Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation

In this paper, we investigate positive viscosity solutions of a third degree homogeneous parabolic equation \begin{document}$ u^{2}u_{t} = \Delta_{\infty}u $\end{document} . We prove a comparison principle, existence and uniqueness of continuous positive viscosity solutions.

1. Introduction. In this paper, we study a nonlinear degenerate parabolic equation u xi u xj u xixj (2) denotes the 1-homogeneous of the very popular infinity Laplace operator, Du and D 2 u denote the spatial gradient vector and Hessian matrix of u respectively, u xi = ∂u ∂xi , u xixj = ∂ 2 u ∂xi∂xj . Our goal is to establish the comparison principle, existence and uniqueness of positive viscosity solutions.
The infinity Laplace equation ∆ ∞ u = 0 is the Euler-Lagrange equation associated with L ∞ -variational problem related to absolutely minimizing Lipschitz extensions of functions defined on the boundary of a bounded domain Ω ⊂ R n . See for details [4,5,6,7,10,13] and the reference therein. The parabolic equation involving the infinity Laplacian has been received a lot of attention in the last decade, notably due to its application to image processing, the main usage being in the reconstructions of damaged digital images [8]- [9]. For numerical purpose it has been necessary to consider also the evolution equation corresponding to the infinity Laplace operator.
Three types of parabolic infinity Laplace equations were described by Lindqvist (page 6 in [16]), that is Theorem 1.1. Let Q T = U × (0, T ), where U ⊆ R n is a bounded domain, and let ψ ∈ C(R n+1 ) and ψ ≥ c > 0 on ∂ p Q T for some positive constant c. Then there exists a unique positive viscosity solution u ∈ C(Q T ∪ ∂ p Q T ) of the problem In Theorem 1.1, ∂ p Q T = (∂U ×[0, T ])∪(U ×{0}) denotes the parabolic boundary of Q T . Here, a viscosity solution u of the problem (4) is a viscosity solution u of the equation (1) satisfying the Dirichlet boundary condition u = ψ on ∂ p Q T pointwise. The definition of a viscosity solution of equation (1) will be given in Section 2. Note that if ψ ≤ c < 0 on ∂ p Q T for some negative constant c in Theorem 1.1, then there exists a unique negative viscosity solution u ∈ C(Q T ∪ ∂ p Q T ) of the problem (4). We state this result as the first corollary of Theorem 1.1. Corollary 1.1. Under the assumptions of Theorem 1.1 with "ψ ∈ C(R n+1 ) and ψ ≥ c > 0 on ∂ p Q T for some positive constant c" replaced by "ψ ≤ c < 0 on ∂ p Q T for some negative constant c ", then there exists a unique negative viscosity solution u ∈ C(Q T ∪ ∂ p Q T ) of the problem (4). Corollary 1.1 follows directly by letting v = −u in Theorem 1.1 and using the third degree homogeneity of equation (1).
The second corollary of Theorem 1.1 is the existence of nonnegative viscosity solutions of the problem (4) when ψ ≥ 0 on ∂ p Q T . Corollary 1.2. Under the assumptions of Theorem 1.1 with "ψ ∈ C(R n+1 ) and ψ ≥ c > 0 on ∂ p Q T for some positive constant c" replaced by "ψ ∈ C(R n+1 ) and ψ ≥ 0 on ∂ p Q T ", then there exists a nonnegative viscosity solution u ∈ C(Q T ∪ ∂ p Q T ) of the problem (4). Moreover, if ψ ≡ 0 on ∂ p Q T , then u ≡ 0 is the unique viscosity solution of (4).
In this work, there are two main difficulties in the equation (1). One is the coefficient u 2 of the term u t , the other is the degeneracy of the infinity Laplace operator ∆ ∞ u itself. Since u 2 is the coefficient of u t , it is difficult to study the comparison principle for positive viscosity solutions of equation (1). In order to overcome this difficulty, we take ρ = log u to transform equation (1) to the equation so that Jensen's method [13] of the comparison principle can be carried out in the usual way. The degeneracy of ∆ ∞ u causes trouble in the proof of the existence for positive viscosity solutions. In order to overcome this difficulty, we introduce the approximating equations of equation (1), that is, where ε ∈ (0, 1] is a constant. Note that such approximations in (6) are different from those in [6,2,18,19]. Then the existence of positive viscosity solutions with positive continuous boundary and initial data is established with the aid of the approximating equations (6) and the uniform estimates. These uniform estimates are derived by using the barrier argument. We remark that it is also standard to prove the existence of viscosity solutions of the Dirichlet problem (4) by Perron's method as in [11]. However, it is not clear the viscosity solution obtained by the Perron's method can satisfy the Dirichlet boundary condition pointwise. This is the reason why we choose the method of elliptic approximations in this paper to prove the existence of viscosity solutions of the Dirichlet problem (4).
Note that it is not clear whether the existence of viscosity solutions to (4) holds for sign changing ψ. It would be interesting to consider the problem (4) with sign changing ψ and study its sign changing viscosity solutions.
The organization of this paper is as follows. In Section 2, we give the notations, definitions of positive (negative, or nonnegative) viscosity supersolutions and viscosity subsolutions of equation (1). We also prove the comparison principle of the positive viscosity solutions. In Section 3, we introduce uniform parabolic approximatons of the problem (4) and prove a weak maximum principle. We derive various uniform estimates including a uniform Lipschitz boundary estimate at t = 0, a full uniform Lipschitz estimate in time and a uniform Hölder estimate at the lateral boundary. Then the main existence result, Theorem 1.1, is proved by the approximating procedure. Finally, the proofs of Corollaries 1.1, 1.2, and Theorem 3.2 (which is a byproduct for equation (5)) are presented.   (1) in Ω if, whenever (x,t) ∈ Ω and test function ϕ ∈ C 2 (Ω) are such that A continuous positive function u : Ω → R is called a positive viscosity solution of (1), if u is both a positive viscosity subsolution and a positive viscosity supersolution.
Remark 2.1. Let Q T = U × (0, T ) and suppose that a function u : Q T → R can be written as u(x, t) = v(x) for some upper semicontinuous function v : U → R. Then u is a viscosity subsolution of equation (1) if and only if −(D 2 v(x)Dv(x)) · Dv(x) ≤ 0 in the viscosity sense.
Before proving the comparison principle, we introduce a transformation of the original equation (1), namely, if u > 0 solves the differential equation in (1) and ρ = log u, then a simple calculation yields In Definition 2.1, discarding "positive" and replacing equations (1) and (7) by equations (8) and we get the definition of viscosity solutions of equation (8). Note that the definitions of viscosity solutions for equations (1) and (8) correspond to the standard definitions of viscosity solutions in [11].
When u > 0, it is easy to check that u is a positive viscosity solution of equation (1) in Q T if and only if ρ is a viscosity solution of equation (8) For a cylinder Q T = U × (0, T ), where U ⊂ R n is a bounded domain, we denote the lateral boundary by S T = ∂U × [0, T ] and the parabolic boundary by Notice that both S T and ∂ p Q T are compact sets. Next, we shall prove a comparison principle of positive viscosity solutions to the initial-boundary problem of equation (1) by using the perturbation argument of Jensen [13].
is a bounded domain. Let u and v be a positive viscosity subsolution and a positive viscosity supersolution of equation (1) in Q T , respectively, such that for all (z, s) ∈ ∂ p Q T . Then Proof. Since (9) holds for all (z, s) ∈ ∂ p Q T , then u and v have positive bounds in Q T from below and above. Such boundedness of u and v can be derived by the maximum principle, (see the case when ε = 0 in Lemma 3.1 in Section 3). Letting ρ(x, t) = log u(x, t) and ζ(x, t) = log v(x, t), then hold in the viscosity sense in Q T , and ρ ≤ ζ on ∂ p Q T . To prove (10), it is enough to prove ρ ≤ ζ for all (x, t) ∈ Q T . For completeness, we give the detailed proof. We argue by contradiction. By replacing ζ with ζ(x, t) = ζ(x, t)+ δ T −t , where δ is a positive constant, we may obtain that ζ is a strict supersolution and ζ(x, t) → ∞ uniformly in x as t → T. Indeed, since ζ(x, t) is a viscosity supersolution of equation and let It follows from (11) and the fact that ρ < ζ on ∂ p Q T that for j large enough x j , y j ∈ U and t j , s j ∈ [0, T ], (see Proposition 3.7 in [11]). From now on, we will consider only such indices j. We divide the proof into two cases.
, has a local minimum at (y j , s j ). Since ζ is a strict supersolution and Dξ(y j , s j ) = 0, we have , has a local maximum at (x j , t j ), and thus 0 ≥ η t (x j , t j ) = j(t j − s j ). We then derive a contradiction.
Case 2. If x j = y j , we also use the parabolic theorem of sums for ω j which implies that there exists n × n symmetric matrices belong to the second order superjet of ρ and the second order subjet of ζ, respectively. See [11,17] for the notation and relevant definitions. Since ζ and ρ are a strict supersolution and a subsolution, respectively, this implies which is also a contradiction.
From cases 1 and 2, we know that (11) can not hold. Then we must have Letting δ → 0 in (12) , then the conclusion (10) follows.
Remark 2.2. As a consequence of Theorem 2.2, the uniqueness of the positive viscosity solutions of the initial and boundary value problem to equation (1) can be readily obtained. Since our proof of the comparison principle in Theorem 2.2 relies strongly on the transformed equation (5), the uniqueness can only apply to positive viscosity solutions. Therefore, as in Corollary 1.2, besides the homogeneous boundary condition case ψ ≡ 0 on ∂ p Q T , it is not clear that the uniqueness result can hold for general nonnegative viscosity solutions.  (1) in Ω.
3. Existence theorem. In this section, we introduce a uniform elliptic regularization for the initial boundary value problem of equation (1). By establishing the uniform estimates for the solutions of the regularized problems, we prove the existence and uniqueness of positive viscosity solutions to equation (1) with the initial boundary data ψ ≥ c > 0. Then the proofs of Corollaries 1.1 and 1.2 are given.
At the end, we also prove the existence and uniqueness of viscosity solutions to the initial boundary value problem of the equation (5). We consider the following approximating problem of (4), where ψ ∈ C 2 (R n+1 ), ψ ≥ c > 0 for some positive constant c, and with , u i = u xi and u ij = u xixj . Similar to Definition 2.1, we can also define the viscosity solution (supersolution, or subsolution) of the equation We first establish the weak maximum principle for viscosity solutions of the equation (15) in the following lemma.
and is a viscosity subsolution of equation (15), then sup and is a viscosity supsolution of equation (15), Proof. First, we choose τ close to T with τ < T . We shall show the weak maximum principle holds in Q τ for any τ < T. Note that Q τ is compactly contained in Q T ∪ ∂ p Q T . Let δ = sup Qτ u ε − sup ∂pQτ u ε , we assume that δ > 0 and obtain a contradiction.

Since sup
Qτ u ε > sup ∂pQτ u ε , it follows that either (i) there is a point (y, s) ∈ Q τ such that u ε (y, s) = M, or (ii) u ε (x, t) < M, for all (x, t) ∈ Q τ , and there is sequence (y k , s k ) ∈ Q τ such that s k → τ (since u ε is a upper semicontinuous function in Q τ ) and lim k→∞ u ε (y k , s k ) = M. In any case, there is a point (y, s) ∈ Q τ such that u ε (y, s) > sup ∂pQτ u ε + 3δ 4 and 0 < s < τ.
The second assertion can be proved similarly by the above argument. We omit its detailed proof.
Note that the viscosity sub and super solutions in Lemma 3.1 has no sign restrictions. We remark that Lemma 3.1 can also hold when ε = 0, since (16) still holds in this case.
Next, we aim to obtain a solution of the problem (4) as a limit of the functions u ε as ε → 0. This amounts to proving uniform estimates for u ε that are independent of 0 < ε ≤ 1. These uniform estimates can be obtained by using the standard barrier argument.
The full Lipschitz estimate in time now follows easily with the aid of Proposition 3.1 and the comparison principle.
Corollary 3.1. Let Q T = U × (0, T ) and u = u ε be as in Proposition 3.1. If ψ ∈ C 2 (R n+1 ) and ψ ≥ c > 0 for some positive constant c, then there exists C ≥ 0 with C = C( D 2 ψ ∞ , ψ t ∞ , Dψ ∞ ), but independent of 0 < ε ≤ 1 such that Moreover, if ψ is only continuous, then the modulus of continuity of u in t on U × (0, T ) can be estimated in terms of ψ ∞ and the modulus of continuity of ψ in x and t.
Proof. Let v(x, t) = u(x, t + τ ), τ > 0, then both u and v are solutions to (15) in for some postive constantC, which implies the Lipschitz estimate (30). The proof for the case when ψ is only continuous is similar. To avoid the repetitions, we omit its proof.
We now discuss the Hölder regularity of u = u ε at the lateral boundary.
Proposition 3.2. (Hölder regularity at the lateral boundary) Let Q T = U × (0, T ), where U ⊆ R n is a bounded domain, and suppose that u = u ε is a positive smooth function satisfying the problem (3.1). If ψ ∈ C 2 (R n+1 ) and ψ ≥ c > 0 for some positive constant c, then for each 0 < α < 1, there exists a constant C ≥ 2 ψ ∞ ≥ 1 depending on α, ψ ∞ , Dψ ∞ , ψ t ∞ but independent of ε such that and ε > 0 sufficiently small (depending on α). Moreover, if ψ is only continuous, then the modulus of continuity of u in x can be estimated in terms of ψ ∞ and the modulus of continuity of ψ in x.
Next we will show that M and C can be chosen so that ω ≥ u on the parabolic boundary of On the other hand, if x ∈ U ∩ ∂B γ (x 0 ) and t 0 − 1 < t ≤ t 0 , we have where the positivity of u and Cγ α = 2 ψ ∞ are used. Finally, we consider the bottom of the cylinder. For x ∈ U ∩ B γ (x 0 ) and t = t 0 − 1, by the positivity of u, In conclusion, we have now shown that if we choose 1, t 0 )) by the comparison principle. In particular, . The other half of the estimate claimed follows by considering the lower barrier ( Case 2. When t 0 < 1, we consider the cylinder Q T ∩ (B γ (x 0 ) × (0, t 0 )) , and notice that since u = ψ on the bottom of this cylinder, The rest of the argument is analogous to the previous case.
For the case when ψ is only continuous, the argument is similar. We omit its proof.
Hence, we have completed the proof of Proposition 3.2.
With the uniform estimates in Corollary 3.1 and Proposition 3.2, we can now give the proof of the main result, Theorem 1.1.
Proof of Theorem 1.1. If ψ ∈ C(R n+1 ) and ψ ≥ c > 0 for some positive constant c, u ε is the unique smooth solution to the problem (13). Corollary 3.1, Proposition 3.2 and the comparison principle imply that the family {u ε } is equi-continuous and uniformly bounded. Hence, the Ascoli-Arzela compactness theorem, up to a subsequence, u ε → u uniformly, and u is a unique viscosity solution to equation (1) with the initial and boundary data ψ by the stability properties of viscosity solutions. The existence for a general continuous data ψ follows by approximating the data by smooth functions and using Corollary 3.1 and Proposition 3.2. The uniqueness follows from the comparison principle, Theorem 2.2. In addition, we obtain that the viscosity solution u of (4) is Lipschitz continuous with respect to the time variable t and Hölder continuous in the space variable x.
Hence, by the third degree homogeneity of equation in (36), u := −v ∈ C(Q T ∪ ∂ p Q T ) is a negative viscosity solution of the problem (4). The uniqueness of negative viscosity solution of the problem (4) follows form Remark 2.3.
Proof of Corollary 1.2. Since ψ ∈ C(R n+1 ) and ψ ≥ 0, let ψ n (x, t) = ψ(x, t) + 1 n for n ∈ N + , where N + denotes the set of positive integers. For a given n, we have ψ n (x, t) ≥ 1 n > 0. By Theorem 1.1, there exists u n ∈ C(Q T ∪ ∂ p Q T ) is the unique positive viscosity solution to u 2 u t = ∆ ∞ u in Q T , u(x, t) = ψ n (x, t) on ∂ p Q T , Since lim n→∞ sup (x,t)∈∂pQ T |ψ n (x, t) − ψ(x, t)| = 0, we have ψ n (x, t) converges uniformly to ψ(x, t). By Lemma 3.1, we have inf Therefore, u n (x, t) uniformly converges to a nonnegative u(x, t) as n → +∞, and u(x, t) ∈ C(Q T ∪∂ p Q T ). We know from the stability properties of viscosity solutions that u is a nonnegative viscosity solution of the problem (4). Moreover, when ψ ≡ 0 on ∂ p Q T , by ε = 0 case of Lemma 3.1, we have u ≡ 0 on Q T ∪ ∂ p Q T .
As a byproduct, we give the existence and uniqueness of the viscosity solution to the initial and boundary value problem of the equation (5).
Theorem 3.2. Let Q T = U × (0, T ), where U ⊆ R n is a bounded domain. Suppose that ψ ∈ C(R n+1 ) and ψ is bounded on ∂ p Q T , then there exists a unique viscosity solution ρ ∈ C(Q T ∪ ∂ p Q T ) of the problem Proof. Letψ := e ψ , by Theorem 1.1, there exists a unique positive viscosity solution u ∈ C(Q T ∪ ∂ p Q T ) of the problem Hence, ρ := log u ∈ C(Q T ∪ ∂ p Q T ) is a viscosity solution of the problem (4). The uniqueness of viscosity solution of the problem (37) follows form the proof of Theorem 2.2.