A remark on a vanishing property for viscosity solutions of fully nonlinear parabolic equations

Nonlinear Analysis 80 (2013) 14–17

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A remark on a vanishing property for viscosity solutions of fully nonlinear parabolic equations Agnid Banerjee ∗ Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States

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Article history: Received 14 October 2012 Accepted 26 November 2012 Communicated by Enzo Mitidieri

abstract We make the observation that, under some natural conditions on F (stated in (A)–(C) in the main text), if a viscosity solution of the fully nonlinear parabolic equation F (D2 u, Du, u) − ut = 0 vanishes to infinite order at (x0 , t0 ), then there is a small spatial neighborhood Br0 (x0 ) × {t0 } of (x0 , t0 ) in which u vanishes identically. The proof is inspired in an essential way by the ideas employed in the recent paper Armstrong and Silvestre (2011) [2], where the unique continuation property for fully nonlinear elliptic equations was established. Published by Elsevier Ltd

1. Introduction We consider the following fully nonlinear parabolic equation: F (D2 u, Du, u) − ut = 0,

(1.1)

under the following assumptions: (A) F is uniformly elliptic and Lipschitz; i.e., there exist constants 0 < λ ≤ Λ and γ , η such that for every M , N ∈ Sn (the space of n × n symmetric matrices) such that N ≥ 0, p, q ∈ Rn and z , w ∈ R, we have

λ∥N ∥ − γ |p − q| − η|z − w| ≤ F (M + N , p, z ) − F (M , q, w) ≤ Λ∥N ∥ + γ |p − q| + η|z − w|;

(1.2)

(B) F is C 1,1 in its arguments (M , p, z ) in a neighborhood of (0, 0, 0); (C) F (0, 0, 0) = 0. A continuous function u defined in a neighborhood U of (x0 , t0 ) is said to vanish to infinite order at (x0 , t0 ) if for each N ≥ 0 there exists CN ≥ 0 such that

|u(x, t )| ≤ CN (|x − x0 |2 + |t − t0 |)N ,

for every (x, t ) ∈ U with t ≤ t0 .

(1.3)

We are interested in the question of whether a viscosity solution u of (1.1) above which vanishes to infinite order at (x0 , t0 ) must vanish identically in a space-like neighborhood of x0 , i.e., there exists a suitable r0 > 0 such that u(x, t0 ) = 0 for every x ∈ Br0 (x0 ). This is a form of space-like strong unique continuation property for viscosity solutions of (1.1). We emphasize that space–time unique continuation (backward in time) does not hold for solutions to parabolic equations since, even for the heat equation, there is an example, given by Frank Jones, of a nontrivial caloric function which is supported on a strip; see [1]. The elliptic counterpart of (1.1) has been studied in the interesting recent paper [2], where a strong unique continuation result was established. The key step in their proof was the use of higher regularity of ‘‘flat’’ solutions (see [3]) and the

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0362-546X/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.na.2012.11.023

A. Banerjee / Nonlinear Analysis 80 (2013) 14–17

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boundary Harnack inequality. This allowed linearizing the equation, and hence reducing the problem to the classical strong unique continuation of divergence form linear equations of the type div(A(x)Du) + ⟨b(x), Du⟩ + c (x)u = 0,

(1.4)

where A is a Lipschitz continuous matrix-valued function with real coefficients. For (1.4), the strong unique continuation property follows either from the Carleman estimates in [4] or from the monotonicity formulas of Garofalo and Lin; see [5,6]. Let us now turn to the parabolic counterpart of (1.4), i.e., div(A(x, t )Du) + ⟨b(x, t ), Du⟩ + c (x, t )u = ut .

(1.5) 1 -Hölder 2

continuous in t, the space-like strong unique For such equations, for when A is Lipschitz continuous in x and continuation has been proved by Escauriaza, Fernandez and Vessella in [7,8], where Carleman estimates are used. We now state the precise parabolic unique continuation result, since it is used crucially in our context. The reader should see [7, Theorem 1], which is stated for backward parabolic operators, but can be adapted into our framework by making the usual change of time variable t = −s. Theorem 1.1. Let u be a solution of the uniformly parabolic equation (1.5) in B2 (0) × (−2, 0], where b, c are bounded and A satisfies |A(x, t ) − A(y, s)| ≤ M (|x − y|2 + |t − s|)1/2 . If u vanishes to infinite order at (0, 0), then u(x, 0) = 0 in B2 (0). Theorem 1.1 was later generalized in [9] where the vanishing to infinite order in space and time was replaced by the assumption that u(., 0) vanishes to infinite order in space only. We now state our main result which is a partial step in the direction of generalizing Theorem 1.1 to the fully nonlinear equation (1.1). Theorem 1.2. Let u be a viscosity solution of the uniformly parabolic equation (1.1) in B2 (0) × (−2, 0]. If u vanishes to infinite order at (0, 0), then there exists r0 > 0, depending on u, such that u(x, 0) = 0 in Br0 (0). In establishing Theorem 1.2 we follow the ideas in [2]. The proof relies on the regularity of flat solutions for parabolic equations obtained in the recent paper [9], which constitutes the parabolic analogue of Savin’s result. It remains a very interesting open problem whether, for u as in Theorem 1.2, one can in fact conclude that u(x, 0) = 0 in the whole of B2 (0), not just in Br0 (0). This latter result would provide us with a complete generalization of Theorem 1.1 to parabolic fully nonlinear equations. Moreover, it also remains an interesting open question whether one could derive the same vanishing property for solutions u of fully nonlinear parabolic equations by just insisting that u(., 0) vanishes to infinite order in space which would be the analogue of the result obtained in [10]. Our proof however uses vanishing in space and time in a crucial way since it relies heavily on the flatness result in [9] which needs smallness assumption in a parabolic cylinder. 2. Proof of Theorem 1.2 Before we start with the proof, we state the previously mentioned flatness result of Yu Wang. Henceforth, we denote the parabolic Hölder spaces by Hk+α , and refer the reader to Chapter 4 in [11] for the precise definitions. Given x0 ∈ Rn , t0 ∈ R and r > 0, we let Qr (x0 , t0 ) = Br (x0 ) × (t0 − r 2 , t0 ]. In this section Qr is to be understood as Qr (0, 0). Theorem 2.1 (See [9, Theorem 1.1]). Let F satisfy (A)–(C). There exists a constant c > 0 such that if u is a viscosity solution of (1.1) satisfying supQ1 |u| ≤ c, then u ∈ H2+α (Q1/2 ). Proof. We are given a solution u to (1.1) which vanishes to infinite order at (0, 0), i.e., (1.3) holds. This implies in particular that for each β > 0, sup |u| ≤ O(r β )

as r → 0.

(2.1)

Qr /2

Thus, (2.1) holds in particular for β = 2 + α when r is small enough. We can now rescale the solution as follows. Let ur (x, t ) =

u(rx,r 2 t ) . r2

We note explicitly that ur solves in Q1

Fr (D2 v, Dv, v) − vt = 0,

(2.2)

where Fr (M , p, z ) = F (M , rp, r 2 z ). Thus, for r < 1, we see that Fr has C the definition of ur ,

∥ur ∥L∞ ≤ Cr α in Q1 .

(2.3) 1 ,1

norm bounded by that of F and therefore it is uniformly bounded. Moreover, from (2.4)

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A. Banerjee / Nonlinear Analysis 80 (2013) 14–17

Hence, for sufficiently small r , ur and Fr satisfy the hypothesis of Theorem 2.1, which implies ur ∈ H2+α (Q1 ), and hence u belongs to H2+α (Qr ). Because of the vanishing to infinite order, we have Du(0, 0) = D2 u(0, 0) = 0. Now, by shrinking r if needed, we may assume that for each (x, t ) ∈ Qr , the triple (u(x, t ), Du(x, t ), D2 u(x, t )) lies in a neighborhood of (0, 0, 0) where F is C 1,1 . By Schauder estimates, it follows that u is in H3+α (Qr ) (see Lemma 14.11 in [11]). Thus, we can now differentiate the equation with respect to xk in Qr to obtain

∂F ∂F ∂F (uk )ij + (uk )i + uk = (uk )t , ∂ Mij ∂ pi ∂z

(2.5)

where hereafter we have let uk = ∂∂xu . By the definition of the parabolic Hölder spaces (see [11]), we have that u, Du, D2 u ∈ k H0+1 (Qr ) with respect to the usual parabolic distance. Therefore, uk satisfies aij (uk )ij + ⟨b, Duk ⟩ + cuk = (uk )t ,

(2.6) 1 -Hölder 2

where aij is Lipschitz in the space variable, and continuous in the time variable, with b and c bounded. Since aij is Lipschitz in the space variable, such an equation can also be written in divergence form. Therefore uk satisfies an equation whose coefficients obey the hypothesis of Theorem 1.1 above. Moreover, since u vanishes to infinite order, we have that uk also vanishes to infinite order at (0, 0). Note that, if u were smooth, we could have concluded this from the Taylor expansion. However, in our case, one way to see this is by using the equation. Since F (0, 0, 0) = 0 and it satisfies (A), (B), by using a simple change of the dependent variable v = eκ t u, for κ suitably chosen (depending on γ in (A) (1.2) above), we see that v satisfies an equation which obeys Hypothesis (SC) in [12, p. 2001]. Now, by Proposition 2.9 in [12], a classical viscosity solution is also an Lp viscosity solution. We note explicitly that v solves an equation of the form

vt + G(t , v, Dv, D2 v) = 0.

(2.7)

Therefore, v also is an Lp viscosity solution of

vt + G(t , 0, Dv, D2 v) = f (x, t ),

(2.8)

where, because of the assumption (A) (1.2) above, we have

|f (x, t )| = |G(t , 0, Dv, D2 v) − G(t , v, Dv, D2 v)| ≤ K |v(x, t )|.

(2.9)

The latter statement can be verified by noting that G(t , z , p, M ) = −κ z − eκ t F (e−κ t z , e−κ t p, e−κ t M ).

(2.10)

Hence, it is seen that K above can be taken to be γ + |κ|, and if we choose |κ| = 3γ , with γ as in (A) (1.2) above, then we can take K = 4γ . We thus see that (SC) in [12] is satisfied. Then, by using Proposition 7.3 in [12] applied to (2.8) (see also Section 4 in [13] for when no lower order terms are present), we have for u sup |Du| ≤ Qr /2

C r k0

sup |u|,

(2.11)

Qr

where k0 depends on n, F . Hence, Du also vanishes to infinite order. Therefore by invoking Theorem 1.1, we conclude that uk (x, 0) = 0 for each x ∈ Br . Since k = 1, . . . , n was arbitrarily fixed, we finally have that u(x, 0) = 0 for x ∈ Br , which is the desired conclusion.  Remark 2.2. The regularity assumptions needed for F in [9] are in fact weaker, as only some modulus of continuation of the first derivatives of F in a neighborhood of the origin is assumed. We would like to emphasize the fact that, although we obtain u(x, 0) = 0 in Br , we cannot assert that u vanishes to infinite order in the sense of (1.3) at any point (x, 0) different from (0, 0) since from the proof we obtain no information about the growth at such points in the t-direction. Consequently, we cannot linearize the equation at such points. This is a major obstruction which does not allow us to conclude that the zeros spread everywhere is space at the fixed time level, as one would ideally like to assert. This is where our result, Theorem 1.2, differs from the elliptic case in [2], where the problem could be reduced to proving the weak unique continuation property. In proving the weak unique continuation property in [2], in order to obtain the flatness required to linearize the equation, the authors used a beautiful argument based on the boundary Harnack principle. In closing we remark that if u were a viscosity solution of F (D2 u) = ut ,

(2.12)

A. Banerjee / Nonlinear Analysis 80 (2013) 14–17

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with F concave and C 1,1 in its domain of definition, then we could appeal to Lemma 14.8 in [11], which asserts existence of classical solutions on boxes for such F ’s with continuous boundary values. This result, combined with the uniqueness of viscosity solutions, implies that any viscosity solution u to Eq. (2.12) is also a classical H2+α solution. This can be thought of as a parabolic analogue of the Evans–Krylov theorem (for the latter see [14]). Then, by the Schauder theory, as mentioned above, we would have u ∈ H3+α . Consequently, we can linearize the equation and assert the space-like strong unique continuation property. The main difficulty in establishing a unique continuation result for fully nonlinear equations thus lies in the fact that, when F is not concave, the best a priori regularity available for viscosity solutions of (1.1) is C 1,α even in the case where F is smooth. This follows from the important works of Nadirashvili and Vlădut; see [15] and the references therein (we thank Prof. Silvestre for bringing [15] to our attention). Acknowledgments The author wishes to thank Prof. N. Garofalo for directing his attention to the recent work by Luis Silvestre and Scott N. Armstrong [2]. The author was supported in part by N. Garofalo’s NSF Grant DMS-1001317, and in part by N. Garofalo’s Purdue Research Foundation Grant ‘‘Gradient bounds, monotonicity of the energy for some nonlinear singular diffusion equations, and unique continuation’’, 2012. We also thank the anonymous referee for his/her careful reading of the manuscript and useful comments, and also for having brought the paper [10] to our attention. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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