepsilon \in (0,1)\) and \(k\in \mathcal{N}\),
Proof
Because \(u^{k,\varepsilon }=0\) on \(\partial \Omega \), we only need the estimate
where \(n=n(z)\in \partial B_1\) denotes the outward unit vector at \(z\in \partial \Omega \).
Let \(v^\varepsilon =(v^{k,\varepsilon }):\overline{\Omega }\rightarrow {\mathbb R}^N\) be the unique solution of the penalized system of the following unilateral obstacle problem.
Due to Lemmas 2.1, 2.2 and 3.1 in [43], we find \(\hat{C}_1>0\), and for each compact \(K\Subset \Omega \), \(\hat{C}_1(K)>0\) such that
We claim that
Indeed, if we suppose \(\Theta :=\max _{\overline{\Omega },\mathcal{N}}(v^{k,\varepsilon }-u^{k,\varepsilon }-\delta w)>0\), where \(\delta >0\) will be sent to 0, and w is the function in Proposition 2.1, then we may suppose \(\Theta = (v^{1,\varepsilon }-u^{1,\varepsilon }-\delta w)(\hat{x})\) for some \(\hat{x}\in \Omega \). Hence, at \(\hat{x}\), we have
provided \(\gamma \ge (\max _{\mathcal{N},\overline{\Omega }}|b^k_1|+\theta )/\theta \). Since \(v^{1,\varepsilon }>u^{1,\varepsilon }\) and \(v^{1,\varepsilon }-v^{2,\varepsilon }\ge u^{1,\varepsilon }-u^{2,\varepsilon }\) at \(\hat{x}\), we immediately obtain a contradiction. Therefore, we have
which concludes the claim by sending \(\delta \rightarrow 0\). Therefore, we have
On the other hand, for each \(k\in \mathcal{N}\), we next let \(w^{k,\varepsilon }\) be solutions of
We claim that for \((\varepsilon ,k)\in (0,1)\times \mathcal{N}\),
Indeed, assuming \(\max _{\overline{\Omega },\mathcal{N}}(u^{k,\varepsilon }-w^{k,\varepsilon }-\delta w)=(u^{1,\varepsilon }-w^{1,\varepsilon }-\delta w)(\hat{x})>0\) for some \(\hat{x}\in \Omega \), at \(\hat{x}\), we have
for large \(\gamma >1\) as before. Hence, the same argument to obtain (3.9) implies
By the same argument as in the proof of Proposition 2.2, we find \(\tilde{C}>0\) such that
which implies
This together with (3.9) and (3.10) concludes the assertion. \(\square \)
Now, we shall use Bernstein method to derive \(W^{1,\infty }(\Omega )\) estimates on \(u^{k,\varepsilon }\).
Proposition 3.5
Assume (1.2), (3.3) and (3.6). Let \(u^\varepsilon =(u^{k,\varepsilon })\in C^1(\overline{\Omega };{\mathbb R}^N)\cap C^3(\Omega ;{\mathbb R}^N)\) be solutions of (3.5). There exists \(\tilde{C}_3>0\) such that
Proof
We shall drop \(\varepsilon \) from \(u^{k,\varepsilon }\). Set
In view of Proposition 3.4, we may suppose that
for some \(\hat{x}\in \Omega \). We shall write u and v for \(u^1\) and \(u^2\), respectively. Furthermore, we shall write \(\beta \), \(\overline{\beta }\) and \(\underline{\beta }\) for \(\beta ^1\), \(\overline{\beta }^1\) and \(\underline{\beta }^1\), respectively.
We then have at \(\hat{x}\in \Omega \),
for large \(\gamma >1\). We use (2.3) to obtain the last inequality in the above.
Since we may suppose the last two terms are non-positive and \(V^1\ge V^2\) at \(\hat{x}\), we have \(\gamma \theta |Du(\hat{x})|^2\le C\), which concludes the assertion. \(\square \)
Since we do not know \(L^\infty \) estimates on \(\beta _\varepsilon (u^k-u^{k+1})\), it seems difficult to find a weak (or viscosity) solution of (3.1) only with \(W^{1,\infty }\) estimates. Thus, we shall switch to unilateral obstacle problems.
3.2 Unilateral Obstacles
In order to show local \(W^{2,\infty }\) estimates on solutions of obstacle problems, we shall restrict ourselves to consider unilateral obstacle ones;
where F is of Bellman type defined in (3.2).
Lenhart in [43] showed the \(W^{2,\infty }_{loc}(\Omega )\) estimates on solutions of (3.11). We will recall the argument here.
We notice that proceeding arguments for \(W^{2,\infty }\) estimates can not be applied to the following unilateral obstacle problem with the same F because the PDE below is of Isaacs type:
Open question 2: Is it possible to obtain (local) \(W^{2,\infty } \) estimates on solutions of (3.12)?
In place of (1.2), we only need to suppose
The penalized system of (3.11) is as follows: for \(u^\varepsilon =(u^{k,\varepsilon })\),
where \(u^{N+1,\varepsilon }:=u^{1,\varepsilon }\).
It is easy to establish the next lemma by following the proofs of Propositions 3.1, 3.2 and 3.5. We note that the Bernstein method with the standard barrier argument can also work for the Bellman equation with unilateral obstacles. We refer to Lemma 2.1 in [43] for the details.
Lemma 3.6
There exists \(\hat{C}>0\) such that
Following the argument in [43] with a bit simpler auxilialy function V below than that there, we establish \(W^{2,\infty }_{loc}(\Omega )\) estimates.
Theorem 3.7
(Local \(W^{2,\infty }\) estimates) Assume (3.3), (3.6) and (3.13). Let \(u^\varepsilon =(u^{k,\varepsilon })\in C^4(\Omega :{\mathbb R}^N)\cap C^1(\overline{\Omega }:{\mathbb R}^N)\) be solutions of (3.14). Then, for each compact \(K\Subset \Omega \), there is \(C_K>0\) such that
Proof
We shall simply write \(u^k\) for \(u^{k,\varepsilon }\) again.
Let \(\zeta \in C_0^\infty (\Omega )\) be the same function as in the proof of Proposition 2.4. Putting \(M^k=\max _{\overline{\Omega }}\zeta |D^2u^k|\), we may suppose \(M=\max _{\mathcal{N}}M^k=\zeta (\hat{z})|D^2u^1(\hat{z})|\ge 1\) for some \(\hat{z}\in \Omega \). By change of variables using the orthogonal matrix B such that \(BA^1(\hat{z})^tB=(\alpha _k\delta _{k\ell })\), we may suppose that
for some \(\alpha _k\ge \theta \). For each \(i\in \mathcal{N}\), setting
we may suppose that \(\max _{\mathcal{N},\overline{\Omega }}V^i=V^{i_0}(\hat{x})>0\) for some \(\hat{x}\in \Omega \) and \(i_0\in \mathcal{N}\).
We note that
Thus, for a fixed \(\gamma >1\), once we obtain
then we have
which concludes the assertion.
We shall write \(a_{ij}\), \(b_i\), c, V, u and v for \(a^{i_0}_{ij}\), \(b^{i_0}_i\), \(c^{i_0}\), \(V^{i_0}\), \(u^{i_0}\) and \(u^{i_0+1}\), respectively, for simplicity. The weak maximum principle yields, at \(\hat{x}\),
Hence, setting \(L_0 v:=-a_{ij}v_{ij}+b_iv_i\), at \(\hat{x}\), we have
By the definition of \(I_2\) and \(I_3\), we have
Moreover, \(I_4\) is estimated by
Hence, these inequalities together with Young’s inequality give
Note \(V^{i_0}\ge V^{i_0+1}\) at \(\hat{x}\). Furthermore, we may suppose \(0\ge -V^{i_0}+\zeta ^2|D^2\psi |^2+\gamma M\zeta ^2\alpha _k\psi _{kk}+\gamma M|D\psi |^2\) at \(\hat{x}\). Thus, taking \(\gamma \ge 2/\theta \), we have
Remembering \(M,\gamma \ge 1\), we have
which implies
provided \(\theta \gamma M\ge 2C\). This yields (3.15). \(\square \)
Open question 3: Is it possible to obtain \(W^{2,\infty }(\Omega )\) or \(W^{2,p}(\Omega )\) estimates for Bellman equations with unilateral obstacles under additional conditions if necessary?
Open question 4: Is it possible to obtain local \(W^{2,\infty }_{loc}(\Omega )\) estimates for Bellman equations with bilateral obstacles?
4 A Fully Nonlinear Operator Case
In Sects. 2 and 3, thanks to Bernstein method, we establish estimates on solutions of approximate PDE (or systems of PDE), which present the existence of (strong) solutions belonging to the associated function spaces (i.e. \(W^{2,\infty }(\Omega )\) or \(W^{2,\infty }_{loc}(\Omega )\)). See [43, 47] for the details. We also refer to [2] for a modern version of Bernstein method.
We note that there is a fully nonlinear uniformly elliptic equation which does not have classical solutions. See [48]. Furthermore, in [49], it is shown that there exists a viscosity solution of a fully nonlinear uniformly elliptic PDE whose second derivative is not bounded. On the other hand, we also know there is a classical solution of a special Isaacs equation consisting of three linear operators in [9].
In this section, we study more general PDE such as Isaacs equations with bilateral obstacles, and with unbounded, possibly discontinuous coefficients and inhomogeneous terms. In fact, to our knowledge, we do not know any regularity results for obstacle problems of Isaacs equations via penalization. In order to see a difficulty in the study of Isaacs equations via penalization, let us consider approximate Isaacs equations with no obstacles via penalization:
where \(u^{M+1,\ell }=u^{1,\ell }\) for \(\ell \in \mathcal{N}\) and \(u^{k,N+1}=u^{k,1}\) for \(k\in \mathcal{M}\). Here, by setting \(\mathcal{M}:=\{ 1,\ldots ,M\}\) and \(\mathcal{N}:=\{1,\ldots ,N\}\), \(u^{k,\ell }:\overline{\Omega }\rightarrow {\mathbb R}\) for \((k,\ell )\in \mathcal{M}\times \mathcal{N}\) are unknown functions, and linear operators are defined by
where given functions \(A^{k,\ell }:\overline{\Omega }\rightarrow S^n\), \(b^{k,\ell }:\overline{\Omega }\rightarrow {\mathbb R}^n\) and \(c^{k,\ell }:\overline{\Omega }\rightarrow [0,\infty )\) satisfy enough regularity.
If we obtain \(L^\infty _{loc}\) estimates on \(\beta _\varepsilon (u^{k,\ell }-u^{k+1,\ell })\) and \(\beta _\varepsilon (u^{k,\ell +1}-u^{k,\ell })\), then it is easy to verify that \(u^{k,\ell }_\varepsilon \) converge to a single limit u as \(\varepsilon \rightarrow 0\) (along a subsequence if necessary), which is a solution of
However, it is difficult to show \(L^\infty \) estimates on the first and second penalty terms. In fact, in a pioneering work [47], we first derive \(W^{2,\infty }\) estimates on solutions of penalized problems for Bellman equations (i.e. \(N=1\)), and then this gives \(L^\infty \) bounds for the penalty term. Moreover, Bernstein method does not work to obtain \(W^{2,\infty }\) estimates on solutions of (4.1). Furthermore, even if we establish \(W^{2,\infty }\) estimates on approximate solutions, since we have two penalty terms with opposite signs in (4.1), we still do not know if solutions of the system (4.1) converge to a single solution of (4.2).
Open question 5: Is it possible to obtain a weak/viscosity solution of (4.2) satisfying (2.5) via penalization?
If we restrict ourselves to try to establish \(C^{1,\gamma }\) estimates on solutions of bilateral obstacle problems for \(\gamma \in (0,1)\), then we can accomplish such estimates even when F is of Isaacs type;
Moreover, since we do not need systems of PDE via penalization, we may deal with compact sets \(\mathcal{M},\mathcal{N}\) in \({\mathbb R}^m\) for some \(m\in \mathbb N\). Furthermore, since we will not differentiate PDE (because it is impossible!), it is possible to treat discontinuous coefficients and inhomogeneous terms. In this procedure, we need to show the existence of weak/viscosity solutions of Isaacs equations with obstacles by a different method. We only refer to [16] and [42] for the existence issue.
This section is based on a recent work by the author and Tateyama in [42].
4.1 Equi-Continuity
Modifying arguments by Duque in [20], we present an idea to apply the weak Harnack inequality to obtain estimates on solutions of obstacle problems when the PDE part may be fully nonlinear. Here, the terminology fully nonlinear means that the mapping \((\xi ,X)\in {\mathbb R}^n\times S^n\rightarrow G(x,r,\xi ,X)\in {\mathbb R}\) is neither convex nor concave for each \((x,r)\in \Omega \times {\mathbb R}\).
In what follows, we suppose that
Here, \(p_0\in [\frac{n}{2},n)\) is the so-called Escauriaza’s constant in [21], and for a fixed \(\theta \in (0,1]\), Pucci operators \(\mathcal{P}^\pm :S^n\rightarrow {\mathbb R}\) are defined as follows:
where \(S^n_\theta :=\{ X\in S^n \ | \ \theta I\le X\le \theta ^{-1}I\}\). Under hypotheses \((i){-}(iv)\) in (4.3), we easily verify that
for \(x\in \Omega \), \(r\in {\mathbb R}\), \(\xi \in {\mathbb R}^n\) and \(X\in S^n\).
In a celebrated paper [10] by Caffarelli, it has turned out that to establish the regularity of viscosity solutions of fully nonlinear uniformly elliptic PDE
instead of this equation, it is essential to study extremal inequalities:
where \(G^\pm (x,r,\xi ,X):=\mathcal{P}^\pm (X)\pm \mu (x)|\xi |\pm c_0(x)r^\pm \).
Furthermore, according to [10] again, the key for the regularity theory is the weak Harnack inequality for supersolutions.
We recall the definition of \(L^p\)-viscosity solutions of
where \(H:\Omega \times {\mathbb R}\times {\mathbb R}^n\times S^n\rightarrow {\mathbb R}\) is given (not necessarily continuous).
Definition 4.1
We say that \(u\in C(\Omega )\) is an \(L^p\) viscosity subsolution (resp., supersolution) of (4.4) if it follows that
whenever for any \(\zeta \in W^{2,p}_{loc}(\Omega )\), \(u-\zeta \) attains its local maximum (resp., minimum) at \(x\in \Omega \). Finally, we say that \(u\in C(\Omega )\) is an \(L^p\) viscosity solution of (4.4) if it is both of an \(L^p\) viscosity subsolution and an \(L^p\) viscosity supersolution of (4.4).
Throughout this section, we at least suppose that
satisfy (1.2). Under hypotheses (4.3), (4.5) and (1.2), we consider
under the Dirichlet condition (2.5). It is immediate to see that if \(u\in C(\Omega )\) is an \(L^p\) viscosity subsolution (resp., supersolution) of (4.6) with this G, then it is an \(L^p\) viscosity subsolution (resp., supersolution) of
in \(\Omega \). We will only use these information in the argument below.
We recall a reasonable result without proof.
Proposition 4.2
(Proposition 2.9 in [42]) Under the same hypotheses as in Theorem 4.3, if \(u\in C(\Omega )\) is an \(L^p\) viscosity subsolution (resp., supersolution) of (4.6), then it follows that
In what follows, we call \(\omega \) a modulus of continuity of functions if
We also use the notation \(A^i\) for the set of interior points of \(A\subset {\mathbb R}^n\).
Theorem 4.3
(Theorem 2.10 in [42]) Assume (4.3), (4.5) and (1.2). Then, there exists a modulus of continuity \(\omega \) such that for any \(L^p\) viscosity solution of (4.6) satisfying (2.5), it follows that
Moreover, if we suppose \(\varphi ,\psi \in C^\alpha (\overline{\Omega })\) for some \(\alpha \in (0,1)\), then there are \(\hat{C}>0\) and \(\hat{\alpha }\in (0,\alpha ]\) such that for any \(L^p\) viscosity solution of (4.6) satisfying (2.5), it follows that
For \(r>0\) and \(x\in {\mathbb R}^n\), we define closed cubes as follows:
Proof
We shall only give a proof for local estimates since we can modify the argument below by using the weak Harnack inequality near \(\partial \Omega \). See [29] for its usage.
Fix any \(K\Subset \Omega \). We shall divide K by
and \(K_0:=K\setminus (K^+\cup K^-)\).
It is standard to show the assertion when \(x,y\in K_0^i\). See [42] for the details.
Let \(\omega _0\) be the modulus of continuity of obstacles;
Fix any \(\hat{x}\in K\). We may suppose \(\hat{x}=0\) by translation. For \(r\in (0,d_0/(2\sqrt{n}))\), where \(d_0:=\text{ dist }(\partial \Omega ,K)\), we set
Notice that \(\varphi (0)+\omega _0(2\sqrt{n}r)\ge \varphi \) and \(\psi \ge \psi (0)-\omega _0(2\sqrt{n}r)\) in \(Q_{4r}\subset \Omega \). It is standard to see that \(\overline{u}\) and \(\underline{u}\) are, respectively, an \(L^p\) viscosity subsolution and supersolution of
For \(s\in (0,d_0)\), set
We then define
It is immediate to see that \(\overline{U}\) and \(\underline{U}\) are nonnegative \(L^p\) viscosity supersolutions of
Since \(\Vert \mu \Vert _{L^n(Q_{4r})}\le \Vert \mu \Vert _{L^q(Q_{4r})}(2\sqrt{n}r)^{1-\frac{n}{q}}\), we can apply Proposition 5.7 in Appendix with the standard scaling to have
where \(\alpha _0:=2-\frac{n}{p\wedge n}\in (0,1]\). By noting \(M_{4r}-m_{4r}=\overline{U}+(\overline{u}-u)+(u-\underline{u})+\underline{U}\le \overline{U}+4\omega _0(2\sqrt{n}r)+\underline{U}\), the above inequalities imply
which gives a decay estimate of oscillations:
Since \(u(x)-u(y)\le \overline{u}(x)-\underline{u}(y)\), it is standard (e.g. in [29]) to obtain equi-continuity of u.
If \(\varphi \) and \(\psi \) are Hölder continuous, then the above estimate implies the Hölder continuity with some exponent. \(\square \)
4.2 \(C^{1,\gamma }\) Estimates
Now, assuming that there is \(\hat{\gamma }\in (0,1)\) such that
we will suppose (4.3) but \(q\ge p>n\) in (v). Under this assumption, we will use the Hölder exponent
For simplicity, we will also suppose
For G in (4.3), we use the notation:
Theorem 4.4
Assume (4.3) replaced by \(q\ge p>n\) in (v), (4.7) and (4.8). For any \(K\Subset \Omega \), there exist \(\hat{C}_K>0\), \(\gamma \in (0,\gamma _0]\), \( r_0\in (0,\text{ dist }(K,\partial \Omega ))\), and \(\delta _0>0\) such that if \(u\in C(\Omega )\) is an \(L^p\) viscosity solution of (4.6), and if
where by setting \(C_K[u]:=\{ x\in K \ | \ u(x)=\varphi (x) \text{ or } u(x)=\psi (x)\}\), we define the non-coincidence set by \(N_K[u]:=\{ x\in K \ | \ \text{ dist }(x,C_K[u])>0\}\), then it follows that
Proof
Following the argument in the proof of Proposition 5.1 in [42], we can find \(\gamma _1\in (0,1)\) such that
In fact, we need some modification of the standard argument in [10] since our PDE contains unbounded ingredients. See Sect. 5.1 in [42] for the details. We only need (4.9) to prove this fact.
We shall show the assertion near the coincidence set. Thus, we shall fix \(z\in K\) such that \(u(z)=\varphi (z)\). Again, we may suppose \(z=0\) by translation. We will show that
which implies that u is differentiable at 0, \(Du(0)=D\varphi (0)\), and moreover,
We refer to [1] for its readable proof.
Setting \(v:=u-\varphi (0)-\langle D\varphi (0),x\rangle +Ar^{1+\hat{\gamma }}\) for large \(A>0\), we claim that v is a nonnegative \(L^p\) viscosity supersolution of
where \(g^-(x):=f^-(x)+|D\varphi (0)|\mu (x)+c_0\{\varphi (0)+\langle D\varphi (0),x\rangle \}\). Considering \(\hat{v}:=v\left( \inf _{Q_r}u+\delta _0^{-1}\Vert g^-\Vert _{L^{p\wedge n}(Q_{4r})}\right)^{-1}\), we note that we may apply Proposition 5.7 to find \(\varepsilon _0>0\) such that
For large \(\nu >1\), it is easy to verify that \(w:=v\vee (\nu Ar^{1+\hat{\gamma }})\) is an \(L^p\) viscosity subsolution of
where \(g^+=f^++|D\varphi (0)|\mu -c_0\{ \varphi (0)+\langle D\varphi (0),x\rangle - Ar^{1+\hat{\gamma }}\}\). In view of Proposition 5.8, we have
where \(\tilde{C}=\tilde{C}(\varepsilon _0)>0\). Hence, by (4.11), we have
The opposite inequality is trivial because Proposition 4.2 yields
for \(|x|\le r\).
Now, we shall combine two cases to establish the estimate. For \(x,y\in N_K\), we may assume \(0<\text{ dist }(y,C_K[u])\le \text{ dist }(x,C_K[u])\). Choose \(\hat{x},\hat{y}\in C_K[u]\) such that
\(\underline{\text {Case 1}: |x-y|<\frac{1}{2} |x-\hat{x}|.}\) In this case, by (4.10), we have
\(\underline{\text {Case 2}: |x-y|\ge \frac{1}{2}|x-\hat{x}|\ge \frac{1}{2}|y-\hat{y}|.}\) We may suppose that \((u-\varphi )(\hat{x})=(u-\varphi )(\hat{y})\) (or \((u-\psi )(\hat{x})=(u-\psi )(\hat{y})\)) because \(\psi (x)-\varphi (y)\ge \tau _0>0\) for \(y\in B_r(x)\cap K\) with small \(r>0\).
Thus, due to the above observation, we have
because \(|\hat{x}-\hat{y}|^{\hat{\gamma }} \le |\hat{x}-x|^{\hat{\gamma }}+|x-y|^{\hat{\gamma }}+|y-\hat{y}|^{\hat{\gamma }}\) and \(\gamma _0\le \hat{\gamma }\). \(\square \)
Open question 6: What is a sufficient condition to obtain \(W^{2,\infty }_{loc}(\Omega )\) or \(W^{2,p}_{loc}(\Omega )\) estimates on solutions of Isaacs equations with obstacles?
5 Appendix
In [38, 39], we established the ABP maximum principle and weak Harnack inequality for \(L^p\) viscosity solutions only when the PDE does not contain 0th order terms for the sake of simplicity. Since in Sect. 4 we obtain the results assuming (4.3), which allows the PDE to admit 0th order terms, we shall give the ABP maximum principle and weak Harnack inequality for those.
The ABP maximum principle can be proved immediately due to known results.
Proposition 5.1
Assume \(\mu \in L^q(\Omega )\), \(f\in L^p(\Omega )\) for \(q>n\) and \(q\ge p>p_0\). Assume also that \(c_0\in C(\overline{\Omega })\) is nonnegative in \(\overline{\Omega }\). Then, there exists a universal constant \(C_0>0\) (depending on \(\Vert \mu \Vert _{L^q(\Omega )})\) such that if \(u\in C(\overline{\Omega })\) is an \(L^p\) viscosity subsolution (resp., supersolution) of
then it follows that
where \(\Omega ^\pm [u]:=\{ x\in \Omega \ | \ \pm u(x)>\max _{\partial \Omega }u^\pm \}\) and \(d_\Omega :=\sup \{ |x-y| \ | \ x,y\in \Omega \}\).
Proof
We shall only show the first assertion. It is immediate to verify that u is an \(L^p\) viscosity subsolution of
Hence, we can apply Proposition 2.8 and Theorem 2.9 in [38] to conclude our proof. \(\square \)
We next show the weak Harnack inequality. We first present a decay of distribution functions of \(L^p\) viscosity supersolutions.
Lemma 5.2
(cf. Theorem 2.3 in [41]) Assume the same hypotheses in Proposition 5.1. There are \(r_0,\delta _0>0\) and \(A\ge 1\) such that for any nonnegative \(L^p\) viscosity supersolution of
if \(\inf _{Q_1}u\le 1\) and \(\Vert \mu \Vert _{L^{p\wedge n}(Q_4)}\vee \Vert f^-\Vert _{L^{p\wedge n}(Q_4)}\le \delta _0\), then we have
Remark 5.3
It is trivial that the conclusion holds true for any \(t>0\) since \(A\ge 1\).
Remark 5.4
The assertion is known in [39] when \(c_0\equiv 0\). In fact, in our case, we do not know if the strong maximum principle holds when the coefficient to the first derivative (i.e. \(\mu \)) is unbounded. Therefore, we will use an auxiliary function \(\varphi _0\), which is a strong solution of PDE with no first derivative terms. We notice that if we add \(\mu |D\varphi _0|\) in the left hand side of (5.3), then we cannot show (5.4) below. We will then have \(\mu \) in the inhomogeneous term which is small in \(L^n\) norm.
Proof
In view of Proposition 2.4 in [40] with some modifications as in the proof of Lemma 4.2 in [39], there exists \(\varphi _0\in W^{2,p'}(Q_4\setminus Q_1)\cap C(Q_4\setminus Q_1^i)\) for any \(p'>n\) such that
Since \(\varphi \) is also an \(L^{p'}\) viscosity solution of the PDE in the above, if we suppose \(\sup _{Q_4\setminus Q_1}\varphi _0>0\) or \(\inf _{Q_4\setminus Q_1}\varphi _0<-1\), then this contradicts to the definition of \(L^{p'}\) viscosity solution. Thus, we have \(-1\le \varphi _0\le 0\) in \(Q_4\setminus Q_1\).
Furthermore, we claim that there is \(\theta _0>0\) such that
Although the proof of (5.4) is known in [33] for instance, we will give a proof of this claim for the reader’s convenience in the end.
Extending \(\varphi _0\) appropriately in \(Q_1\), for large \(\lambda >1\), we may suppose that \(\varphi :=\lambda \varphi _0\in W^{2,p'}(Q_4)\) is an \(L^{p'}\) strong solution of
such that \(\varphi \le -2\) in \(Q_3\), where \(\xi \in L^q(Q_1)\) satisfies \(\xi =0\) in \(Q_4\setminus Q_1\).
We observe that \(w:=u+\varphi \) is an \(L^p\) viscosity supersolution of
Hence, setting \(\Omega :=\{ x\in Q_4^i \ | \ w(x)<0\}\), by Proposition 5.1, we have
Therefore, for a fixed \(\delta _0>0\), we can find \(\theta _1\in (0,1)\) such that
where \(M:=\max _{Q_4}(-\varphi )>1\). It is now standard by an induction argument to see that
which implies the decay of distribution function of u. Therefore, we conclude the assertion by the standard argument. See [39] for the details.
Proof of claim (5.4) (cf. Theorem 1 in [33]) It is enough to show that \(\varphi _0(x)<0\) for \(x\in Q_4^i \setminus Q_1\). Setting \(K_0:=\{ x\in Q_4^i\setminus Q_1 \ | \ \varphi _0(x)=0\}\), we may suppose \(K_0\ne \emptyset \). We can choose \(R>0\), \(z\in K_0\) and \(\hat{z}\in \Omega _0:= (Q_4^i\setminus Q_1)\cap K_0^c\) such that
Setting an open annulus \(A_0:=\{ x\in {\mathbb R}^n \ | \ R/2< |x-\hat{z}|<R\}\), we introduce \(\zeta (x):=\varepsilon (e^{-\beta R^2/2}-e^{-\beta |x-\hat{z}|^2/2})\le 0\), where \(\beta >1\) and \(\varepsilon \in (0,1)\) will be chosen later. Furthermore, we have
We also note that \((\varphi _0-\zeta )(x)<0\) if \(x\in \partial B_R(\hat{z})\setminus \{ z\}\). Now, setting \(\theta _0:=\min _{x\in \partial B_{R/2}(\hat{z})}(-\varphi _0(x))>0\) and \(\varepsilon :=\theta _0/2\), we observe that
Next, assume that \(\varphi _0-\zeta \) attains its maximum at \(\hat{x}\in A_0\). Since \(\varphi _0\) is a viscosity subsolution of
we have
Following an argument in p. 20 of [12], since \(\mathcal{P}^-(I-\beta (\hat{x}-\hat{z})\otimes (\hat{x}-\hat{z})) \ge -\frac{n-1}{\theta } + \left(\frac{\beta R^2}{4}-1\right)\theta \ge 1\) provided \(\beta \ge \beta _0\) for some \(\beta _0>1\), we have
which yields a contradiction when \(\beta >\beta _0+\max _{x\in \overline{\Omega }}c_0\). Therefore, because \((\varphi _0-\zeta )(z-he)\le (\varphi _0-\zeta )(z)=0\) for small \(h>0\), where \(e:=(z-\hat{z})/|z-\hat{z}|\), we have
Sending \(h\rightarrow 0+\), we have \(\langle D\varphi _0(z),e\rangle =0\ge \varepsilon e^{-\frac{\beta R^2}{2}}\beta R>0\), which is a contradiction. Hence, we have \(K_0=\emptyset \). \(\square \)
Remark 5.5
It is possible to give precise functions \(\varphi _0\) by considering larger ball \(B_{2\sqrt{n}}\supset Q_4\). See [30] for such a function.
Remark 5.6
Concerning the strong maximum principle for PDE of divergence type with 0th order terms, we refer to [51] and references therein.
Now, we present our weak Harnack inequality.
Proposition 5.7
(cf. Theorem 3.1 in [39]) Assume the same hypotheses in Proposition 5.1. There are \(\varepsilon _0>0\), \(\delta _0>0\) and \(\hat{C}>0\) such that for any nonnegative \(L^p\) viscosity supersolution of
if \(\Vert \mu \Vert _{L^{p\wedge n}(Q_4)}\le \delta _0\), then we have
Proof
In place of u, considering
where \(\varepsilon >0\) will be sent to 0 in the end, and \(\delta _0>0\) will be fixed later, we may suppose \( \Vert f^-\Vert _{L^{p\wedge n}(Q_4)}\le \delta _0\) and \(\inf _{Q_1}u\le 1\).
In view of Lemma 5.2, we easily verify that for any \(\varepsilon _0\in (0,r_0)\), there is \(\hat{C}=\hat{C}(\varepsilon _0)>0\) such that
which implies the conclusion by sending \(\varepsilon \rightarrow 0\). \(\square \)
In order to establish the Harnack inequality, we combine the weak Harnack inequality with the next local maximum principle.
Proposition 5.8
(Theorem 3.1 in [41]) Assume the same hypotheses in Proposition 5.1. For any \(\varepsilon >0\), there is \(\hat{C}_\varepsilon >0\) such that for any \(L^p\) viscosity subsolution of
we have
Since we have unbounded coefficient \(\mu \), we cannot use the standard argument as in [29]. We follow the idea of the proof of Lemma 4.4 in [12] with some modifications. We first prepare the following lemma:
Lemma 5.9
(cf. Theorem 2.3 in [41]) For \(q\ge p>p_0\) and \(q>n\), let \(f\in L^p(Q_4)\) and \(\mu \in L^q(Q_4)\) be nonnegative. Assume that \(u\in C(Q_4)\) is an \(L^p\) viscosity subsolution of (5.5) satisfying
where the constants \(A\ge 1\) and \(r_0>0\) are from Lemma 5.2. Then, there are an integer J, \(\nu >1\) and \(\ell _j>0\) \((j\ge J)\) such that \(\sum _{j=J}^\infty \ell _j<\infty \), and if \(u(x_0)\ge \nu ^{j-1}\) for \(j\ge J\) and \(x_0\in Q_{\frac{1}{2}}\), then \(sup_{Q_{\ell _j}(x_0)}u\ge \nu ^j\).
Proof
We will fix \(\nu >1\), \(J\in \mathbb {N}\) and \(\ell _j\in (0,1)\) for \(j\ge J\). Suppose
then we will obtain a contradiction.
Setting \(x=x_0+\frac{\ell _j}{4}y\) for \(y\in Q_4\), we define
where \(\alpha :=\nu (\nu -1)^{-1}\) (or \(\nu =\alpha (\alpha -1)^{-1}\)). Thus, we immediately verify that \(v\ge 0\) in \(Q_4\), and \(\inf _{Q_3}v\le v(0)\le \alpha (1-\nu ^{-1})=1\).
We next set
and
Choose \(J_0\in \mathbb N\) such that
Notice that \(\ell _j<1\) for \(j\ge J_0\). We next choose \(J_1\ge J_0\) such that
We then see that v is a nonnegative \(L^p\) viscosity supersolution of
where
Because of our choice of \(\alpha >1\), \(\ell _j\) and \(J_1\in \mathbb N\), we verify that for \(j\ge J_1\),
and
Finally, we choose \(J_2\ge J_1\) such that \(\Vert \hat{\mu }\Vert _{L^{p\wedge n}(Q_4)}\le \delta _0\), where \(\delta _0>0\) is the constant in Lemma 5.2.
In view of Lemma 5.2, we have
which yields
However, (5.6) implies
Hence, we have
which implies a contradiction to the definition of \(\ell _j\). \(\square \)
Proof of Proposition 5.8. We first consider the case of \(\varepsilon =r_0\), where \(r_0>0\) is the constant from Lemma 5.2.
Choose \(z\in Q_{\frac{1}{4}}\) such that \(u(z)=\max _{Q_{\frac{1}{4}}}u\). Setting \(v(y):=u(z+sy)\) for \(s>0\), we observe that v is an \(L^p\) viscosity subsolution of
where \(\hat{\mu }(y):=s\mu (z+sy)\) and \(\hat{f}(y):=s^2f^+(z+sy)\).
Since we may suppose \(\displaystyle \int _{Q_1}(v^+)^{r_0}dy>0\), by setting
it is immediate to see that
Furthermore, we verify that w is an \(L^p\) viscosity subsolution of
where \(g(y):=\delta _0 \hat{f}(y)\Vert \hat{f}\Vert _{L^{p\wedge n}(Q_4)}^{-1}\).
Let \(\nu >1\), \(J\in \mathbb N\) and \(\ell _j\in (0,1)\) be from Lemma 5.9. There is \(\hat{J}\ge J\) such that
We claim that \(\sup _{Q_{\frac{1}{4}}}w\le \nu ^{\hat{J}-1}\). Indeed, if \(w(x_0)\ge \nu ^{\hat{J}-1}\) for some \(x_0\in Q_{\frac{1}{4}}\), then thanks to Lemma 5.9, we can choose \( x_j\in Q_{\ell _j}(x_0) \) (for \(j\ge \hat{J}\)) such that
Since \(x_j\in Q_{\frac{1}{2}}\) for \(j\ge \hat{J}\), this contradicts to the continuity of \(w\in C(Q_4)\). Hence, we have
In case when \(\varepsilon >r_0\), instead of the above w, consider
Thus, we have
Therefore, Lemma 5.9 implies the conclusion.
On the other hand, if \(0<\varepsilon <r_0\), then considering
we have
Hence, Lemma 5.9 concludes the proof in this case. \(\square \)
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Koike, S. (2021). Regularity of Solutions of Obstacle Problems –Old & New–. In: Koike, S., Kozono, H., Ogawa, T., Sakaguchi, S. (eds) Nonlinear Partial Differential Equations for Future Applications. PDEFA 2017. Springer Proceedings in Mathematics & Statistics, vol 346. Springer, Singapore. https://doi.org/10.1007/978-981-33-4822-6_6
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