Abstract. We prove a strong maximum principle for semicontinuous viscosity subsolutions or supersolutions of fully nonlinear degenerate elliptic PDE's,.
On the strong maximum principle for fully nonlinear degenerate elliptic equations Martino Bardi and Francesca Da Lio Dipartimento di Matematica P. e A., Universita di Padova, via Belzoni 7, 35131 Padova, Italy. Abstract
We prove a strong maximum principle for semicontinuous viscosity subsolutions or supersolutions of fully nonlinear degenerate elliptic PDE's, which complements the results of [17]. Our assumptions and conclusions are di erent from those in [17], in particular our maximum principle implies the nonexistence of a dead core. We test the assumptions on several examples involving the p-Laplacian and the minimal surface operator, and they turn out to be sharp in all cases where the existence of a dead core is known. We can also cover equations that are singular for p = 0 and very degenerate operators such as the 1-Laplacian and some rst order Hamilton-Jacobi operators.
1. Introduction.
In this note we investigate the validity of the Strong Maximum Principle and the Strong Minimum Principle (SMaxP and SMinP in the following) for semicontinuous viscosity subsolutions and supersolutions of fully nonlinear partial di erential equations F(x; u; Du; D2u) = 0
(1)
that are proper in the sense of [7], that is, F(x; r; p; X) F(x; s; p; Y )
if r s and Y X;
(2)
where
F: RN (RN n f0g) S(N) ! R;
is an open set of RN , S(N) denotes the set of N N symmetric matrices, and Y X means that X ? Y is nonnegative semide nite. By Strong Maximum and Minimum Principle we mean the following properties. SMaxP: any upper semicontinuous viscosity subsolution of (1) in an open set
that attains a nonnegative maximum in is constant. Partially
supported by M.U.R.S.T., project \Problemi nonlineari nell'analisi e nelle applicazioni siche, chimiche e biologiche".
1
SMinP: any lower semicontinuous viscosity supersolution of (1) in an open set
that attains a nonpositive minimum in is constant. Our main result is that the SMaxP (resp., SMinP) holds if the function F is lower semicontinuous (resp., upper semicontinuous) and satis es at any x0 2
the following nondegeneracy property with respect to any small vector 6= 0 F(x0; 0; ; I ? ) > 0
8 > 0 ;
(3)
(resp.
F(x0; 0; ; ? I) < 0 8 > 0 ); (4) where is the matrix de ned by ( )ij := ij , and the scaling property that for some '() > 0 F(x; s; p; X) '()F(x; s; p; X); 8 2 (0; 1]; (5) (resp. F(x; s; p; X) '()F(x; s; p; X); 8 2 (0; 1]: (6) The nondegenaracy conditions (3) and (4) are both satis ed, for instance, if F(x0; 0; p; X ? Y ) F(x0; 0; p; X) + (p; TraceY ) (7) for all p 6= 0 and Y 0, with such that lim sup (p; t) > F(x0; 0; p; I); (8) t!+1
which is a much weaker assumption than ellipticity. An even more degenerate operator satis ng (3) and (4) is the 1-Laplacian 1 u := Du D2 uDu: (9) The scaling properties (5) and (6) of the operator F both hold, for instance, for equations of the form c(x)jujk?1u ? a(x)G(Du; D2 u) = 0 (10) if G is positively homogeneous of degree k and c; a 0. Therefore we get the SMaxP and SMinP for any equation of the form (10) where G is the m-Laplace operator, or the 1-Laplace operator, or a Pucci extremal operator, provided k is larger than the degree of homogeneity of G, or c 0, and a > 0. This threshold for k is sharp for the equation
jujk?1u ? m u = 0;
(11) where m is the m-Laplacian with m > 1; which has nonnegative solutions with compact support if and only if k < m ? 1 (see [4] for the usual Laplacian with m = 2 and [10, 22] for the general case), so the SMinP cannot hold for such values of k. The existence of such solutions (in a weak sense di erent from 2
viscosity) for quasilinear equations was studied in depth in the 80s, as well as the properties of the dead core, that is, the set where the solution vanishes. Our SMinP can be viewed as an extension to viscosity solutions of fully nonlinear equations of the results of nonexistence of a dead core in [22, 9, 11] and the references therein. Let us remark that the homogeneity can be avoided if F can be approximated by Fe satis ng the previous assumptions in the sense that
e "s; "p; "X) + o('(")); as " ! 0+; F(x; "s; "p; "X) F(x;
(12)
(resp. ), for p; X bounded. For instance,we get the SMaxP and SMinP for equation (10) when G is the minimal surface operator and k 1, c 0, a > 0 (and the bound on k is optimal, see [11]). Moreover SMaxP or SMinP are satis ed by some very degenerate equations, even of rst order; for instance, the eikonal equation jDuj ? f(x) = 0 (13) veri es SMinP for all f > 0. In Section 4 we give a long list of examples with more comments. We note also that (3) and (5) (resp., (4) and (6)) imply lim sup F(x0; 0; p; X) 0; (resp., (p;X lim)!inf F(x0; 0; p; X) 0 ), (0;0) (p;X )!(0;0)
so if F is de ned and continuous in (x0 ; 0; 0; 0) we have F(x0; 0; 0; 0) 0 (resp., F(x0; 0; 0; 0) 0 ), (14) which is a very natural condition for the SMaxP (resp., SMinP) (cfr. Remark 3 in [17]). We also prove some related results. We begin with a Hopf boundary lemma when the nondegeneracy assumptions (3) and (4) hold with = n(x0 ), the exterior normal to at the point x0, then we show the local propagation of maxima from the interior point x0 2 into the halfspace opposite to if (3) holds for a single vector , and from this we easily deduce the SMaxP. The methods of this paper can be used also for parabolic operators and the Levi equation. More re ned results on the propagation of maxima for linear degenerate operators, such as those of Bony [5] and Stroock and Varadhan [20], are extended to viscosity solutions of Hamilton-Jacobi-Bellman operators (i.e., max or min of a parametrized family of linear degenerate operators) in our paper [2]. Before making more comments on the related literature let us mention that sometimes di erent results are called Strong Maximum Principle. Usually various forms of Minimum Principle are named SMaxP, and the same happens to what we call the Strong Comparison Principle (SComP in short), that is, if a subsolution u and a supersolution v of (1) are such that u v and they coincide at some point, then u v. The reason is that for classical solutions of linear equations the three statements are equivalent. For nonlinear equations, however, this is not necessarily true. The SMaxP and SMinP are still consequences 3
of SComP, because any nonnegative (resp. nonpositive) constant is a supersolution (resp. subsolution) of (1) by (14) and (2), but the converse is false in some cases that satisfy our assumptions. An example is
?u ? jDujm = 0 in a ball with m 2]0; 1[, which satis es the SMinP but not the SMaxP (see the counterexample in [17]), and therefore neither the SComP. Note that this operator is uniformly elliptic, quasilinear, and satis es both necessary conditions in (14), but it is not a Lipschitz function of Du. A degenerate example with Lipschitz dependence on Du is the eikonal equation (13); a counterexample to the SComP in the case f 1, =]0; 1[, is given by the solutions v(x) := 1 ? x and u(x) := 1=2 ? jx ? 1=2j, since they coincide in [1=2; 1] but not in ]0; 1=2[. The Strong Maximum Principle for linear equations and the Strong Comparison Principle for nonlinear equations go back to the work of E. Hopf in the 20s, see also [19] and [14]. For weak solutions of quasilinear equations of mLaplacian type see [22, 9], and for minimal surface type equations [18, 11]. G. Diaz [8] studied the SMaxP and the existence of a dead core for W 2;p solutions of fully nonlinear uniformly elliptic equations. We recall that for these equations uniform ellipticity means the existence of ; > 0 such that TraceY F(x; r; p; X ? Y ) ? F(x; r; p; X) TraceY:
(15)
The theory of viscosity solutions produced a large number of comparison and uniqueness results starting with the work of M.G. Crandall and P.L. Lions for 1st order equations, and of P.L. Lions and R. Jensen for 2nd order equations, see [15], the survey paper [7] and the references therein. These results require regularity assumption on F instead of the non degeneracy and scaling properties (3)-(6). For the uniformly elliptic case Trudinger [21] proved a SComP between Lipschitz sub- and supersolutions and Ca arelli and Cabre [6] a SMinP for continuous supersolutions (for equations independent of u and Du). Kawohl and Kutev [17] studied a slightly weaker form of SMaxP, namely, the property that semicontinuous subsolutions that attain a strictly positive interior maximum are constant. This allows the existence of a dead core for a nonpositive solution, and in fact it holds for the equation (11) with the usual Laplacian for all k > 0. Their ellipticity assumptions is weaker than uniform but stronger than (7) (8), and it does not cover (11) for any m 6= 2. On the other hand they can deduce from their maximum principle a new comparison result. Very recently Giga, Ohnuma and Sato [13] proved a special form of strong maximum principle for the mean curvature ow equation. Our proof of the SMaxP and SminP does not need any previous result and it is rather simple: it combines the classical Hopf's arguments with the very de nition of viscosity subsolution. 2. Main Results. For S RM we will use the following notations: USC(S) = fupper semicontinuous functions u: S ! Rg; 4
LSC(S) = flower semicontinuous functions u: S ! Rg: Note that the equation (1) can be singular for p = 0; the de nition of subsolution(resp. supersolution) we use is weaker then those in [12] and in [7]: a function u 2 USC( ) (resp. LSC( )) is a subsolution (resp. supersolution) of (1) in if, for every ' 2 C 2( ) and x maximum (resp. minimum) point of u ? ' such that D'(x) 6= 0 we have F(x; u(x); D'(x); D2'(x)) 0; (resp. 0): The basic assumptions of this Section are the nondegeneracy condition (3) and (H0) F 2 LSC( R (RN n f0g) S(N)) and is proper; and, given x0 2 ; (H1) for all > 0; there exists a function ': (0; +1) ! (0; +1) such that (5) holds for all x 2 B(x0 ; ); s 2 (?1; 0]; jpj; jjX jj : We will also approximate F with operators Fe in the sense that, (A) for some > 0; (12) holds uniformly for x 2 B(x0 ; ); s 2 (?1; 0); jpj; jjX jj : The following Theorem is an extension of the Hopf Boundary Lemma (see, for istance, [14]) to viscosity subsolution of (1). It is proved under the assumption that the nondegeneracy property (3) of F holds for an exterior normal to at x0 2 @ : Theorem 1 Let O RN be an open set, x0 2 @O; u 2 USC(O Sfx0g) be a viscosity subsolution of (1) in O such that i) u(x0) > u(x); 8x 2 O and u(x0 ) 0; T ii) there exists a ball B := B(y; R) : B O; B @O = fx0g; y and some Assume that either F satis es (H0 ); (H1) and (3) for := jxx0 ? ? 0 yj e
0 ; or there exists F with these properties and satisfying (A): Then, for any w 2 RN n f0g such that w < 0; ? u(x0 ) < 0: (16) lim sup u(x0 + sw) s s!0 Proof. We introduce an auxiliary function v by de ning v(x) := e? R2 ? e? jx?yj2 ; (17) where is a positive constant yet to be determined. Note that v(x0 ) = 0 and ?1 < v(x) < 0; for jx ? yj < R: If F satis es (H0); (H1) and (3), we have F(x0; v(x0); Dv(x0 ); D2v(x0 )) = F(x0; 0; 2 e? R2 ; 2 e? R2 (I ? 2 )) '(2 e? R2 )F(x0; 0; ; I ? 2 ) C; 5
for some C > 0 and for all > 0 : By the lower semicontinuity of F in x0, T there exists2 r > 0 such that for any x 2 X := B(x0 ; r) B(y; R); Dv(x) = 2 e? jx?yj (x ? y) does not vanish and F(x; v(x); Dv(x); D2 v(x)) C > 0: (18) Moreover, by (H1) there exists " > 0 such that 8" "; "v is a strict supersolution of (1) in X as well. Now we claim that "v is a strict supersolution of (1) in X; for " small enough, also in the case that F satis es (12) with Fe as above. In fact e v(x); Dv(x); D2v(x)) + o('(")); (19) F(x; "v(x); "Dv(x); "D2 v(x)) '(")F(x;
for " ! 0+ ; so we use inequality (18) for Fe to get (20) F(x; "v(x); "Dv(x); "D2 v(x)) C2 ; 8" ": Let " " be such that u(x) ? u(x0) "v(x); 8x 2 @X: We claim that u(x) ? u(x0 ) "v(x); 8x 2 X: Suppose by contradiction that there exists x 2 X such that u(x) ? u(x0 ) ? "v(x) = maxX (u ? u(x0 ) ? "v) > 0: Since "v 2 C 1 (RN ); we can use the de nition of viscosity subsolution, the assumptions that u(x0 ) 0 and F is proper to get F(x; "v(x); "Dv(x); "D2 v(x)) F(x; u(x); "Dv(x); "D2 v(x)) 0 that contradicts (18) and proves the claim. Thus, for any w 6= 0 such that w < 0; ? u(x0) "Dv(x ) w = lim sup u(x0 + sw) 0 s s!0 = 2 e? jx0 ?yj2 (x0 ? y) w < 0: In the next Theorem we prove a local propagation of maxima of a viscosity subsolution of (1) from an interior point x0 2 into the half-space opposite to a vector for which (3) holds. Theorem 2 Let RN be an open set and let u 2 USC( ) be a viscosity subsolution of (1) in that attains a nonnegative maximum at x0 2 : Assume F satis es (H0); (H1) and for some w 2 RN n f0g; 0 > 0; for all = jwwj with 0 ; (3) holds for some 0 ; or, alternatively, there exists Fe with these w ; R); with R small properties and satisfying (A): Then any ball B := B(x0 ? R jw j enough, contains xR 6= x0 with u(xR ) = u(x0): Proof. Suppose by contradiction there exists a ball B := B(x0 ? R jwwj ; R)
such that u(x) < u(x0); for all x 2 B: Now we introduce the auxiliary function 6
v de ned in (17). It is not restrictive to suppose that R 0 ; otherwise we can change the centre of the ball along the direction ofT the vector : Using the same arguments of the proof of Theorem 1 in O := fx 2 RN : (x ? x0) < 0g we get that, for T some " > 0; "v is a strict classical supersolution of (1) in X := B(x0 ; r) B(y; R) O and u(x) ? u(x0) "v(x); 8x 2 X: Let us consider in B(x0 ; r) the function (x) := u(x) ? "v(x); and note that has a maximum in x0: Since u is a subsolution of (1) in B(x0 ; r); "v 2 C 1 (RN ); v(x0 ) = 0 u(x0); and F is proper we get F(x0; "v(x0 ); "Dv(x0 ); "D2 v(x0 )) 0; but this is in contradiction with the fact that "v is a strict classical supersolution of (1) in x0: So we conclude. Now we can derive the Strong Maximum Principle for the viscosity subsolution of (1) under the assumptions (H0); (H1) and the following (H2) For all 2 RN n f0g; there exists 0 such that (3) holds. Corollary 1 (Strong Maximum Principle) Let RN be an open set and let u 2 USC( ) be a viscosity subsolution of (1) in that achieves a nonnegative maximum in : Assume that either F satis es (H0) ? (H2); or there exists Fe with these properties and satisfying (A): Then u is a constant. Proof. Let M := max u and de ne the set K := fx 2 : u(x) = M g: Suppose by contradiction that K 6= : Then thereTexist x0 2 @K; y 2 n K and R > 0 such that B(y; R) n K and B(y; R) K = fx0g: By Theorem 2 this ball contains xR 6= x0 such that u(xR ) = M and this is a contradiction. Remark 1 We recall that if w 2 LSC( ) is a viscosity supersolution of (1) in
then u := ?w is a viscosity subsolution of ?F(x; ?u(x); ?Du(x); ?D2u(x)) = 0: So the conditions under which the Strong Minimum Principle holds for F can be easily deduced from the assumptions of Strong Maximum Principle for G(x; r; p; X) := ?F(x; ?r; ?p; ?X): Remark 2 By looking at the proofs, it can be seen that the results of this section remain true if (H1 ) is replaced by the following property of conservation of sign: for any x0 2 ; > 0; there exists > 0 such that for all x 2 B(x0 ; ); s 2 (?1; 0]; jpj kX k : F(x; s; p; X) > 0 =) F(x; s; p; X) > 0; 8 2 (0; ]: (21) 7
Remark 3 By the arguments of the proof of Theorem 3 in [17] it is easy to prove a Strong Comparison Principle when either the subsolution or the supersolution is C 2, for operators satisfying for all x; s; q; Y and for all = 6 0 F(x; s; q + ; Y ? ) > 0 8 > 0 ; and
F(x; r; q; Y ) 0 and F(x; r + s; q + p; Y + X) > 0 =) F(x; r + s; q + p; Y + X); 8 2 (0; 1]:
3. Examples.
In this section we list several equations that satisfy the assumptions of Strong Maximum or Minimum Principles of the previous section. We begin with equations of the form (10) where G : (RN n f0g) S(N) ?! R is positively homogeneous of degree 2 R, i.e., G(p; X) = G(p; X); 8 > 0; p 2 RN n f0g; X 2 S(N): In Examples 1{5 we assume the coecients c; a are continuous and satisfy c 0; a > 0; (22) either c 0 or k and k > 0: (23) Example 1 The m-Laplacian is the quasilinear operator m u := div(jDujm?2Du) = jDujm?2TraceD2 u + (m ? 2)jDujm?4Du D2 uDu; so we de ne G(p; X) := jpjm?2TraceX + (m ? 2)jpjm?4p Xp for all p 2 RN if m 2 and for p 6= 0 if 1 < m < 2. The operator G = m is positively homogeneous of degree = m ? 1, and it coincides with the usual Laplacian if m = 2. It is easy to see that (7) is satis ed, thus the SMaxP and the SMinP hold for the equation (10). This result was proved by Vazquez [22] for weak solutions in the sense of distributions, the threshold k m ? 1 is known to be sharp by the results in [4, 10, 22, 9]. Example 2 For the 1-Laplacian (9) we de ne G(p; X) := p Xp that is homogeneous of degree = 3 and satis es (3) and (4) because G(p; I ? p p) = ?G(p; p p ? I) = jpj2 ? jpj4 : We recall that the equation 1 u = 0 is satis ed in the viscosity sense by functions with prescribed boundary data and minimizing the 1-norm of the gradient, it was studied by Jensen [16] who proved a comparison and uniqueness theorem for the Dirichlet problem. 8
Example 3 The Pucci minimal and maximal operators are, for xed > 0; Gm (X) = TraceX + (1 ? N) minfeigenvalues of X g; (24) GM (X) = TraceX + (1 ? N) maxfeigenvalues of X g; (25) see [14]. They are fully nonlinear, uniformly elliptic and positive homogeneous of degree 1. In this case the SMinP for continuous supersolutions falls also within the theory of uniformly elliptic operators in [6] if c 0:
Example 4 The operator
G(p; X) = Trace((I ? pjp j2p )X)
(26)
arises in the mean curvature ow equation, see [12, 7, 13]. Note that is degenerate elliptic, unde ned for p = 0; and cannot be put in divergence form. Moreover it is homogeneous of degree = 1 and G(; I ? ) = ?G(; ? I) = N ? 1; 8 2 R: Therefore the SMaxP and the SminP hold for equation (10) with this G:
Example 5 The minimal surface operator is G(p; X) = TraceX(1 + jpj2)?1=2 ? p Xp(1 + jpj2)?3=2:
(27)
In this case the equation (10) satis es the assumption (A) because c"k jsjk?1s ? aG("p; "X) = = c"k jsjk?1s ? a"TraceX + O("3 ); as " ! 0+ ; for p; X in a bounded set. Since the operator
e s; ; X) = c(x)jsjk?1s ? a(x)TraceX F(x;
satis es (H0); (H1); (H2) under the assumptions (22), (23) with = 1 (see example 1), we obtain the SMaxP and SMinP for (10) when G is the minimal surface operator under the same assumptions. In particular we cover the equation of capillarity c(x)u(1 + jDuj2)3=2 ? u(1 + jDuj2) + Du D2 uDu = 0:
Example 6 The rst order equation c(x)jujk?1u + a(x)jDujm ? f(x) = 0
(28)
is related to the eikonal equation and to deterministic optimal control theory, see [1], where c 0; k = 1 and a; f > 0: The SMaxP is false because, for istance, 9
?jxj is a viscosity solution of jDuj? 1 = 0; in RN : The nondegeneracy condition (4) for the SMinP is satis ed for
f(x) ; jpj < 0 := inf
a(x) if 0 > 0: On the other hand the scaling property (4) is satis ed for k m; or c 0; thus the SMinP holds under these conditions. Example 7 The uniformly elliptic equation c(x)jujk?1u ? u + b(x)jDujm = 0 (29) arises in stochastic optimal control theory (with c 0 or k = 1). For c 0 the scaling property (5) for the SMaxP is satis ed if either b 0 and m 1; or b 0 and m 1; (30) and either c 0 or k 1; (31) while the scaling property (6) for the SMminP is satis ed if (31) holds and either b 0 and m 1; or b 0 and m 1: (32) The condition m 1 in (30) is sharp if b ?1 by an example in [17]. Uniqueness and nonuniqueness of solutions of the Dirichlet problem for the equation (29) with nonzero right-hand side and c 0; b 1; are studied in [3]. Example 8 A very degenerate equation satisfying the SMaxP is ?uxx + juy j + 1 = 0; in R2; that has a stochastic control interpretation. It easy to check that the scaling property (5) holds with '(") = " and the nondegeneracy property (3) with
0 = 0: Acknowledgements. We wish to thank Bernd Kawohl for the preprints [17] and [13] and for useful discussions.
References
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