Nonlinear elliptic-parabolic problems

Transcription

1 Nonlinear elliptic-parabolic problems Inwon C. Kim and Norbert Požár Abstract We introduce a notion of viscosity solutions for a general class of elliptic-parabolic pase transition problems. Tese include te Ricards equation, wic is a classical model in filtration teory. Existence and uniqueness results are proved via te comparison principle. In particular, we sow existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data. Tese results are new even in te linear case, were we also sow tat viscosity solutions coincide wit te regular weak solutions introduced in [AL]. 1 Introduction Let Ω R n be a smoot bounded domain and T > 0. Let us denote Q = Ω (0, T ]. We are interested in te following problem: find a function u(x, t), u : Q R, tat solves t b(u) F (D 2 u, Du, u) = 0 in Q, u = g 1 on Ω [0, T ], (1.1) u(, 0) = u 0 on Ω, were Du denotes te spatial gradient of u, D 2 u is te spatial Hessian, and F (M, p, z) : S n R n R R is a fully nonlinear, uniformly elliptic operator (see Section 1.1 for precise assumptions on F ). For te function b : R R we assume tat (a) b is increasing and Lipscitz, (b) b(s) = 0 for b 0, b C(R) C 1 ([0, )), (c) tere exists a constant c > 0 suc tat b (s) > c for s (0, ). Te nonlinear operators F we consider include: te uniformly elliptic operator of non-divergence form F (D 2 u, Du, u) = tr(a(du)d 2 u) + H(u, Du), (1.2) were A satisfies te uniform ellipticity condition λ q 2 A(p)q q Λ q 2 for all p, q R n, for some λ, Λ > 0, (1.3) Dept. of Matematics, UCLA, USA. Partially supported by NSF DMS Dept. of Matematics, University of Tokyo, Japan. Version: , fbc183a 1

2 and as well as te Bellman-Issacs operators arising from stocatic optimal control and differential games F (D 2 u, Du, u) = inf {L αβ u}, sup α A β B were L αβ is a two-parameter family of operators of te form (1.2) satisfying (1.3); we refer to [CIL, FS] for furter examples. a divergence-form operator; to simplify our discussion, we restrict our attention to operators of te form F (D 2 u, Du, u) = (Ψ(b(u))Du), (1.4) were Ψ C 1 ([0, )) is a positive function. Te class of operators given in (1.4) is of particular interest since in tat case te problem (1.1) represents te well-known Ricards equation, wic serves as a basic model for te filtration of water in unsaturated soils (see e.g. [DS, MR, R]). Our aim is to study te well-posedness of (1.1). Note tat, due to te regularity teory for uniformly elliptic nonlinear operators ([CC,W2]), solutions of (1.1) satisfy interior C 1,α estimates in te sets {u > 0} and {u < 0}. Hence te callenge in te study of te problem lies in te beavior of a solution near te transition boundary between te positive and te negative pases: for example, as illustrated in Example 1.2, discontinuities of solutions in time across te set {u = 0} are generic. Problem (1.1) can be understood as te limiting equation for te evolution of two pases wit different time scales of diffusion and wit te tresold value at u = 0. In particular, as verified in section 6, te problem can be viewed as a singular limit of a family of uniformly parabolic problems (6.1), were b(s) in (1.1) is regularized. Hence it is expected tat a maximum principle olds for te solutions, and te teory of viscosity solutions may be applicable for te study of pointwise beavior of solutions near te transition boundary {u = 0}. Tis is indeed our approac: in tis paper we will introduce te notion of viscosity solutions for (1.1) and discuss existence, uniqueness and stability properties, and compare tem to te notion of weak solutions (see te discussion below). Suc results ave been establised for Stefan-type problems (see [CS, KP], for example), but significant callenges in te analysis arise due to te implicit nature of te boundary motion law in (1.1) and te generic nature of discontinuities in u (as in Example 1.2); see an extended discussion on tis in section 3. We aim to present te proof of te comparison principle for fully nonlinear operators in more detail, bot to illustrate te flexibility of te viscosity solution approac and to make te results readily available for applications in a general context. It sould be pointed out tat our approac treats te transition boundary Γ := {u > 0} between te elliptic and parabolic regions as a free boundary and constructs barriers based on its movement. Hence our approac may not be optimal for oter problems suc as fast diffusion (wen b (0) = 0) were te transition boundary moves wit infinite speed (see [V], for example), but strong regularity properties for te solutions are expected (see [BH]). Altoug our presentation is mainly focused on te problem wit fully nonlinear operators wose structure assumption requires a linear growt in terms of Du, te results of tis paper extend to te case of divergence-form operators of te form (1.4) as well. We point out te necessary modifications in te text were appropriate. 2

3 Literature review Let us briefly discuss previously known results on (1.1): all of tem concern divergenceform operators including (1.4). Te weak solutions are defined via integration by parts in te important paper of Alt & Luckaus [AL], wic sows existence ([AL, Teorem 1.7]) of weak solutions for te general class of elliptic-parabolic pase transition problems wit divergence-form operators. Uniqueness results are, owever, rater limited. For F given by (1.4), wen Ψ(s) is a positive constant (tat is, wen F is linear), te autors prove tat regular weak solutions (in te sense t b(u) L 2 ) can be constructed by te Galerkin metod ([AL, Teorem 2.3]), and sow tat regular weak solutions satisfy te comparison principle and are unique given te initial data ([AL, Teorem 2.2]). Indeed, wen Ψ(s) is a positive constant, it is known tat b(u) is continuous in a local setting as long as u is bounded (see [DG]). Continuity of b(u) seems to be te optimal result for tis problem wit general initial data, since u may become discontinuous in time in te elliptic pase (see Example 1.2, and also [DG]). Te proof of continuity in [DG] is based on te weak Harnack inequality, making use of te linearity of te elliptic operator wit respect to u. We also refer to [BW] and [C], wo use an entropy solution approac to define weak solutions as well as to prove comparison principle in L 1 for te relevant model; tis approac, wile powerful, does not fit into our setting were we ave a non-vanising elliptic pase wit operator F unbounded wit respect to Du. Even for te quasi-linear F given in (1.4), tere are no uniqueness or stability results except for te aforementioned linear case and for te one-dimensional case (see [BH, vdp]): tis serves as a motivation of our analysis in tis paper. In one dimension, Mannucci & Vazquez [MV] studied viscosity solutions of (1.1) for divergence-form operators. Teir approac avoids possible complications at te transition boundary {u > 0} by relying on previously known regularity properties of weak solutions in one dimension. Summary of te main results In tis section we summarize te main results obtained in tis paper. In all statements F is assumed to be eiter a fully nonlinear operator satisfying te assumptions in Section 1.1 or a quasilinear divergence-form operator of te form (1.4). Our main teorem is te following comparison principle: Teorem 1 (Teorem 3.1 and Teorem 3.24). Let u be a viscosity subsolution and v a viscosity supersolution of (1.1) on Q = Ω (0, T ] for some T > 0. If u < v on te parabolic boundary P Q, ten u < v on Q. Equipped wit te comparison principle, we use Perron s metod to sow te following existence and stability teorem. Teorem 2. For initial data u 0 P (see te definition of P in section 4, by (4.1) (4.2)), te following olds: (a) (Teorem 4.2) Tere exists a minimal and a maximal viscosity solution u and u of (1.1) wit initial data u 0. (b) (Teorem 4.3) u and u are stable under perturbations of initial data wit appropriate ordering. 3

4 (c) (Corollary 6.3) u and u can be obtained as a limit of solutions solving te regularized parabolic equation (6.1). Aforementioned teorem states tat te maximal and minimal viscosity solutions are stable. Unfortunately, we are only able to sow te uniqueness of general viscosity solutions (i.e. te coincidence of minimal and maximal viscosity solutions) in several restricted settings. Te coincidence of te minimal and te maximal viscosity solutions wit general initial data remains open, except for te linear case. Teorem 3. For given initial data u 0 P, te following olds: (a) (Teorem 5.6) If F is linear, i.e. if F (M, p, z) = F (M) = tr M, ten tere exists a unique viscosity solution u wit initial data u 0, and u coincides wit te unique weak solution defined in [AL]. (b) (Teorem 5.8) If u 0 is eiter star-saped or if u decreases at t = 0, ten tere exists a unique viscosity solution of (1.1). 1.1 Remark. As mentioned above, our approac may not be optimal if b(s) degenerates at s = 0. On te oter and, we expect tat our approac can be extended to non-lipscitz b(s) and produce results similar to te above. Te difficulty in te analysis lies in te corresponding degeneracy of te elliptic operator in te positive pase, wen we write te equation in terms of b(s). 1.2 Example (Discontinuous solution). Here we briefly discuss an example wic illustrates discontinuities in te solutions. Set b(u) = u + := max(0, u), and consider (1.1) wit negative boundary data, and initial data tat are positive on some open set. As te solution evolves, te positive pase disappears in finite time, and ten te solution jumps to te stationary solution. Neverteless, one expects b(u) to be continuous. We refer te reader to [AL, p. 312] for an explicit formula. 1.1 Assumptions on te nonlinear operator F Let S n be te space of symmetric n n matrices. For given 0 < λ Λ, we define te Pucci extremal operators M ± : S n R as in [CC, W1]: M + (M) = sup tr AM, M (M) = inf tr AM, (1.5) A [λi,λi] A [λi,λi] were [λi, ΛI] = {A S n : λi A ΛI}. Alternatively, te Pucci operators can be expressed using te eigenvalues e 1,..., e n of matrix M: M + (M) = Λ e i>0 e i + λ e i<0 e i, M (M) = λ e i>0 e i + Λ e i<0 e i. Wit te Pucci operators at and, we sall assume te following structural condition on te operator F (M, p, z) : S n R n R R : (i) Tere exist 0 < λ < Λ and δ 0, δ 1 0 suc tat M (M N) δ 1 p q δ 0 z w F (M, p, z) F (N, q, w) for all M, N S n, p, q R n and z, w R. M + (M N) + δ 1 p q + δ 0 z w (1.6) 4

5 (ii) F is proper, i.e. z F (M, p, z) is nonincreasing in z. (1.7) (iii) Finally, to guarantee tat u 0 is a solution of bot te parabolic and te elliptic problems, we assume tat 1.2 Notation F (0, 0, 0) = 0. (1.8) In tis paper, we work in a fixed space dimension n 2. For a point x R n and time t R, te pair (x, t) R n R represents a point in space-time. For given r > 0, we define te open balls { B r := (x, t) : x 2 + t 2 < r 2}, te space disk D r := B n r {0} = {(x, 0) : x < r}, B n r := {x : x < r}, and te flattened set E r := { (x, t) : x 3 + t 2 < r 2}. Finally, we define te domain tat is used in te definition of regularizations of solutions, Ξ r := D r + E r. Here + is te Minkowski sum. Note tat Ξ r C 2 (in contrast wit D r + B r, wic is only C 1,1 ). We ask te reader to forgive a sligt abuse of notation: D r := D r D r = B n r {0} = {(x, 0) : x = r}. It will also be advantageous to introduce te (open) top and bottom flat pieces of Ξ r, Ξ r := {(x, r) : x < r}, Ξ r = {(x, r) : x < r}, and te (open) lateral boundary of Ξ r, L Ξ r := Ξ r \ Ξ r Ξ r. Tese sets are sketced in Figure 1. Te translation of a set A R n R by a vector (x, t) R n R will be denoted as A(x, t) := (x, t) + A. Te translation A(x) of a set A R n is defined similarly. 5

6 L Ξ r t Ξ r x = r t = r D r x Ξ r t = r Figure 1: Te boundary Ξ r of te set Ξ r We will often need to consider timeslices (cross sections at a fixed time) of a given set. To simplify te notation, let us define te timeslice of a set A R n R at time t as We often write A t if tere is no ambiguity. A t := {x : (x, t) A}. Let E R n R. Ten USC(E) and LSC(E) are respectively te sets of all upper semicontinuous and lower semi-continuous functions on E. For a locally bounded function u on E we define te semi-continuous envelopes u,e := inf v USC(E) v u v, u,e := sup v LSC(E) v u v. (1.9) Tese envelopes are used trougout most of te article wit E = Q, and terefore we simply write u and u if te set E is understood from te context. 2 Viscosity solutions In tis section we define te notion of viscosity solutions of problem (1.1). Formally, viscosity solutions are te functions tat satisfy a local comparison principle on parabolic neigboroods wit barriers wic are te classical solutions of te problem. We refer te reader to [CIL, CV, CS, KP] and references terein for oter examples of tis approac. 2.1 Definition (Parabolic neigborood and boundary). A nonempty set E R n R is called a parabolic neigborood if E = U {t τ} for some open set U R n R and some τ R. Let us denote P E := E \ E, te parabolic boundary of E. (See [W1] for a more general definition.) 2.2 Definition (Classical subsolution). Let E be a parabolic neigborood. Function ϕ is called a classical subsolution of problem (1.1) in a parabolic neigborood E if ϕ C(U) on an open set U R n R suc tat E = U {t τ} for some τ R, and te following olds: (i) ϕ C 2,1 x,t ({ϕ > 0}) and C 2,1 x,t ({ϕ < 0}), (ii) {ϕ = 0} {ϕ > 0} {ϕ < 0} and Dϕ ± > 0 on {ϕ = 0}, 6

7 (iii) b(ϕ) t F (D 2 ϕ, Dϕ, ϕ) 0 on {ϕ > 0} and {ϕ < 0}, (iv) Dϕ + Dϕ on {ϕ = 0}. Here {ϕ > 0} := {(x, t) U : ϕ(x, t) > 0} etc., and Dϕ ± (ξ, τ) := lim Dϕ(x, t). (x,t) (ξ,τ) (x,t) {±ϕ>0} We say tat ϕ is a strict classical subsolution if te inequalities in (iii) and (iv) are strict. Classical supersolutions are defined similarly by flipping te inequalities in Definition 2.2 (iii) (iv). At last, we define viscosity solutions. Note tat we set g 1 on Ω [0, T ] trougout te paper. 2.3 Definition (Viscosity subsolution). Let Q = Ω (0, T ] be a parabolic cylinder. Function u USC(Q) is a viscosity subsolution of (1.1) in Q if u(, 0) u 0 on Ω, u g on Ω [0, T ], and if u < ϕ on E for any strict classical supersolution ϕ on any parabolic neigborood E Q for wic u < ϕ on P E. One can define viscosity supersolutions accordingly, as a function in LSC(Q), by switcing te direction of inequality signs in te previous definition. 2.4 Definition (Viscosity solution). Locally bounded function u is a viscosity solution of (1.1) on Q if u,q is a viscosity subsolution on Q and u,q is a viscosity supersolution on Q. 2.5 Remark. We only test viscosity solutions by strict classical barriers. It is terefore possible to narrow te coice of E in Definition 2.3 to only include parabolic cylinders of te form Q = Ω (t 1, t 2] Q, were Ω as a smoot boundary, instead of all parabolic neigtboroods. Indeed, suppose tat E Q is a parabolic neigborood, ϕ is a strict classical supersolution on E, u < ϕ on P E, but u ϕ at some point in E. Define τ := sup {σ : u < ϕ on E {t σ}} R. Te set A := {x : (x, τ) E, u ϕ} is compact and terefore δ := dist(a {τ}, P E) > 0. Define te parabolic cylinder Q = (A + B δ/2 ) (τ δ/2, τ]. Clearly Q E and u < ϕ on P Q. Te boundary of A + B δ/2 can be easily regularized. Tis observation will be useful in section 5, were we will sow tat regular weak solutions are viscosity solutions. 3 Comparison principle Tis section is devoted to te proof of te following weak comparison principle. 3.1 Teorem. Let u be a viscosity subsolution and v a viscosity supersolution of (1.1) on Q = Ω (0, T ] for some T > 0, and assume tat u < v on P Q. Ten u < v on Q. To simplify te exposition of te proof of tis teorem, we sall assume tat T =. In fact, it is always possible to extend u and v from Ω (0, T ] to Ω (0, ). Moreover, we will only consider b(u) of te form b(u) = u + := max(u, 0). Te problem (1.1) wit a fully nonlinear operator F and a more general b can be always rewritten in tis way. Indeed, te differentiation b(u) t = b (u)u t for u > 0 is justified by te regularity of b, and b (u) can be absorbed into F. 7

8 Heuristic arguments Te comparison principle for classical subsolutions and supersolutions u and v of (1.1) can be proved using te following formal argument: we would like to sow tat {u(, t) > 0} {v(, t) > 0} for all t > 0, (3.1) since ten te conclusion follows due to te standard elliptic and parabolic comparison principle. Hence suppose (3.1) fails at some time. Since Du, Dv > 0 on te boundary of teir respective positive pases, te sets {u(, t) > 0} and {v(, t) > 0} ave smoot boundaries and evolve continuously in time wit respect to te Hausdorff distance. Terefore it follows tat tere exists te first time t 0 > 0 were {u(, t) > 0} intersects {v(, t) > 0}, let s say at x = x 0. Since {u > 0} {t t 0 } {v > 0} {t t 0 } and u 0 < v 0, te comparison principle applied to u and v respectively in te sets {u > 0} and in {v < 0} yields tat u v up to t = t 0. In particular, since u(x 0, t 0 ) = v(x 0, t 0 ) = 0, tis means tat Du + (x 0, t 0 ) Dv + (x 0, t 0 ) and Dv (x 0, t 0 ) Du (x 0, t 0 ). Furtermore, due to te regularity of {u(, t > 0} and {v(, t) > 0} and Hopf s lemma for uniformly parabolic and elliptic operators, it turns out tat te above inequalities are in fact strict. Tis contradicts te flux-matcing condition for te classical sub- and supersolution (Definition 2.2(iv)), i.e. te fact tat Du + Du and Dv + Dv at (x 0, t 0 ). Unfortunately, te rigorous version of te above euristic argument is rater lengty. Tere are many difficulties one faces in te general setting, were u and v are merely semi-continuous functions. As it is always te case in te teory of weak solutions, one sould translate te above euristics onto appropriate test functions or, to be more precise in our case, barriers. Our argument relies on a certain regularization procedure (see subsection 3.2) wic ensures tat, at a contact point (x 0, t 0 ) of te regularized solutions, te pase boundaries of eac regularized solution are bot locally C 1,1 in space. Suc regularity of te pase boundary would enable us to construct appropriate barriers wic would allow one mimic te euristic argument above. Tis tecnique was pioneered by Caffarelli and Vázquez ([CV]) in teir treatment of viscosity solutions for te porous medium equation. It was later applied to several one pase free boundary problems in [BV, K1, K2], and later extended to two-pase Stefan problems by te autors in [KP] (see also [CS]). In contrast to te aforementioned results for free boundary problems, te analysis of our problem presents several new callenges. Te most obvious callenge arises from te flux matcing condition of (1.1) on te transition boundary. Wile te regularization procedure provides regularity information in space variables, one sould still sow te finite propagation property of te pase boundary. In te aforementioned free boundary problems, te free boundary motion law prescribes te normal velocity of te free boundary in terms of te gradient of te solution, wic links te space regularity to te time regularity of te solution. Here one does not ave suc a direct relationsip between time and space regularity of te transition boundary. Indeed, te flux matcing condition of (1.1) turns out to be more 8

9 delicate tan te prescribed gradient condition in flame propagation type problems, since one as to account for te possibility tat te fluxes from bot sides will eiter degenerate to zero or diverge to infinity. Tis is overcome by te observation tat wit te regularization we can rule out te scenarios of a sudden extinction of te elliptic pase (Lemma 3.12), or a sudden srinkage/discontinuous expansion of te parabolic pase (Lemmas 3.13 and 3.19) at a contact point. Note, owever, tat even wit regularized solutions, te elliptic pase migt instantly become extinct away from a contact point. We point out tat, to allow for te regularization procedure, te strict ordering of u and v on P Q in te statement of Teorem 3.1 is necessary. We also point out tat a proof via doubling of variables, a classical tool in te teory of viscosity solutions (see [CIL]), is not available for pase transition-type problems, including (1.1). We sall present te proof by spliting it in a number of smaller intermediate results, wic will be later collected in 3.6 below. 3.1 Properties of solutions In tis section we clarify wat we mean by te parabolic and te elliptic problems. We refer te reader to [CIL] for te precise definitions and te detailed overview of te standard viscosity teory. 3.2 Definition. We say tat w(x, t) is a solution (resp. subsolution, supersolution) of te parabolic problem in an open set Q R n R if w is te standard viscosity solution (resp. subsolution, supersolution) of w t F (D 2 w, Dw, w) = 0 in Q. Similarly, we call w(x) a solution (resp. sub/supersolution) of te elliptic problem in an open set Ω R n if w is te standard viscosity solution (resp. sub/supersolution) of F (D 2 w, Dw, w) = 0 in Ω. Now wit te elp of te previous definition, we can rigorously express te intuitive fact tat te problem (1.1) is parabolic in te positive pase and elliptic in te negative pase. 3.3 Lemma. If u is a viscosity subsolution of (1.1) in Q ten u + is a subsolution of te parabolic problem in Q. Similarly, if v is a viscosity supersolution of (1.1) in Q ten v (, t) is a supersolution of te elliptic problem in Ω for eac t > 0. Proof. 1. Te first claim follows easily since u + = max {u, 0} and 0 solves (see (1.8)). u t F (D 2 u, Du, u) = 0 2. To prove te second claim for v := min {v, 0}, suppose it is not a supersolution of F (D 2 u, Du, u) = 0 at time t = t 0 > 0. Let us denote ζ(x) = v (x, t 0 ). Ten tere is a function ϕ C 2, ϕ < 0, and an open (space) ball B {ζ < 0} suc tat F (D 2 ϕ, Dϕ, ϕ) < 0 on B, ϕ ζ in B and ϕ < ζ on B, but ϕ = ζ at some point in B. 9

10 Due to te continuity of F, tere exists η 0 > 0 suc tat F (D 2 ϕ, Dϕ, ϕ η) < 0 in B for all η [0, η 0 ]. For any δ > 0 we set Q δ := B (t 0 δ, t 0 ]. Since v is lower semi-continuous, tere exists small δ > 0 suc tat Q δ {v < 0}, ϕ < v(, t) on B for t [t 0 δ, t 0 ], and v(x, t) ϕ(x) > η 0 on Q δ. Let us define te barrier ψ(x, t) = ϕ(x) + η0 δ t. Observe tat F (D2 ψ, Dψ, ψ) < 0 and ψ < 0 in Q δ and terefore it is a strict classical subsolution of (1.1). Furtermore, ψ < v on P Q δ, wile ψ = v at some point in Q δ. Tis contradicts te fact tat v is a supersolution of (1.1). 3.2 Regularizations In tis section, we define regularizations of te subsolution u and te supersolution v, and prove some of teir properties tat are applied in te proof of te comparison teorem. For given r (0, 1), define te regularizations Z(x, t) := W (x, t) := sup u, Ξ r(x,t) inf v. Ξ r(x,t) (3.2) Functions Z and W are well-defined on te parabolic cylinder Q r, Q r := { (x, t) Q Ξ r (x, t) Q } = Ω r (r, ), were Ω r := { x Ω dist(x, Ω c ) > r + r 2/3}. 3.4 Remark. Ω r as te uniform exterior ball property (wit radius r + r 2/3 ) and terefore it is a regular domain for te elliptic problem. 3.5 Remark. Te main advantage of regularizing over te set Ξ r instead of te ball B r is tat te appropriate level sets of Z and W ave bot space-time and space interior balls, see Proposition Remark. Defining Ξ = D + E, instead of D + B as in te previous paper [KP], as te consequence tat te parabolic boundary of Ξ is not C 1,1/2 x,t at te top flat piece Ξ, and tus it is not a regular set for te eat equation (see [CS]). We can terefore expect tat te gradient of a positive caloric function in Ξ wic is zero on te lateral boundary of Ξ will blow up at te lateral boundary as we approac Ξ, see Lemma A similar effect of vanising gradient is expected wen te solution is positive on Ξ c but zero in Ξ, see Lemma Tis is te necessary new ingredient required in te proof of te finite speed of propagation in Lemma Proposition. Suppose tat v is a visc. supersolution on Q = Ω (0, ). Ten W is a visc. supersolution on Q r. Similarly, if u is a visc. subsolution on Q ten Z is a visc. subsolution on Q r. Te strict separation of u and v on te parabolic boundary of Q allows us to separate Z and W on te parabolic boundary of Q r. 3.8 Proposition. Suppose tat u USC and v LSC in Q suc tat u < v on P Q. Ten tere exists r 0 > 0 suc tat Z < W on P Q r for all 0 < r r 0. 10

11 Proof. Standard from semicontinuity. Arguably te most important feature of regularizations Z and W is te interior ball property of teir level sets, as well as of te time-slices of teir level sets. We formalize tis fact by introducing te notion of dual points. 3.9 Definition. Let r > 0, u USC(Ω) and let Z be its sup-convolution. Let P Q r. We say tat P Q is a sup-dual point of P wit respect to u if P Ξ(P ) and u(p ) = Z(P ). Let us define Π u (P ) to be te set of all sup-dual points of P wit respect to u, Π u (P ) = { P Ξ r (P ) : u(p ) = Z(P ) }. Similarly we can define inf-dual points Π v (P ) for v LSC(Ω) by Π v (P ) := Π v (P ). In wat follows, we sall use te following convenient notation for various level sets of u and Z: wic is contrasted wit {u 0} = { (x, t) Q : u(x, t) 0 }, {u < 0} = {(x, t) Q : u(x, t) < 0}, {Z 0} = { (x, t) Q r : Z(x, t) 0 }, {Z < 0} = {(x, t) Q r : Z(x, t) < 0}. Sets {v 0}, {v > 0}, {W 0} and {W > 0} are defined in a similar fasion. Tis coice guarantees tat sets wit and are closed, wile sets wit < and > are open. We first make a few simple observations about Π u and Π v Proposition. Let u USC(Q). Ten for all P Q r : (i) Π u (P ), (ii) Ξ r (P ) Q r {Z Z(P )} for all P Π u (P ), (iii) if P {Z 0}, ten Π u (P ) Ξ r (P ) {u 0} and Ξ r (P ) {u < 0}. Since te closed sets {W 0} and {Z 0} ave closed space and space-time interior balls at eac point, tey don t ave any points tat are isolated from teir interior: 3.11 Lemma. Te level sets of te functions W and Z defined above ave te following properties: int {W 0} = {W 0}, int {Z 0} = {Z 0}. Moreover it is true for every time-slice t (wit space closure and interior). Proof. Tis follows from te interior ball property in Proposition Fix time t and for simplicity define E = {W 0} t. Pick any point x E. Ten tere is an open ball B = B r (y) suc tat x B and B E. But B is open so B int E and terefore x B int E. 11

12 {W 0} σ ξ ˆξ B i1 {v 0} σ x i1 x im B im γ Ω r Ω Figure 2: Space cross section of te situation in te proof of Lemma 3.12 Anoter important property of te regularized supersolution W is tat eac point of te time-slice {W 0} t is connected to te boundary Ω r wit nonpositive values of W by a wide trunk of finite lengt: 3.12 Lemma. Let W be te inf-convolution of a supersolution v. Ten for every (ξ, σ) {W 0} tere is a piecewise linear continuous curve γ : [0, 1] Ω r, wit finite lengt suc tat γ(1) Ω c r and ξ B r/2 (γ(s)) Ω r {W 0} t=σ. s [0,1] Proof. Pick P = (ξ, σ) {W 0}. By Proposition 3.10, tere exists a point P = (ξ, σ ) on Π v (P ) suc tat (ξ, σ) Ξ r (ξ, σ ) {W 0}. Terefore we can also find ˆξ suc tat ξ B r/2 (ˆξ) and B r/2 (ˆξ) {W 0} σ. Te situation is depicted in Figure 2. Let us denote H = {v 0} t=σ. Since H is compact we can select a finite subcover from te open cover H B r/2 (x). x H Let us denote te balls in te finite subcover by B 1,..., B k, wit centers x 1,..., x k. Suppose tat tere exists a permutation (j 1,..., j k ) of (1,..., k) and q N suc tat B j l Ω B j l =. 1 l q q<l k Let us denote G = 1 l q Bj l and C = H G. Note tat G is open and C is nonempty. By definition, v > 0 on G {σ } and terefore tere is δ > 0 suc tat v > 0 on G (σ δ, σ ] ({v > 0} is open by v LSC). In particular, since v (, t) is a supersolution of te elliptic problem for every t (Lemma 3.3), an application of te elliptic comparison principle (Proposition A.1) yields tat v 0 in G (σ δ, σ ]. And a straigtforward barrier argument (barrier constructed in Lemma B.4) sows tat in fact v > 0 on G {σ }. Tis is a contradiction wit v(x j1, σ ) 0 at x j1 G. 12

13 t (0, 0) 0 1 a 2 0 a 0 Q 0 Q 1 2 a 2 1 a 1 Q 2 a 2 2 Ξ 3 γa 2 2 a 2 x v > 0 Q 3 Figure 3: Iterations in te proof of lower bound in Lemma 3.13 Terefore we conclude tat we can find i 1,..., i m distict wit ξ B i1, B im Ω and B i l B i l+1, 1 l < m. Since we observe tat B r (x i ) {W 0} σ for all 1 i k, we can coose te curve γ as te piecewise linear curve connecting points ˆξ, x i1,..., x im. Now we present two results, Lemmas 3.13 and 3.16, tat justify te sape of te domain Ξ r cosen in te definition of te regularizations Z and W Lemma. Let v be a visc. supersolution of (1.1) on Ξ, Ξ = Ξ r (ξ, σ) for some (ξ, σ) R n R, and v > 0 in Ξ. Ten tere exists f C([0, r]), f(0) = 0, f > 0 on (0, r] and as s 0+ suc tat f(s) s v(x, t) f(r x ξ ) for x ξ < r, t [σ + r/2, σ + r]. Proof. We only prove tat v(x, σ + r) f(r x ξ ). Te full result follows from te simple observation Ξ r/2 (x, t) Ξ r for x r [ 2, t r 2, r ]. 2 Let us fix a point (ζ, σ + r) D r (ξ, σ + r). Since te argument is invariant under translation and space rotation, we can assume tat (ζ, σ + r) = (0, 0). Let s = r ζ ξ. Our goal is to sow tat v(0, 0) = v(ζ, σ + r) f(s) for some function f. Te situation is depicted in Figure 3. It is straigtforward to estimate te distance of te lateral boundary L Ξ at eac time, Ξ t, from te origin x = 0 for t ( r, 0): dist(0, Ξ t ) = s + 3 r 2 (t + r) 2 = s + 3 2rt t 2 > s + 3 rt. (3.3) Since v LSC is positive in Ξ, it as a positive minimum on te compact set K := { (x, t) : x ξ r + 3 rt, t [ 7r/8, r 2 /256] } 13

14 as K Ξ. Let us set and let us define te barrier { } 1 γ := min, 1 16nλ + 8δ 1 + 2δ 0 ϕ(x, t) = ( 1 ) 2γ t 4 x (3.4) + Clearly ϕ is a viscosity subsolution of te parabolic problem due to (1.6), and ϕ(0, γ) = 1 2 ϕ(0, 0) = 1 2. Let us define a sequence of parabolic cylinders were Q j = B aj ( j (1 + γ)a 2 j, j ), j 0, j 1 j = a 2 i, j 1, 0 = 0. a j s are defined in accordance wit (3.3) as a j = 3 { r j + s, j 1, a 0 = min s, r }. 16 Writing j+1 = j a 2 j, we can derive te recurrence relation i=0 a j+1 = 3 ra 2 j + (a j s) 3 + s, j 0. (3.5) Note tat a j s were cosen in suc a way tat Q j Ξ for all j for wic j+1 < r due to (3.3). First assume tat s < r/16. From (3.5) we estimate wic yields a j+1 r 1/3 a 2/3 j, j 0, ( s ) ( 2 3) j a j r r as j. (3.6) r We want to estimate te minimal k = k(s) tat guarantees B Q k = D ak (0, k (1 + γ)a k ) K. We first observe tat if a j r 2 + s for some j 0 ten k j. Terefore (3.6) yields te upper bound [ ] log s r log log k 1 2 log (3.7) 2 14

15 To sow tat suc k indeed exists, suppose tat j > r 2 /8 for some j. Ten a j < r/2+s and j (1 + γ)a 2 j r2 8 2 ( r 2 + s ) 2 = r 2 8 r2 2 2rs 2s2 7r 8, since s < r/16 and γ 1. If s r/16 ten we set k = 0. were Next we iteratively define, for j = k,..., 0, te rescaled barriers ϕ j (x, t) = κ j ϕ(a 1 j κ j = x, a 2 (t j ) γ), j { mink v > 0 j = k, min ψ Baj /2 j+1(, j+1 γa 2 j ) j < k, and te functions ψ j, j = k,..., 0, are te unique solutions of te parabolic problem wit boundary data ψ j = ϕ j on P Q j. We observe tat ψ j ϕ j in Q j due to te parabolic comparison (Prop. A.2) since ϕ j is a subsolution of te parabolic problem. Now te parabolic Harnack inequality (Prop. A.4 applied wit t 1 = γ and t 2 = 1 + γ) yields κ j 1 = inf ψ j (, j γa 2 j 1) inf ψ j B aj 1 /2 B aj /2 [ j γa 2 j 1,j] 1 c sup ψ j (, j+1 ) 1 B aj 2c sup ψ j (, j+1 γa 2 j) /2 B aj /2 (3.8) = 1 2c κ j. Finally, we realize tat, by definition, v ϕ k = ψ k on P Q k and tus an application of te parabolic comparison sows tat v ψ k in Q k. Since we cose ψ j so tat ψ j ψ j+1 on Q j Q j+1, we can apply te comparison principle iteratively to conclude tat v ψ j in Q j for j = k,..., 1. Te case j = 0, owever, as to be considered separately, because we only know tat v > 0 in Ξ, i.e. in Q 0 {t < 0}. Terefore te parabolic comparison can only sow tat v ψ 0 in Q 0 {t < 0}. Neverteless, since ψ 0 (0, 0) > 0 and v is bounded from below on Ξ, a straigtforward barrier argument using a strict classical subsolution of (1.1), tat can be constructed in te form similar to (3.4), extended in negative pase using a strict subsolution of te elliptic problem, sows tat v(0, 0) ψ 0 (0, 0). Now let α = 1 2c. Ten a simple induction of (3.8) using te bound (3.7) yields te lower bound v(0, 0) ψ 0 (0, 0) α ( log s ) log α log r 2 3 log 1 2 min v =: f(s), s (0, r]. K A straigtforward computation verifies tat f(s) 0 and f(s)/s as s 0+ since α < 1. 15

16 3.14 Corollary (Continuous expansion of {v 0}). If v is a visc. supersolution of (1.1) ten te set {v 0} cannot expand discontinuously, i.e. for any s. {v 0, t < s} = {v 0, t s}, Proof. Suppose tat te claim is not true, i.e. tere is a point Tat means tat tere is ρ > 0 suc tat (ξ, τ) {v 0, t τ} \ {v 0, t < τ}. Ξ ρ (ξ, τ ρ) {v 0, t < τ} = and tus v > 0 in Ξ ρ (ξ, τ ρ). Lemma 3.13 ten yields v(ξ, τ) > 0, a contradiction Remark. Note tat te set {u 0} of a subsolution u can expand discontinuously wen a part of te set {u < 0} is pinced off by a collision of two fingers of {u 0} Lemma. Let u 0 be a bounded subsolution of te parabolic problem in a parabolic neigborood of Ξ, wit Ξ = Ξ r (ξ, σ) for some (ξ, σ) R n R and r (0, 1). Assume tat u = 0 on Ξ. Ten tere exists ε > 0 and g C 1 ([0, r + ε)) wit g = g = 0 on [0, r] suc tat 0 u(x, t) g( x ξ ) for x ξ < r + ε, t [σ, σ + r]. Proof. Let us coose (y, s) B r (x) [σ, σ + r]. For given ε > 0, η (0, ε), we define te barrier on Note tat: ψ ε,η (x, t) := 4M ε (4nΛt + x 2 + η), ( E ε,η := B ε 1/2 ( ε ) 8nΛ, 0] {ψ ε,η (x, t) > 0}. (i) Due to (1.6), ψ ε,η is a strict supersolution of te parabolic problem in E ε,η as long as ε 1/2 < 2nΛ 3(δ, and ψ 0+δ 1) ε,η 2M on B ε 1/2 [ ε 8nΛ, 0]. (ii) ψ ε,η > 0 if and only if t > η 4nΛ or x > 4nΛt η. (iii) 3 rt > 4nΛt wen t ( r 8nΛ, 0) and 3 rε 8nΛ > ε1/2 for small ε > 0. (i) (iii) verify tat for small ε > 0 and all η (0, ε), we ave te ordering u(x, t) ψ ε,η (x y, t s) on P E ε,η + (y, s) and ence by te parabolic comparison (Prop. A.2) te ordering olds in E ε,η + (y, s). Terefore tere exists a constant ε > 0 suc tat for (x, t) B r+ε [σ, σ + r]. u(x, t) g( x ξ ) := inf η>0 ζ B r ψ ε,η (x ζ, 0) 16

17 3.3 Z and W cross We proceed wit te te proof of te comparison principle (Teorem 3.1) for te regularized solutions Z and W, in place of u and v. To argue by contradiction, we investigate te situation wen Z and W cross in Q r, i.e. tere is a finite first crossing time t 0, defined by t 0 := sup {τ Z(, t) < W (, t) for 0 t τ}. (3.9) Now since {W 0} cannot expand discontinuously due to Lemma 3.14, we can prove tat a certain ordering of level sets of W and Z is preserved up to te crossing time t Lemma. Let Z and W be te regularized solutions defined in (3.2) and t 0 be te crossing time defined in (3.9). Ten {Z 0} t0 {W 0} t0 = ( {Z 0} t0 ) ( {W 0}t0 ) (3.10) In particular, int ( {Z 0} t0 ) int ( {W 0}t0 ) =. (3.11) Proof. First observe tat {W 0} t {Z < 0} t for t < t 0 (or equivalently {Z 0} t {W > 0} t for t < t 0 ). Step 1. We claim tat int {W 0} t0 {Z < 0} t0. Pick any ξ int {W 0} t0. Due to continuous expansion of {W 0} (Corollary 3.14), tere exists δ > 0 suc tat B δ (ξ) int {W 0} t for a sort time before t 0, t [t 0 δ, t 0 ]. Lemma 3.12 yields tat B δ (ξ) is connected to Ω r wit non-positive values of W (and tus negative Z) troug a wide trunk G = (γ([0, 1]) + B r/2 ) Ω r, and B δ (ξ) G {W 0} t0. Te continuous expansion of {W 0} again guarantees tat G can be cosen so tat G {W 0} t for a sort time before t 0. Finally, we recall tat Z < W in G [0, t 0 ). Terefore we can construct a strict classical supersolution of (1.1) up to te time t 0 tat will stay above Z and is negative in B δ (ξ). Indeed, we solve te elliptic problem in G wit zero on G Ω r and negative data on G Ω r. Since G as a uniform interior ball property at points G Ω r, we can proceed as in te proof of Lemma 4.1. We conclude Z(ξ, t 0 ) < 0. Step 2. Since {Z 0} Q r = {Z < 0} c Q r, we ave wic togeter wit Lemma 3.11 also gives int ( {W 0} t0 ) {Z 0}t0 =, (3.12) int ( {Z 0} t0 ) {W 0}t0 =. (3.13) Step 3. Now (3.12) clearly implies {Z 0} t0 {W 0} t0 {W 0} t0 \ int ( {W 0} t0 ) = ( {W 0}t0 ). Using (3.13) symmetrically concludes te proof of te lemma. 17

18 3.18 Corollary. Let Z, W and t 0 be defined as above and let ξ {Z 0} t0 {W 0} t0. Ten tere is a unique unit vector ν suc tat ν is te unit outer normal to {Z 0} t0 and te unit outer normal vector to {W > 0} t0 at ξ. at ξ Proof. Due to Lemma 3.17, ξ ( {Z 0} t0 ) ( {W 0}t0 ). Moreover tere are dual points (ξ u, σ u ) Π u (P ) {u 0} and (ξ v, σ v ) Π v (P ) {v 0} (see Proposition 3.10). Let B u = int (Ξ r (ξ u, σ u ) ) t=t0 and B v = int (Ξ r (ξ v, σ v ) ) t=t0. Due to te definition of Ξ r, B u and B v are balls in R n, centered at ξ u and ξ v, respectively, of radius greater tan or equal to r. Observe tat B u int {W 0} t0 and B v int {Z 0} t0. Terefore B u B v ( int {W 0} t0 ) ( int {Z 0}t0 ) =, wile B u B v = {ξ}. Tis is true for an arbitrary coice of dual points. Terefore te outer unit normal of ν of B u (resp. inner unit normal of B v ) at ξ is uniquely determined, and can be taken as ν = ξ ξ u ξ ξ u = ξ v ξ ξ v ξ. 3.4 Finite speed of expansion Our goal in tis section is to use te ordering of te support to prove te ordering of te functions Z and W at te contact time t = t 0. Tis needs a careful analysis since Z and W are merely semi-continuous. From Lemma 3.3 we know tat Z + is a parabolic subsolution, and W is an elliptic supersolution for eac time, and also W is a parabolic supersolution in {W > 0} (open set) and Z is an elliptic subsolution for eac time in {Z < 0} (open set). Terefore we can invoke te standard comparison principle (Propositions A.1 and A.2) in te open sets {W > 0} and {Z < 0} t, if we know tat te functions are ordered on te boundaries of tese sets. Tis is not completely obvious, owever, and we ave to pay special attention to te situation at te contact point Lemma (Finite speed of expansion at te contact point). Let Z, W be te regularizations and t 0 be te crossing time as defined in (3.9). Ten for any ξ {Z 0} t0 {W 0} t0 Π u (ξ, t 0 ) Π v (ξ, t 0 ) L Ξ r (ξ, t 0 ). Proof. We split te proof into a number of sorter steps. Here we use te simplified notation Ξ = Ξ r (ξ, t 0 ) (and analogously for Π u and Π v ). Step 1. Π v Ξ =. For te sake of te argument, suppose tat tis does not old and tere indeed is a point P = (ξ, σ ) Π v Ξ. Note tat σ = t 0 + r. We observe tat v > 0 in Ξ due to 18

19 (ζ, σ ) t K (ξ, σ ) Π v ν Σ t = σ Ξ v > 0 x ν ε ϕ > 0 ε 1 ϕ < 0 ε t = σ ε ρ 0 Figure 4: Construction of a test function ϕ in Lemma 3.19 (3.10) and Proposition 3.10(iii), and tus we can apply Lemma 3.13 for v on Ξ. Te first consequence is ξ ξ = r, i.e. P D r (ξ, t 0 + r). Let ν be te unit normal from Corollary We coose ρ 0 = min( r 4, ˆρ c) (ˆρ c is defined in Prop. B.3) and set ζ = ξ ρ 0 ν, ˆσ = σ, â = 2, ˆb = 1 and ˆω = 0. Proposition B.3 provides us wit a radially symmetric subsolution of (1.1) wit parameters â, ˆb and ˆω. We denote ˆϕ its translation by (ζ, ˆσ) wic is defined on te cylinder K = {ρ 0 ε < x ζ < ρ 0 + ε, t [ˆσ ε, ˆσ]}, (3.14) for some ε > 0. Te situation is depicted in Figure 4, wic is a part of te larger picture in Figure 5. Let us define ϕ = µ ˆϕ, were µ > 0 is picked large enoug to ensure tat min K ϕ < min K v. Furtermore, since v(x, t) f(r x ξ ) for x ξ < r, wit f(s)/s as s 0+, tere exists ε 1 > 0 for wic v > ϕ on We sall compare v and ϕ on a cylinder Σ, {ρ 0 ε 1 x ζ < ρ 0, t [ˆσ ε, ˆσ]}. Σ := {ρ 0 ε 1 < x ζ < ρ 0 + ε, t (ˆσ ε, ˆσ]} K. So far we ave sown tat v > ϕ on all of P Σ except on {(x, ˆσ ε) : ρ 0 x ζ < ρ 0 + ε} Ξ, were clearly ϕ 0 wile v > 0. Terefore ϕ < v on P Σ wile ϕ v at P Σ, and we arrive at a contradiction since ϕ is a strict classical subsolution of (1.1) on Σ. Step 2. Π u Ξ = 19

20 Te proof of step 2 is similar to tat of step 1. Coose P Π u Ξ. Due to (3.11), Lemma 3.12 and Hopf s lemma (Prop. A.3), we ave lim inf 0+ Terefore tere is 0 > 0 for wic Z(ξ + ν) = 2c > 0. u(x, t) Z(ξ + ν, t 0 ) < c < 0 for 0, (x, t) Ξ r (ξ + ν, t 0 ). (3.15) Proceeding as in te proof of step 1, we ave ˆϕ, ε (0, 0 ) and K, tis time wit ζ = ξ +ρ 0 ν, ˆσ = σ. In contrast wit step 1, we define ϕ = µ ˆϕ, and (3.15) lets us coose µ > 0 small for wic ϕ > u on { x ζ = ρ 0 ε, t [ˆσ ε, ˆσ]} {ρ 0 ε x ζ ρ 0, t = ˆσ ε}. Now we find ε 1 > 0 suc tat ϕ > u on { x ζ = ρ 0 + ε 1, t [ˆσ ε, ˆσ]}, tis is again possible tanks to Lemma 3.16 applied to u on Ξ. Finally, define Σ := {ρ 0 ε < x ζ < ρ 0 + ε 1, t (ˆσ ε, ˆσ]}. Since clearly ϕ 0 and u < 0 on {ρ 0 x ζ < ρ 0 + ε, t = ˆσ ε}, we succeeded in sowing tat ϕ > u on P Σ wile 0 = ϕ u at P Σ, a contradiction. Step 3. Π u Ξ ten Π v Ξ = (and also wen if we swap Π u and Π v ) Indeed, tis follows from (3.11) and te definition of t 0 in (3.9), wic togeter yield tat for any P u Π u and P v Π v we ave Ξ r (P u ) Ξ r (P v ) {t t 0 } = (ξ, t 0 ) = P. Step 4. To finis te proof, we simply realize tat steps 2 and 3 used togeter imply tat Π u Ξ =, and similarly steps 1 and 3 imply Π v Ξ = Remark. Lemma 3.19 sows tat te situation at any P = (ξ, t 0) {Z 0} {W 0} looks like Figure 5. Now we ave enoug regularity to sow te following: 3.21 Lemma. For Z, W, t 0 defined above, we ave Z = W = 0 on {Z 0} t0 {W 0} t0. Proof. We will only sow tis result for Z. A similar, simpler argument applies to W as well. See also [KP, Lemma 3.6]. Let M = 2 max {t t0} Z < and P = (ξ, t 0 ) {Z 0} {W 0}. We can also coose P v Π v (P ). Let ν be te unit normal from Corollary Lemma 3.19 guarantees tat tere exists m v R suc tat (ν, m v ) is an interior normal of Ξ(P v ) at P, see Figure 5. Tis is wy we call Lemma 3.19 finite speed of expansion. We recall tat Z < W 0 in Ξ(P v ) {t < t 0 }. Furtermore, te argument from step 2 of te proof of Lemma 3.19, using (3.11), Lemma 3.12 and Proposition A.3, verifies tat Z < 0 in Ξ(P v ) {t = t 0 } as well. 20

21 Ξ(P u ) P u {u 0} (ν, m u ) P ν (ν, m v ) t = t 0 t x ν {Z 0} (ν, m u ) {W 0} Ξ(P ) P v {v 0} Ξ(P v ) Figure 5: Situation at P = (ξ, t 0 ) {Z 0} {W 0}. (ξ, t 0 ) (ν, m v ) ν ρ 0 (ν, 2m v ) t 0 (ζ, t 0 ) t Ξ(P v ) Σ τ ρ 0 ˆωτ x ν t 0 τ Figure 6: Construction of a test function ϕ in Lemma 3.21 As in te proof of Lemma 3.19, we can construct a strict classical supersolution of (1.1) wit te elp of Proposition B.3. Let us again coose ρ 0 small enoug so tat for â = 2, ˆb = 1, ˆω = max {0, 2m v }, ζ = ξ + ρ 0 ν, tere is a strict classical supersolution ˆϕ( ζ, t 0 ), defined on te set for some ε > 0. For τ > 0 and = τ 2, let us define K = {ρ 0 ε x ζ ρ 0 + ε, t (t 0 ε, t 0 ]}, Σ τ = {ρ 0 < x ζ (t t 0 )ˆω < ρ 0 +, t (t 0 τ, t 0 ]}, see Figure 6. We can take τ small enoug so tat (a) Σ τ K, (b) Σ τ {t = t 0 τ} Ξ(P ). 21

22 Now we find µ > 0 large enoug so tat µ ˆϕ > 2M on { x = ρ tˆω, t [ τ, 0]}. For η > 0 set ϕ η (x, t) = µ ˆϕ(x ζ ην, t t 0 ). Since Z is a subsolution of (1.1) and Z < ϕ on P Σ + (ην, 0) for small η, we also ave Z < ϕ in Σ + (ην, 0). We conclude by sending η 0, Z(ξ, t 0 ) lim η 0+ ϕ η(ξ, t 0 ) = ϕ 0 (ξ, t 0 ) = 0. Lemmas 3.17 and 3.21 ave te following consequence: 3.22 Corollary. Let Z, W and t 0 be as above. Ten Furtermore, Z W at t = t 0. {Z = 0} t0 {W = 0} t0, Proof. We apply te comparison principle for elliptic equations (Prop. A.1) and compare Z(, t 0 ) and W (, t 0 ) on te open set {Z < 0} t0. Indeed, we can set Z = 0 on {Z < 0} to ensure tat Z 0. Te modified Z(, t 0 ) is clearly in USC({Z < 0} t0 ) and a subsolution of te elliptic problem in int {Z < 0} t0 (= {Z < 0} t0 by Lemma 3.11). Ten Z W on ( {Z < 0} t0 ) due to Lemma 3.21 and te elliptic comparison applies. Te same can be done using te comparison principle for te parabolic equation (Proposition A.2) in te parabolic neigborood {W > 0, t t 0 }. Finally, if {Z = 0} t0 {W = 0} t0 =, ten in te view of Lemma 3.17 and Lemma 3.21, {Z 0, t t 0 } {W 0, t t 0 } =. Since tese sets are compact, tey ave positive distance and te comparison in te first part of te proof yields Z < W at t = t 0, a contradiction wit te definition of t 0 and Z W USC. 3.5 Ordering of gradients at te contact point Here we will sow tat, at a contact point P, te gradients of Z (resp. W ) follow te flux ordering given in Definition 2.2(iv) (resp. its supersolution counterpart). As mentioned in te beginning of section 3, suc ordering would readily yield a contradiction wit te fact tat Z crosses W from below at P Lemma. Let Z, W and t 0 be as defined above, let P = (ξ, t 0 ) {Z 0} {W 0} be a contact point at time t 0, and let ν be te unique spatial unit normal vector at P obtained in Corollary Ten lim inf 0+ lim sup 0+ Z(ξ ν, t 0 ) W (ξ ν, t 0 ) + lim inf 0+ + lim sup 0+ Z(ξ + ν, t 0 ) 0, W (ξ + ν, t 0 ) 0. 22

23 Proof. We prove te result for Z. Te proof for W is similar. Let us again write Ξ = Ξ r (P ) Ξ r (ξ, t 0 ) trougout te proof. Denote a := lim inf 0+ Z(ξ ν, t 0 ), b := lim inf 0+ Z(ξ + ν, t 0 ). Note tat b 0 a. A simple barrier argument, following te one in te proof of Lemma 3.21 and taking advantage of te room provided by Lemma 3.19, sows tat a <. Furtermore, te inequality a + b 0 is satisfied trivially if b = 0. Tus we may replace b by 2a if necessary, and assume tat < b < 0. Suppose tat te result does not old, in oter words, a + b < 0. In tat case, we set κ := a b 2 > 0, and we note tat a < κ, Let us coose η > 0 suc tat b < κ, a + 3η < κ, b + 3η < κ. We sall use tis to construct a barrier tat crosses u from above at an arbitrary fixed point P u = (ξ, σ ) Π u (P ), yielding a contradiction. As in te proof of Lemma 3.21, since P u L Ξ tanks to Lemma 3.19, tere is m u R suc tat (ν, m u ) R n R is an interior normal vector to Ξ at P u (see Figure 5). Following te proof of Lemma 3.19, we can construct a radial supersolution of (1.1) wit te elp of Proposition B.3. Let us again coose ρ 0 small so tat for â = a + 3η and ˆb = b + 3η, ˆω = max(0, 2m), ζ = ξ + ρ 0 ν, and ˆσ = σ, tere is ϕ = ˆϕ, a strict classical supersolution of (1.1) translated by (ζ, ˆσ), defined on K as in (3.14), and ϕ = 0 at P u. From te definition of a and b, and Z, for every τ (0, 1) tere are 1, 2 (0, τ 2 ) suc tat For a given coice of τ, 1, 2, let us define sup u = Z(ξ 1 ν, t 0 ) < (a + η) 1, (3.16) Ξ r(ξ 1ν,t 0) sup u = Z(ξ + 2 ν, t 0 ) < (b + η) 2. (3.17) Ξ r(ξ+ 2ν,t 0) Σ τ := {ρ 0 2 < x ζ (t ˆσ)ˆω < ρ 0 + 1, t (ˆσ τ, ˆσ]}, see Figure 4 for a sketc of a similar construction. We sall coose τ (0, 1) small enoug so tat te following olds: (a) Σ τ K, (b) ϕ > (a + η) 1 on A := { x ζ (t ˆσ)ˆω = ρ 0 + 1, t [ˆσ τ, ˆσ]}, (c) ϕ > (b + η) 2 on P Σ τ \ A, (d) A Ξ r (ξ 1 ν, t 0 ), (e) P Σ τ \ A Ξ r (ξ + 2 ν, t 0 ). 23

24 Indeed, ϕ satisfies (b) and (c) for small τ due to its smootness in te positive and negative pases, and (a), (d) and (e) are a consequence of te coice of ˆω and te definition of Σ τ. Since ϕ > u on P Σ τ (tanks to (b)+(d)+(3.16) and (c)+(e)+(3.17)) wile 0 = ϕ = u at P u Σ τ, we get a contradiction. 3.6 Proof of Teorem 3.1 Now we are ready to prove Teorem 3.1. Te proof proceeds by sowing te comparison for te regularizations Z and W defined in (3.2). We coose r > 0 suc tat Z < W on P Q r (Proposition 3.8). Suppose tat tere is a point in Q were u v. Since u Z and W v, we see tat tere must be a point in Q r were Z W. Ten t 0 in (3.9) is finite. We recall tat Z W at t = t 0 by Corollary Corollary 3.22 also guarantees te existence of a contact point at t = t 0 : P = (ξ, t 0 ) {Z 0} {W 0} wit Z(P ) = W (P ) = 0. At te point P, te boundaries {Z 0} and {W 0} are C 1,1, in te sense tat tey ave space-time and space balls from bot sides (Proposition 3.10(iii)). Let ν be te unique unit normal vector to te space balls at ξ given by Corollary Set Ω to be te connected component of te open set {Z < 0} t0 boundary. Lemma 3.12 guarantees tat tat contains ξ on its Z(, t 0 ) W (, t 0 ) on Ω. We recall tat Z(, t 0 ) is a subsolution of te elliptic problem on Ω and W (, t 0 ) is a supersolution of te elliptic problem (Lemma 3.3). We can apply te elliptic Hopf s lemma (Proposition A.3) to Z(, t 0 ) and W (, t 0 ) on Ω at ξ, wic yields lim inf 0+ Z(ξ + ν, t 0 ) > lim inf 0+ W (ξ + ν, t 0 ). Finally, te weak gradients are ordered at ξ by Lemma 3.23 wic leads to a contradiction (we write Z( ) instead of Z(, t 0 ), all limits are 0+): lim inf Z(ξ ν) Z(ξ + ν) Z(ξ + ν) lim sup lim inf Hopf W (ξ + ν) W (ξ ν) > lim inf lim sup Z(ξ ν) lim inf. Terefore t 0 cannot be finite and we conclude tat u < v in Q. Tis finises te proof of Teorem

25 3.7 Remarks on te proof of Teorem 3.1 for divergence-form operators Te arguments in te proof of Teorem 3.1 also old for te divergence-form operator F given by (1.4). Te only difference lies in te references on te properties of solutions of te parabolic and elliptic problems, wic ave been used trougout section 3 and wic are proved or referred to in Appendix A, as well as te specific barriers constructed in Appendix B.1. Since te barriers for te divergence-form operator are constructed in Appendix B.2, ere we only point out te references for te regularity properties of solutions of te parabolic and elliptic problems (in te sense of Definition 3.2). More precisely, we used te parabolic Harnack inequality (Proposition A.4) and te elliptic Hopf s lemma (Proposition A.3). For a divergence-form operator F given by (1.4), Proposition A.4 is sown in [LSU, Capter V] and Proposition A.3 is sown in [DDS]. Having all of te above properties and test functions, one can proceed as in section 3 to prove te following: 3.24 Teorem. Teorem 3.1 olds for F given in (1.4). 4 Existence and stability of viscosity solutions Let F and Ω be given as before. In tis section we sow te existence of viscosity solutions of (1.1) via te Perron s metod. We say tat a continuous function u 0 : Ω R is in P if te following conditions old: u 0 = 1 on Ω and F (D 2 u 0, Du 0, u 0 ) = 0 in {u 0 < 0}. (4.1) Γ(u 0 ) := {u 0 > 0} = {u 0 < 0} is locally a C 1,1 -grap. (4.2) Note tat (4.2) and te boundedness of Ω guarantee tat tere exists R 0 > 0 suc tat {u 0 > 0} as bot an interior and an exterior ball of radius R 0 at eac point. First we ensure, by constructing suitable barriers, tat te solutions wit initial data u 0 P evolve continuously at te initial time. 4.1 Lemma. Suppose u 0 P. Ten tere exist at least one viscosity subsolution U and one supersolution V of (1.1) wit initial data u 0. Moreover, tere exists a constant C > 0 and a small time t 0 > 0 depending only on te maximum of u 0, N, d and R 0 suc tat for 0 < t < t 0 we ave d(x, Γ), d(y, Γ) t 1/4 for x {U(, t) > 0} and y {V (, t) < 0}. Proof. 1. First we construct a supersolution V. Let us set O + (t) := {x : d(x, {u 0 > 0}) < t 1/4 }, and let V (x, t) solve V t F (D 2 V, DV, V ) > 0 in O + (t) V = 0 on Γ(t) := O + (t) F (D 2 V, DV, V ) > 0 in O (t) := Ω \ O + (t) 25