A comparison principle for vector valued minimizers of semilinear elliptic energy, with application to dead cores
aa r X i v : . [ m a t h . A P ] F e b A COMPARISON PRINCIPLE FOR MINIMIZERS OF SEMILINEARELLIPTIC ENERGY, WITH APPLICATION TO DEAD CORES
PANAYOTIS SMYRNELIS
Abstract.
We establish a general comparison principle for vector valued minimizers of afunctional, associated when the potential is smooth, to elliptic gradient systems. As a con-sequence, we give a sufficient condition for the existence of dead core regions, where theminimizer is equal to one of the minima of the potential. In the scalar case, this conditionis also necessary, due to the availability of a strong maximum principle. Our results providevariational versions of several classical theorems well-known for solutions of scalar semilinearelliptic PDE.
MSC2020 : Primary 35B51; 35J50; Secondary 35B50.
Keywords : comparison principle,dead core, minimizer, semilinear elliptic energy. Introduction
The scope of this paper is to establish a general comparison principle providing accurateupper bounds for the modulus of vector valued minimizers of the energy functional(1.1) E ω ( v ) := ˆ ω h |∇ v ( x ) | + W ( v ( x )) i d x, v ∈ W , ( ω ; R m ) , ω ⊂ R n , n, m ≥ , where W : R m → [0 , ∞ ) is a nonnegative, lower semicontinuous potential. Concerning thebehaviour of W in a neighbourhood of one of its zero, supposed to be located at the origin,we shall only make two basic assumptions (cf. H below). Namely, that in a neighbourhoodof 0: • W rad ( | u | ) ≤ W ( u ), where W rad : [0 , q ] → [0 , ∞ ) is a nondecreasing, lower semicontin-uous function , • and u W ( u ) − W rad ( | u | ) is nondecreasing on the rays emanating from the origin.Thus our result applies to a large class of potentials, including for instance the interestingparticular case of the characteristic function of R m \ { } . Phase transition problems involvingnonsmooth potentials are often considered in the literature. We mention in particular thework [4] on free boundaries; the density estimates obtained in [6] (resp. [1]) in the scalar(resp. vector) case; the properties of minimal surfaces and minimizers studied in [13]; theheteroclinic orbit problem examined in [14]; the structure of minimizers described in [7] in theone dimensional case n = m = 1. Although the potential W may be a very rough function,we recall that the minimizers of (1.1) are continuous maps (cf. Lemma 4.2).Comparison principles are useful in phase transition problems, to study the convergence ofa solution to the minima of the potential. The most typical situation occurs (cf. [2, Lemma We shall see in Theorem 2.2 that the upper bound obtained for the modulus of the minimizer, onlydepends on the profile of the function W rad . .4]) when W : R m → [0 , ∞ ) is a smooth potential such that(1.2) W ≥ , W (0) = 0 , and ∇ W ( u ) · u ≥ c | u | , holds for | u | ≤ q (i.e. the minimum 0 is nondegenerate), and u ∈ C (Ω; R m ) is a smooth solution to ∆ u ( x ) = ∇ W ( u ( x )) in Ω ⊂ R n , such that | u | ≤ q holds in Ω. Then, in view of the inequality(1.3) ∆ | u | ( x ) ≥ ∇ W ( u ( x )) · u ( x ) ≥ c | u ( x ) | , ∀ x ∈ Ω , the maximum principle implies that(1.4) | u ( x ) | ≤ Φ( x ) , ∀ x ∈ Ω , where Φ : Ω → R is the solution of the problem (1.5) ∆Φ = 2 c Φ in Ω , and Φ = q on ∂ Ω . On the other hand, we would also like to recall a classical result (cf. [10, Theorem 7.2.]), onthe existence of dead core solutions in the scalar case. Let W ∈ C ([0 , q ]; R ) be a potentialdefined on the interval [0 , q ], and assume also that(1.6a) W is convex , (1.6b) W (0) = W ′ (0) = 0 , and W ′ > , q ] , (1.6c) ˆ q d s p W ( s ) < ∞ . Then, in every ball B R := { x ∈ R n : | x | < R } , the equation(1.7) ∆ u ( x ) = W ′ ( u ) , x ∈ B R , admits a nonnegative dead core solution , that is, a solution of (1.7) satisfying(1.8a) u ≡ ω such that ω ⊂ B R , (1.8b) u > B R \ ω. We refer to [10, 11, 12] and the references therein, for general statements of maximum andcomparison principles, as well as for the theory of dead core solutions.As a consequence of our comparison principle, we shall give in the vector case, a sufficientcondition for the existence of dead core regions where the minimizer is identically equal toone of the zeros of the potential. We shall see in Theorem 2.4 that the conditions • W rad ( | u | ) > ⇔ u = 0 in a neighbourhood of 0, • and ´ q s √ W rad ( s ) < ∞ ,imply the existence of dead core regions. Note that the convexity of W assumed in (1.6a) isnot needed. In particular, whenever the function W rad is discontinuous at 0, dead core regionsappear. We refer again to [2, Appendix A] for the decay properties of Φ. The investigation of dead cores was initiated in [5] (cf. also [15]). . Main results
Now, we shall state more precisely our assumptions and main results. Let us assume that B q ⊂ R m is the open ball of radius q > W : B q → [0 , ∞ ) isa potential such that H W ≥ W (0) = 0, H W is lower semicontinuous and bounded on B q , H W ( u ) = W rad ( | u | ) + W ( u ), with W rad : [0 , q ] → [0 , ∞ ) a nondecreasing, lower semi-continuous function, and W : B q → [0 , ∞ ) a function such that W ( rξ ) ≤ W ( sξ )holds for every 0 ≤ r ≤ s ≤ q , and every unit vector ξ ∈ R m .Our comparison principle applies to maps u ∈ W , (Ω; R m ) defined in an open set Ω ⊂ R m ,such that(2.1) k u k L ∞ (Ω; R m ) ≤ q, and u is a local minimizer of the energy functional (1.1), for perturbations satisfying (2.1).That is, for every bounded open set ω with Lipschitz boundary, such that ω ⊂ Ω, and everyperturbation v = u + ξ such that ξ ∈ W , ( ω ; R m ) and k v k L ∞ ( ω ; R m ) ≤ q , we assume that(2.2) E ω ( u ) ≤ E ω ( v ) . For instance, if Ω is a smooth domain, and if we extend W on the whole space R m by setting˜ W ( u ) = ( W ( u ) when | u | ≤ q,W ( qu | u | ) when | u | ≥ q, one can check that assumptions (2.1) and (2.2) hold, for every minimizer u of˜ E Ω ( v ) := ˆ Ω h |∇ v ( x ) | + ˜ W ( v ( x )) i d x, in the class of maps v ∈ W , (Ω; R m ) satifying the boundary condition v = φ on ∂ Ω , with φ ∈ W , (Ω; R m ) , and k φ k L ∞ (Ω; R m ) ≤ q. In the following Theorem 2.2, we shall establish an upper bound for the modulus of the localminimizer u . Our comparison function Ψ R is defined in Proposition 2.1.
We assume that W rad : [0 , q ] → [0 , ∞ ) is a bounded, nondecreasing, lowersemicontinuous function. Let B R ⊂ R n be the open ball of radius R > centered at the origin,let ˜ W rad ( r ) = ( W rad ( | r | ) when | r | ≤ q,W rad ( q ) when | r | ≥ q, and let J B R ( h ) := ˆ B R h |∇ h ( x ) | + ˜ W rad ( h ( x )) i d x. Then, there exists a unique minimizer Ψ R (resp. Ψ R ) of J B R in the class A R := { h ∈ W , (Ω; R ) : h = q on ∂B R } , satisfying the following properties: (i) Ψ R (resp. Ψ R ) is radial (i.e. Ψ R ( x ) = Ψ R, rad ( | x | ) , Ψ R ( x ) = Ψ R, rad ( | x | ) , ∀ x ∈ B R ),and continuous on B R , ii) the function Ψ R, rad (resp. Ψ R, rad ) is nondecreasing on the interval [0 , R ] , (iii) if ψ R is another minimizer of J B R in the class A R , then we have Ψ R ≤ ψ R ≤ Ψ R in B R . Remark 1.
In general, the minimizer of J B R in the class A R is not unique. For instance, letus assume that n = 2, and ˜ W rad ( r ) = ( r = 0 , r > . Then, a trivial computation shows that when R = R := √ e q , J B R admits exactly tworadial minimizers in the class A R , namely Ψ R = q , andΨ R ( x ) = ( | x | ≤ √ q, q ln( r √ q ) if √ q ≤ | x | ≤ R . On the other hand, when
R > R (resp. R < R ), J B R admits only one radial minimizer inthe class A R , namely Ψ R ( x ) = Ψ R ( x ) = ( | x | ≤ R √ e , q ln( √ e | x | R ) if R √ e ≤ | x | ≤ R, (resp. Ψ R = Ψ R = q ). Thus, Proposition 2.1 implies that this is the only minimizer of J B R in the class A R .We refer to Lemma 3.3 below, for further properties of the comparison functions Ψ R andΨ R . In particular, we study their dependence on R , and we establish that the minimizer of J B R in the class A R is unique, for every R ∈ (0 , ∞ ) \ D , where D is a countable subset of(0 , ∞ ).Next, we state the comparison principle: Theorem 2.2.
We assume that hypotheses H - H hold, and that the map u ∈ W , (Ω; R m ) satisfies (2.1) and (2.2) . Let Ψ R be the radial minimizer defined in Proposition 2.1. Then,for every closed ball B R ( x ) contained in Ω , we have (2.3) | u ( x ) | ≤ Ψ R ( x − x ) , on B R ( x ) . Remark 2.
In the case where m = 1, W ≡
0, and W ( u ) = W rad ( | u | ), the bound provided byTheorem 2.2 is optimal, since the function Ψ R : B R → [0 , q ] is a minimizer of (1.1) satisfying(2.1). Remark 3.
Theorem 2.2 covers the case where the profile of W is not uniform along the raysemanating from the origin. For instance, if we take W ( u ) = | u | α ( u/ | u | ) in the unit ball B , with α : S m − → (0 , ∞ ) a continuous function, then setting α := max S m − α , and α := min S m − α ,we can apply Theorem 2.2 in the ball of radius q := e − α − , with W rad ( s ) = s α , ∀ s ∈ [0 , q ],since the functions [0 , q ] W ( sξ ) − W rad ( s ) are nondecreasing, for every ξ ∈ S m − . Remark 4.
Let W rad : [0 , q ] → [0 , ∞ ) (resp. V rad : [0 , q ] → [0 , ∞ )) be two bounded,nondecreasing, lower semicontinuous functions, and let Ψ R (resp. Φ R ) be the comparisonfunctions provided by Proposition 2.1. If moreover we assume that the function V rad − W rad s nondecreasing on [0 , q ], then an application of Theorem 2.2 with u = Φ R , and W ( u ) = V rad ( | u | ), shows that Φ R ≤ Ψ R holds on B R . Thus the optimal comparison function Ψ R provided by Proposition 2.1, is obtained by choosing the greatest function W rad satisfying H . This also explains why the profile of the comparison function Ψ R , corresponding to W rad ( s ) = s α ( α > α decreases.We also have the following useful version of Theorem 2.2 at the boundary of Ω: Theorem 2.3.
We assume that hypotheses H - H hold. Let Ω ⊂ R n be a bounded, open setwith Lipschitz boundary, and let u ∈ W , (Ω; R m ) be a map satisfying (2.1) , and (2.2) forevery v ∈ W , (Ω; R m ) such that k v k L ∞ (Ω; R m ) ≤ q . Then, if the ball B R ( x ) intersects ∂ Ω ,and if u = 0 on B R ( x ) ∩ ∂ Ω , we have (2.4) | u ( x ) | ≤ Ψ R ( x − x ) , on B R ( x ) ∩ Ω . In Lemma 3.4 below, we determine the conditions implying the existence of dead coreregions for the comparison function Ψ R . As a consequence of Theorem 2.2 and Lemma 3.4,we give in Theorem 2.4 a sufficient condition for the existence of dead core regions , in thecase of vector minimizers: Theorem 2.4.
In addition to hypotheses H - H , we assume that H W rad ( s ) > , ∀ s ∈ (0 , q ] , and I q := ´ q s √ W rad ( s ) < ∞ .Then, if the map u ∈ W , (Ω; R m ) satisfies (2.1) and (2.2) , we have u ( x ) = 0 , provided that d ( x, ∂ Ω) ≥ (4 n + √ I q . In the case of scalar minimizer of (1.1), hypothesis H is also a necessary condition for theexistence of dead core regions. Indeed, we have the following version of the strong maximumprinciple for scalar minimizers of (1.1): Proposition 2.5.
Let m = 1 , and let ω be a bounded open set with Lipschitz boundary, suchthat ω ⊂ Ω . We assume that hypotheses H - H hold for W ( u ) = W rad ( | u | ) ( W ≡ ), andmoreover that (2.5) ˆ q d s p W rad ( s ) = ∞ . Then, if the function u ∈ W , (Ω; R ) satisfies (2.1) , (2.2) , and u > on ∂ω , we also have u ( x ) > , ∀ x ∈ ω . Proposition 2.5 may be compared to the classical strong maximum principle for nonnegativesolutions of ∆ u ( x ) = W ′ ( u ) (cf. [10, Theorem 1.1]), holding under assumptions (1.6a), (1.6b),and (2.5).The plan of the next sections is as follows. In section 3 we give the proofs of Propositions2.1 and 2.5, as well as Theorems 2.2 and 2.3. In section 4 we recall that the minimizers of(1.1) are continuous, and we also establish the validity of Pohozaev identity for minimizersof (1.1). This identity is crucial in the proof of Lemma 3.4. As a consequence of Theorem 2.3 and Lemma 3.4, we also deduce the existence of dead core regions atthe boundary of Ω. . Proofs of Propositions 2.1 and 2.5, and Theorems 2.2 and 2.3
We first establish the existence of a radial minimizer of J B R in the class A R . Lemma 3.1.
Under the assumptions of Proposition 2.1: • There exists a minimizer ψ R of J B R in the class A R , which is radial (i.e. ψ R ( x ) = ψ R, rad ( | x | ) , ∀ x ∈ B R ), and continuous on B R . • For such a radial minimizer, the function ψ R, rad is nondecreasing on the interval [0 , R ] .Proof. Let ˜ ψ be a minimizer of J B R in the class A R := { h ∈ W , (Ω; R ) : h = q on ∂B R } .We first notice that 0 ≤ ˜ ψ ≤ q , since otherwise the competitor min( ˜ ψ + , q ) ∈ A R has lessenergy. We also know that ˜ ψ is continuous in B R (cf. Lemma 4.2). Starting from ˜ ψ , we canconstruct ˜ ψ (1)0 ( x ) = ˜ ψ ( | x | , x , . . . , x n ) , which is another minimizer of J B R in A R , invariant by the reflection ( x , x , . . . , x n ) ( − x , x , . . . , x n ). Indeed, we have J B R ∩{ x > } ( ˜ ψ ) = J B R ∩{ x < } ( ˜ ψ ), since otherwise either thecompetitor x ˜ ψ ( −| x | , x , . . . , x n ) or the competitor ˜ ψ (1)0 has less energy than ˜ ψ . Similarly,we can construct a minimizer ˜ ψ (1)1 ( x ) = ˜ ψ ( | x | , | x | , . . . , x n ) , which coincides with ˜ ψ on { x ∈ B R : x > , x > } , and is invariant by the reflections( x , x , . . . , x n ) ( − x , x , . . . , x n ) and ( x , x , . . . , x n ) ( x , − x , . . . , x n ). By repeatingthis process, we obtain for every k ≥
2, a minimizer ˜ ψ (1) k , which coincides with ˜ ψ on { x ∈ B R : 0 < x < tan( π/ k ) x } , and is invariant by the dihedral group D k generated by thereflections with respect to the hyperlanes x = 0, and x = tan( π/ k ) x . It is clear that k ˜ ψ (1) k k W , ( B R ; R ) is uniformly bounded, thus (up to subsequence) we have˜ ψ (1) k ⇀ ˜ ψ (1) ∞ in W , ( B R ; R ) , and ˜ ψ (1) k → ˜ ψ (1) ∞ a.e. in B R . By the weakly lower continuity of the L norm, it follows that(3.1a) ˆ B R |∇ ˜ ψ (1) ∞ | ≤ lim inf k →∞ ˆ B R |∇ ˜ ψ (1) k | , while by Fatou’s lemma and the lower semicontinuity of ˜ W rad , we get(3.1b) ˆ B R ˜ W rad ( ˜ ψ (1) ∞ ) ≤ ˆ B R lim inf k →∞ ˜ W rad ( ˜ ψ (1) k ) ≤ lim inf k →∞ ˆ B R ˜ W rad ( ˜ ψ (1) k ) . As a consequence, ˜ ψ (1) ∞ is another minimizer of J B R in A R . By construction, given x ∈ B R ,such that l := p x + x , we have | ˜ ψ (1) k ( x , x , x , . . . , x n ) − ˜ ψ ( l , , x , . . . , x n ) | ≤ sup n | ˜ ψ ( z , z , x , . . . , x n ) − ˜ ψ ( l , , x , . . . , x n ) | : q ( z − l ) + z ≤ πl k o . Therefore, letting k → ∞ , it follows that ˜ ψ (1) ∞ ( x , x , x , . . . , x n ) = ˜ ψ ( p x + x , , x , . . . , x n ). ext, we proceed by induction, and starting from ˜ ψ (1) ∞ , we consider for every k ≥ ψ (2) k , which coincides with ˜ ψ (1) ∞ on { x ∈ B R : 0 < x < tan( π/ k ) x } , andis invariant by the dihedral group D k generated by the reflections with respect to thehyperlanes x = 0, and x = tan( π/ k ) x . As previously ˜ ψ (2) ∞ := lim k →∞ ˜ ψ (2) k is a mini-mizer of J B R in A R , such that ˜ ψ (2) ∞ ( x , x , x , x , . . . , x n ) = ˜ ψ (1) ∞ ( p x + x , x , , x , . . . , x n ) =˜ ψ ( p x + x + x , , , x , . . . , x n ). The process terminates after a finite number of exactly n − ψ R := ˜ ψ ( n − ∞ of J B R in A R , such that ψ R ( x , x , . . . , x n ) =˜ ψ ( p x + . . . + x n , , . . . , ψ R of J B R in the class A R , we can easily see by contra-diction that the function ψ R, rad is nondecreasing on the interval [0 , R ]. Indeed, assume that ψ R, rad ( r ) > ψ R, rad ( s ) holds for some 0 ≤ r < s ≤ R . Then, the competitor(3.2) ζ ( x ) := ( ψ R ( x ) if s ≤ | x | ≤ R, min( ψ R ( x ) , ψ R, rad ( s )) if | x | ≤ s, has less energy than ψ R , which is impossible. Finally, in view of the monotonicity of ψ R, rad ,the continuity of ψ R up to B R is clear. (cid:3) In the next Lemma, we consider a perturbation of the functional J B R for λ ∈ (0 ,
1) (cf.(3.3)). We shall use the corresponding comparison functions ψ λR provided by Lemma 3.1, toobtain an upper bound for the modulus of the local minimizer u considered in Theorem 2.2. Lemma 3.2.
We assume that hypotheses H - H hold, and that the map u ∈ W , (Ω; R m ) satisfies (2.1) and (2.2) . Given λ ∈ (0 , , let (3.3) J λB R ( h ) := ˆ B R h |∇ h ( x ) | + λ ˜ W rad ( h ( x )) i d x, and consider a radial minimizer ψ λR of J λB R in the class A R , provided by Lemma 3.1. Then,for every closed ball B R ( x ) contained in Ω , we have (3.4) | u ( x ) | ≤ ψ λR ( x − x ) , on B R ( x ) . Proof.
Without loss of generality, we assume that x = 0. We recall that u is continuous onΩ (cf. Lemma 4.2), and consider on the open set Ω := { x ∈ Ω : u ( x ) = 0 } the polar form:(3.5) u ( x ) = ρ ( x ) n ( x ) , with ρ ( x ) := | u ( x ) | , n ( x ) := u ( x ) | u ( x ) | . An easy computation shows that(3.6) |∇ u ( x ) | = |∇ ρ ( x ) | + | ρ ( x ) | |∇ n ( x ) | on Ω . Next, we define on B R the comparison map:(3.7) ˜ u ( x ) = ( u ( x ) when ρ ( x ) ≤ ψ λR ( x ) ψ λR ( x ) n when ρ ( x ) > ψ λR ( x ) , where ψ λR is a radial minimizer of J λB R in the class A R , provided by Lemma 3.1. It is obviousthat u = ˜ u on ∂B R . One can also check that | ˜ u | ≤ | u | holds on B R , and ˜ u is continuouson B R . Our claim is that ˜ u ∈ W , ( B R ; R m ). Let U := { x ∈ B R : ψ R ( x ) > } . We otice that either U = B R , or U = { x : R ′ < | x | < R } , for some R ′ ∈ (0 , R ). Now, given x ∈ Ω ∩ U , it is clear that ˜ u ( x ) = min( ψ R ( x ) ,ρ ( x )) ρ ( x ) u ( x ) holds in an open neighbourhood V x of x , where ρ ( x ) ≥ ǫ >
0. As a consequence, ˜ u ∈ W , ( V x ; R m ), as a product of mapsbelonging to W , ( V x ; R m ) ∩ L ∞ ( V x ; R m ). Otherwise, if u vanishes for some x ∈ U , we have˜ u = u in a neighbourhood of x . This proves that ˜ u ∈ W , ( U ; R m ). Moreover, setting˜ ρ ( x ) = | ˜ u ( x ) | = min( ψ R ( x ) , ρ ( x )), we compute ˆ U |∇ ˜ u | = ˆ Ω ∩ U ( |∇ ˜ ρ | + ˜ ρ |∇ n | ) ≤ ˆ Ω ∩ U ( |∇ ρ | + ρ |∇ n | ) + ˆ B R |∇ ψ R | = ˆ U |∇ u | + ˆ B R |∇ ψ R | < ∞ , thus ˜ u ∈ W , ( U ; R m ). Finally, in the case where U = B R i.e. U = { x : R ′ < | x | < R } , wehave ˜ u ≡ B R ′ . This proves our claim that ˜ u ∈ W , ( B R ; R m ).At this stage, we utilize the minimality of u to deduce that E B R ( u ) = E B R ∩{ ρ>ψ λR } ( u ) + E B R ∩{ ρ ≤ ψ λR } ( u ) ≤ E B R ∩{ ρ>ψ λR } (˜ u ) + E B R ∩{ ρ ≤ ψ λR } ( u ) = E B R (˜ u ) , or equivalently(3.8) E B R ∩{ ρ>ψ λR } ( u ) = ˆ B R ∩{ ρ>ψ λR } h |∇ ρ | | ρ | |∇ n | W rad ( ρ ) + W ( ρ n ) i ≤ ˆ B R ∩{ ρ>ψ λR } h |∇ ψ λR | | ψ λR | |∇ n | W rad ( ψ λR ) + W ( ψ λR n ) i = E B R ∩{ ρ>ψ λR } (˜ u ) . Similarly, by the minimality of ψ λR , and since ( ρ − ψ λR ) + ∈ W , ( B R ), it follows that J λB R ( ψ λR ) = J λB R ∩{ ρ>ψ λR } ( ψ λR ) + J λB R ∩{ ρ ≤ ψ λR } ( ψ λR ) ≤ J λB R ∩{ ρ>ψ λR } ( ρ ) + J λB R ∩{ ρ ≤ ψ λR } ( ψ λR ) = J λB R ( ψ λR + ( ρ − ψ λR ) + ) , or equivalently J λB R ∩{ ρ>ψ λR } ( ψ λR ) = ˆ B R ∩{ ρ>ψ λR } h |∇ ψ λR | λW rad ( ψ λR ) i (3.9) ≤ ˆ B R ∩{ ρ>ψ λR } h |∇ ρ | λW rad ( ρ ) i = J λB R ∩{ ρ>ψ λR } ( ρ ) . Gathering the previous results from (3.8) and (3.9), we conclude that(3.10a) I + I + I + I ≤ I := ˆ B R ∩{ ρ>ψ λR } h |∇ ρ | λW rad ( ρ ) − |∇ ψ λR | − λW rad ( ψ λR ) i ≥ , (3.10c) I := ˆ B R ∩{ ρ>ψ λR } ( | ρ | − | ψ λR | )2 |∇ n | ≥ , I := ˆ B R ∩{ ρ>ψ λR } ( W ( ρ n ) − W ( ψ λR n )) ≥ H ) , (3.10e) I := (1 − λ ) ˆ B R ∩{ ρ>ψ λR } ( W rad ( ρ ) − W rad ( ψ λR )) ≥ H ) . Consequently, we have(3.11a) I := ˆ B R ∩{ ρ>ψ λR } ( | ρ | − | ψ λR | )2 |∇ n | = 0 , (3.11b) I := (1 − λ ) ˆ B R ∩{ ρ>ψ λR } ( W rad ( ρ ) − W rad ( ψ λR )) = 0 . Now, let us assume by contradiction that the open set V := B R ∩ { ρ > ψ λR } is nonempty,and let ˜ V be a nonempty connected component of V . It follows from (3.11a) that ∇ n ≡ V , thus we have n ≡ n in ˜ V , for a unit vector n ∈ R m , as well as u = ρ n in ˜ V .Our next claim is that(3.12) W rad ( ψ λR ) = W rad ( ρ ) ≡ Const . in ˜ V .
Indeed, let us first assume by contradiction that W rad ( ψ λR ( x )) + 2 ǫ < W rad ( ρ ( x )) holdsfor some x ∈ ˜ V , and ǫ >
0. Then, by the lower semicontinuity of W rad ( ρ ), we have W rad ( ψ λR ( x )) + ǫ < W rad ( ρ ) in an open neighbourhood ˜ V x ⊂ ˜ V of x . On the other hand,since W rad is nondecreasing on [0 , q ], while | x | 7→ ψ λR ( | x | ) is nondecreasing on [0 , R ], it isclear that W rad ( ψ λR ) ≤ W rad ( ψ λR ( x )) holds on the set S := { x ∈ ˜ V x : | x | ≤ | x |} (whichhas positive Lebesgue measure). As a consequence, we have W rad ( ρ ) − W rad ( ψ λR )) ≥ ǫ > S , in contradiction with (3.11b). This proves that W rad ( ψ λR ) ≡ W rad ( ρ ) in ˜ V . Next,we assume by contradiction that W rad ( ψ λR ( x )) < W rad ( ψ λR ( x )) holds for some x , x ∈ ˜ V .Let q := ψ λR ( x ), q := ψ λR ( x ) (with q < q , since W rad is nondecreasing), and let s := max { r ∈ [0 , q ] : W rad ( r ) = q } . We notice that s ∈ [ q , q ], thus in view of thecontinuity of ψ λR , there exists x ∈ ˜ V such that ψ λR ( x ) = s . By definition of s , we have W rad ( ψ λR ( x )) < W rad ( ρ ( x )), which is a contradiction. This establishes (3.12).To prove the bound(3.13) | u ( x ) | ≤ ψ λR ( x ) , on B R , it remains to show that(3.14) ∆ ψ λR ≤ , and ∆ ρ ≥ V .
Indeed, since the boundary condition ρ − ψ λR ≤ ∂ ˜ V , the maximum principlewould give that ρ ≤ ψ λR holds in ˜ V , in contradiction with our assumption that ˜ V is nonempty.To check (3.14), we utilize the minimality of u and ψ λR , as well as (3.12). Given x ∈ ˜ V , let s := ψ λR ( x ), t := ρ ( x ), and 2 κ := t − s >
0. In view of (3.14) is is clear that W rad isconstant on [ s, t ]. Let also ˜ V x ⊂ ˜ V be an open neighbourhood of x , such that ψ λR ≤ s + κ nd ρ ≥ t − κ hold in ˜ V x . Now, given φ ∈ C ( R n ; R ), such that supp φ ⊂ ˜ V x , and 0 ≤ φ ≤ κ ,we have for every ǫ ∈ (0 , J λB R ( ψ λR + ǫφ ) − J λB R ( ψ λR ) ǫ = ˆ B R |∇ ψ λR + ǫ ∇ φ | − |∇ ψ λR | ǫ ≥ , (3.15b) E B R ( u − ǫφ n ) − E λB R ( u ) ǫ = ˆ B R |∇ ρ − ǫ ∇ φ | − |∇ ρ | ǫ + ˆ B R ( W (( ρ − ǫφ ) n ) − W ( ρ n )) ≥ . Finally, since W (( ρ − ǫφ ) n ) ≤ W ( ρ n ), we let ǫ → ˆ B R ∇ ψ λR · ∇ φ ≥ , i.e. ψ λR is superharmonic in ˜ V x ,(3.16b) ˆ B R ∇ ρ · ∇ φ ≤ , i.e. ρ is subharmonic in ˜ V x .This establishes (3.14), and completes the proof of (3.13). (cid:3) Now, we are able to complete the proofs of Proposition 2.1, as well as Theorems 2.2 and2.3.
Proof of Proposition 2.1.
For every λ >
0, let ψ λR be a radial minimizer of J λB R in the class A R , provided by Lemma 3.1. We first notice that ψ λR is uniformly bounded in W , ( B R ),provided that λ remains bounded. Thus, as λ → λ <
1, we have (up to subsequence):(3.17) ψ λR ⇀ ζ in W , ( B R ; R ) , and ψ λR → ζ a.e. in B R . In addition, by the weakly lower continuity of the L norm, it follows that(3.18a) ˆ B R |∇ ζ | ≤ lim inf λ → − ˆ B R |∇ ψ λR | , while by Fatou’s lemma and the lower semicontinuity of W rad , we get(3.18b) ˆ B R W rad ( ζ ) ≤ ˆ B R lim inf λ → − W rad ( ψ λR ) ≤ lim inf λ → − ˆ B R λW rad ( ψ λR ) . Finally, in view of the minimality of ψ λR , we deduce that(3.19a) J λB R ( ψ R ) ≥ J λB R ( ψ λR ) , (3.19b) J B R ( ψ R ) = lim inf λ → − J λB R ( ψ R ) ≥ lim inf λ → − J λB R ( ψ λR ) ≥ J B R ( ζ ) . That is, ζ is a minimizer of J B R in the class A R . Moreover, by construction ζ is radial, andnondecreasing as a function of | x | . It remains to establish that ζ also satisfies property (iii) ofProposition 2.1. Indeed, if ˜ ψ is another minimizer of J B R in A R , we have in view of Lemma3.2 applied to ˜ ψ instead of u :(3.20) ˜ ψ ( x ) ≤ ψ λR ( x ) , ∀ x ∈ B R , ∀ λ ∈ (0 , ⇒ ˜ ψ ( x ) ≤ ζ ( x ) , ∀ x ∈ B R . Therefore, ζ is the minimizer Ψ R described in Proposition 2.1, which is uniquely determinedby property (iii). imilarly, by taking the limit of the minimizers ψ λR , as λ → λ >
1, we obtain a radialminimizer ζ of J B R in the class A R . By construction, we have (up to subsequence):(3.21) lim λ → + ψ λR = ζ a.e. in B R . It remains to establish that ζ also satisfies property (iii) of Proposition 2.1. To see this, let˜ ψ be another minimizer of J B R in A R . As in the proof of Lemma 3.1, we can construct forevery unit vector ν ∈ R n , a radial minimizer ˜ ψ ν of J B R in A R , such that ˜ ψ ( sν ) = ˜ ψ ν ( sν ), ∀ s ∈ [0 , R ]. Next, we apply Lemma 3.2 with the radial comparison function ˜ ψ ν , and thepotential W ( u ) = λW rad ( | u | ) , W ( u ) = ( λ − W rad ( | u | ) , λ > , to u = ψ λR , ( λ > ψ λR ( x ) ≤ ˜ ψ ν ( x ) , ∀ x ∈ B R , ∀ λ > , ∀ ν ∈ S n − ⇒ ζ ( x ) ≤ ˜ ψ ( x ) , ∀ x ∈ B R . Therefore, ζ is the minimizer Ψ R described in Proposition 2.1, which is uniquely determinedby property (iii). (cid:3) Proof of Theorem 2.2.
The desired bound (2.4) follows by letting λ → λ <
1) in(3.4), and using (3.17). (cid:3)
Proof of Theorem 2.3.
We consider on B R ∩ Ω the comparison map:(3.23) ˜ u ( x ) = ( u ( x ) when ρ ( x ) ≤ ψ λR ( x ) ψ λR ( x ) n when ρ ( x ) > ψ λR ( x ) , and reproduce the arguments in the proof of Theorem 2.2. (cid:3) From Lemma 3.2 and Proposition 2.1, we also deduce the following useful result:
Lemma 3.3.
For every
R > , and λ > , we consider the functional J λB R ( h ) := ˆ B R h |∇ h ( x ) | + λ ˜ W rad ( h ( x )) i d x, and the corresponding comparison functions Ψ λR and Ψ λR provided by Proposition 2.1. Then,we have (a) Ψ λR ( x ) = Ψ κ λ Rκ ( xκ ) , and Ψ λR ( x ) = Ψ κ λ Rκ ( xκ ) , ∀ x ∈ B R , ∀ κ, λ > . (b) Ψ µR ≤ Ψ µR ≤ Ψ λR ≤ Ψ λR , provided that < λ < µ . (c) Ψ Rκ ( xκ ) ≤ Ψ R ( x ) , ∀ x ∈ B R , ∀ κ ∈ (0 , . (d) There exists a countable set D ⊂ (0 , ∞ ) , such that for every R ∈ (0 , ∞ ) \ D , we have Ψ R = Ψ R , and thus the minimizer of J B R in the class A R is unique.Proof. (a) follows from a simple rescaling argument. On the other hand, when λ ∈ (0 , B R , with Ψ λR (instead of ψ λR ), and u = Ψ R , gives the inequalityΨ R ≤ Ψ λR , from which we derive (b) in view of the rescaling in (a). The proof of (c) is obviousfrom (a) and (b). Next, let Q be a countable dense subset of the unit ball B . If Ψ λ = Ψ λ ,for some λ >
0, then there exists x ∈ Q , such that the function (0 , ∞ ) ∋ λ Ψ λ ( x ) s discontinuous at λ . Let ˜ D be the set of λ > λ Ψ λ ( x ) isdiscontinuous at λ , for some x ∈ Q . We notice that ˜ D is countable, since the functions(0 , ∞ ) ∋ λ Ψ λ ( x ) are nonincreasing. Thus, for λ ∈ (0 , ∞ ) \ D , we have Ψ λ ≡ Ψ λ ⇔ Ψ √ λ ≡ Ψ √ λ , and this proves (d). (cid:3) Lemma 3.4.
In addition to the assumptions of Proposition 2.1, we suppose that W rad ( s ) > , ∀ s ∈ (0 , q ] . Then, • if ´ q s √ W rad ( s ) = ∞ , we have Ψ R > , ∀ x ∈ B R , • if I q := ´ q s √ W rad ( s ) < ∞ , the function Ψ R vanishes in the ball B R −√ I q , providedthat R ≥ (4 n + √ I q .Proof. In the case where I q := ´ q s √ W rad ( s ) < ∞ , we define the function γ : [0 , q ] → [0 , I q ] , γ ( s ) = ˆ s √ W rad . Since γ is strictly increasing, we denote its inverse function by β := γ − , β : [0 , I q ] → [0 , q ],and it is clear that s − s ≤ p W rad ( q )( γ ( s ) − γ ( s )) holds for 0 < s ≤ s ≤ q . Thus, β belongs to W , ∞ (0 , I q ). In addition, we have γ ′ ( s ) = 1 p W rad ( s ) for a.e. s ∈ (0 , q ) , and β ′ ( t ) = p W rad ( β ( t )) for a.e. t ∈ (0 , I q ) . Next, we consider the restriction of the minimizer Ψ R to the ball B r ⊂ R n (with I q < r < R ),and setting(3.24) φ r ( x ) := ( β ( | x | − r + γ (Ψ R ( r ))) if r − γ (Ψ R ( r )) ≤ | x | ≤ r, | x | ≤ r − γ (Ψ R ( r )) , we obtain a function φ r ∈ W , ( B r ) such that φ r = Ψ R on ∂B r . A computation shows that J B r ( φ r ) = | S n − | ˆ γ (Ψ R ( r ))0 ( t + r − γ (Ψ R ( r ))) n − W rad ( β ( t ))d t ≤ | S n − | W rad (Ψ R ( r )) I q r n − , where | S n − | denotes the measure of the unit sphere S n − ⊂ R n . On the other hand, Pohozaevidentity (4.1) applied to Ψ R in the ball B r implies that | S n − | r n (cid:0) W rad (Ψ R, rad ( r )) − | Ψ ′ R, rad ( r ) | (cid:1) ≤ nJ B r (Ψ R ) ≤ nJ B r ( φ r ) , for a.e. r ∈ ( I q , R ) , where in the last inequality we have used the minimality of Ψ R . Therefore, we deduce that (cid:0) W rad (Ψ R, rad ( r )) − | Ψ ′ R, rad ( r ) | (cid:1) ≤ nI q r − W rad (Ψ R, rad ( r )) , for a.e. r ∈ ( I q , R ) . In particular, for a.e. r ∈ (4 nI q , R ), we have W rad (Ψ R, rad ( r )) ≤ | Ψ ′ R, rad ( r ) | . Now, let ( l, R )be the intersection of the intervals (4 nI q , R ) and { r ∈ (0 , R ) : Ψ R, rad ( r ) > } . Since Ψ R, rad is trictly increasing on the interval ( l, R ), we denote its inverse function by χ R : ( δ, q ) → ( l, R ).Proceeding as previously, we can see that given 0 < ǫ ≪
1, the function χ R is Lipschitzon ( δ + ǫ, q ), and moreover the inequality χ ′ R ( s ) ≤ √ W rad ( s ) holds for a.e. s ∈ ( δ, q ). As aconsequence, it follows that R − r ≤ ˆ q Ψ R, rad ( r ) √ W rad ≤ √ I q , ∀ r ∈ ( l, R ) . This proves that the function Ψ R vanishes in the ball B R −√ I q , provided that R ≥ (4 n + √ I q .Conversely, we are going to establish that when n = 1, the existence of a dead core regionfor β R := Ψ R implies that I q := ´ q s √ W rad ( s ) ≤ R . In view of Pohozaev identity (4.1), wehave ˆ r (cid:0) W rad ( β R ) − | β ′ R | (cid:1) = r (cid:0) W rad ( β R ( r )) − | β ′ R ( r ) | (cid:1) for a.e. r ∈ (0 , R ) , which implies that 12 | β ′ R ( r ) | − W rad ( β R ( r )) = H for a.e. r ∈ (0 , R ) , (3.25)for some constant H . By assumption β vanishes on a small interval [0 , ǫ ], thus it followsfrom (3.25), that actually12 | β ′ R ( r ) | = W rad ( β R ( r )) for a.e. r ∈ (0 , R ) . (3.26)Let ( l, R ] be the interval where β R >
0. Since β R is strictly increasing on the interval ( l, R ),we denote its inverse function by γ R : (0 , q ) → ( l, R ). As previously, we can see that γ R islocally Lipschitz on (0 , q ), and that γ ′ R ( s ) = √ W rad ( s ) holds for a.e. s ∈ (0 , q ). Therefore, weconclude that I q := ´ q s √ W rad ( s ) ≤ R .So far we have proved that ´ q s √ W rad ( s ) = ∞ , implies that β R ( r ) >
0, for every
R >
0, and r ∈ (0 , R ). Actually, the functions β R are positive on the whole interval [ − R, R ]. Indeed, inview of Lemma 3.3 (c), we have β R ( r ) ≥ Ψ R (2 r ), ∀ r ∈ [ − R, R ]. Next, Theorem 2.2 appliedin Ω = ( − R, R ), with u ( s ) = β R (2 R + s ) and Ψ R ( s ), gives the inequality β R (2 R + s ) ≤ Ψ R ( s ), ∀ s ∈ [ − R, R ], from which we deduce that 0 < β R (2 R +2 r ) ≤ β R ( r ), ∀ r ∈ ( − R, R ).To complete the proof of Lemma 3.4, it remains to establish that the condition ´ q s √ W rad ( s ) = ∞ , also implies the positivity of the functions Ψ R , in higher dimensions n ≥
2. To see this,we apply Theorem 2.2 in Ω = ( − R, R ) n , to the minimizer u ( x , x , . . . , x n ) = β R ( x ), and weget 0 < β R ( r ) ≤ Ψ R, rad ( r ), ∀ r ∈ [0 , R ]. Finally, in view of Lemma 3.3 (c), we conclude thatthe functions Ψ R are positive. (cid:3) Remark 5.
In view of Lemma 3.3, if Ψ R or Ψ R has a dead core, then Ψ S and Ψ S have alsoa dead core for every S > R . As a consequence, assuming that I q < ∞ , there exists a criticalvalue R such that Ψ R has a dead core for R > R , while Ψ R does not have a dead core for We point out that (3.25) expresses the conservation of the total mechanical energy for the solutions ofthe Hamiltonian system u ′′ ( x ) = ∇ W ( u ( x )). This property still holds for minimizers of (1.1), in the case ofnonsmooth potentials < R . Lemma 3.4 establishes that R ≤ (4 n + √ I q holds in any dimension n . On theother hand, in Remark 1 we have determined the value of R , when n = 2 and W rad is thecharacteristic function of R \ { } . We refer to [12, Section 8.4.] for the general theory ofdead cores in the smooth case, and in particular to [12, Theorems 8.4.2., 8.4.3., 8.4.4.] for theproperties of the function Ψ R . In [12, Section 8.4.], several explicit examples of dead coresare also provided. Proof of Proposition 2.5.
In view of the continuity of u (cf. Lemma 4.2), we have u ≥ ǫ on ∂ω , for some ǫ >
0. Let B R be a ball containing the domain ω . By increasing R , wemay assume that the functional J B R admits a unique minimizer Φ in the class B := { h ∈ W , (Ω; R ) : h = ǫ on ∂B R } (cf. Proposition 2.1, and Lemma 3.3). In addition, it is clearthat (Φ − u ) + , ( u − Φ) − ∈ W , ( ω ). Thus, in view of the minimality of Φ and u , we have onthe one hand(3.27a) E B R (Φ) ≤ E B R (Φ − ( u − Φ) − ) ⇔ E { Φ >u } (Φ) ≤ E { Φ >u } ( u ) , and on the other hand(3.27b) E ω ( u ) ≤ E ω ( u + (Φ − u ) + ) ⇔ E { Φ >u } ( u ) ≤ E { Φ >u } (Φ) . That is, E { Φ >u } (Φ) = E { Φ >u } ( u ), which means that Φ − ( u − Φ) − is a minimizer of J B R in theclass B . By uniqueness of the minimizer Φ, we conclude that u ≥ Φ on ω . In the case where W rad ( s ) > ∀ s ∈ (0 , q ], we have seen in Lemma 3.4 that Φ >
0. Otherwise, if W rad ( η ) = 0,for some η ∈ (0 , q ), it is straightforward that Φ ≥ η (cf. the proof of Lemma 3.1). (cid:3) Pohozaev identity and continuity for minimizers
Pohozaev identity is commonly used for smooth solutions of semilinear elliptic systems (cf.for instance [2, Remark 3.1]). We prove below that the identity also holds for minimizers of(1.1).
Lemma 4.1.
Let W be a nonnegative, bounded and lower semicontinuous function definedon B q , and let u ∈ W , (Ω; R m ) be a map defined in the domain Ω ⊂ R n , and satisfying (2.1) as well as (2.2) . Then, given a ball B R ( x ) ⊂ Ω , we have (4.1) ˆ B r ( x ) (cid:16) n − |∇ u | + nW ( u ) (cid:17) = r ˆ ∂B r ( x ) (cid:16) |∇ u | + W ( u ) − (cid:12)(cid:12)(cid:12) ∂u∂ν (cid:12)(cid:12)(cid:12) (cid:17) , for a.e. r ∈ (0 , R ) , where ν stands for the outer nornal to the ball B r ( x ) .Proof. Without loss of generality, we may assume that x = 0. Let r ∈ (0 , R ), and s ∈ ( r, R )be fixed. Given x ∈ B s \ { } , we write x = tσ , with t := | x | , and σ ∈ S n − . Moreover, we set | u t ( t, σ ) | := | ∂u∂t ( t, σ ) | , and |∇ σ u ( t, σ ) | = |∇ u ( t, σ ) | − | u t ( t, σ ) | . Next, we consider in B s ,the comparison map(4.2) ˜ u ( x ) = ˜ u ( t, σ ) = ( u ( xκ ) when 0 ≤ t = | x | ≤ κr,u ( r + ( s − r ) t − κrs − κr , σ ) when κr ≤ t ≤ s, where κ ∈ (0 , sr ) is fixed. It is clear that ˜ u = u on ∂B s , thus by the minimality of u , we have(4.3) E B s (˜ u ) − E B s ( u ) ≥ , ∀ κ ∈ (cid:0) , sr (cid:1) . etting f ( κ ) := E B s (˜ u ), a long but otherwise trivial computation shows that f ( κ ) = ˆ B r h κ n − |∇ u | + κ n W ( u ) i + ˆ sr ˆ S n − (cid:0) ( s − κr ) t − rs − r + κr (cid:1) n − s − rs − κr | u t ( t, σ ) | t d σ + ˆ sr ˆ S n − (cid:0) ( s − κr ) t − rs − r + κr (cid:1) n − t s − κrs − r |∇ σ u t ( t, σ ) | t d σ + ˆ sr ˆ S n − (cid:0) ( s − κr ) t − rs − r + κr (cid:1) n − s − κrs − r W ( u ( t, σ ))d t d σ, and(4.4) f ′ (1) = ˆ B r h n − |∇ u | + nW ( u ) i + rs − r ˆ B s \ B r (cid:0) ( n − st − ( n − (cid:1) | u t ( t, σ ) | rs − r ˆ B s \ B r (cid:0) ( n − st − ( n − (cid:1) |∇ σ u t ( t, σ ) | rs − r ˆ B s \ B r (cid:0) ( n − st − n ) (cid:1) W ( u ( t, σ )) . Therefore, in view of (4.3), and since f (1) = E B s ( u ), we deduce that f ′ (1) = 0. Finally,letting s → r in (4.4), we obtain for a.e. r ∈ (0 , R ):0 = ˆ B r h n − |∇ u | + nW ( u ) i + r ˆ ∂B r (cid:0) | u t | − |∇ σ u | − W ( u ) (cid:1) . (cid:3) Remark 6.
Proceeding as in [2, page 91], one can also derive from Pohozaev identity themonotonicity formula dd r ( r − ( n − E B r ( x ) ( u )) ≥
0, holding for a.e. r ∈ (0 , R ), under the as-sumptions of Lemma 4.1. We refer to the expository papers [8, 9] for a detailed account ofmonotonicity formulae.Next, we recall the continuity of bounded minimizer of (1.1). This property is crucial inthe proof of Theorem 2.2. Lemma 4.2.
Let W be a nonnegative, bounded and lower semicontinuous function definedon B q , and let u ∈ W , (Ω; R m ) be a map satisfying (2.1) and (2.2) . Then, u is continuousin Ω .Proof. We refer to [3, Lemma 2.1] where a logarithmic estimate is established for the localminimizer u , implying in particular its H¨older continuity. (cid:3) Acknowledgments
This research is supported by REA - Research Executive Agency - Marie Sk lodowska-CurieProgram (Individual Fellowship 2018) under Grant No. 832332, by the Basque Governmentthrough the BERC 2018-2021 program, by the Ministry of Science, Innovation and Univer-sities: BCAM Severo Ochoa accreditation SEV-2017-0718, by project MTM2017-82184- Rfunded by (AEI/FEDER, UE) and acronym “DESFLU”. eferences [1] Alikakos, N. D., Fusco, G.: Density estimates for vector minimizers and applications. Discrete andcontinuous dynamical systems No. 12, 5631–5663 (2015), Special issue edited by E.Valdinoci[2] Alikakos, N. D., Fusco, G., Smyrnelis, P.: Elliptic systems of phase transition type. Progress in NonlinearDifferential Equations and Their Applications , Springer-Birkh¨auser (2018).[3] Alikakos, N. D., Gazoulis, D., Zarnescu, A.: Existence of entire solutions of the Allen-Cahn systempossessing free boundaries. Preprint.[4] Alt, H. W., Caffarelli, L., Friedman, A.: Variational problems with two phases and their free boundaries.Trans. Amer. Math. Soc. , 431–461 (1984)[5] Bandle C., Sperb R. and Stakgold I.: Diffusion and reaction with monotone kinetics, Nonlinear Analysis , 321–333 (1984)[6] Caffarelli, L., C´ordoba, A.: Uniform convergence of a singular perturbation problem. Comm. Pure Appl.Math. , 1–12 (1995)[7] Dr´abek, P., Robinson, S. B.: Continua of local minimizers in a non-smooth model of phase transitions.Z. Angew. Math. Phys. , 609–622 (2011)[8] Evans, L. C.: Monotonicity formulae for variational problems, Phil. Trans. R. Soc. A : 20120339.[9] Fried, E., Lussardi, L.: Monotonicity formulae for smooth extremizers of integral functionals. RendicontiLincei - Matematica e Applicazioni , No. 2 365–377 (2019)[10] Pucci, P., Serrin, J.: The strong maximum principle revisited. J. Differential Equations , 1–66 (2004)[11] Pucci, P., Serrin, J.: Dead cores and bursts for quasilinear singular elliptic equations. SIAM J. Math.Anal. , No. 1 259–278 (2006)[12] Pucci, P., Serrin, J.: The maximum principle. Progress in Nonlinear Differential Equations and TheirApplications , Springer-Birkh¨auser (2007)[13] Savin, O.: Minimal Surfaces and Minimizers of the Ginzburg Landau energy. Cont. Math. Mech. AnalysisAMS , 43–58 (2010)[14] Smyrnelis, P.: Connecting orbits in Hilbert spaces and applicatons to P.D.E. Comm. Pure Appl. Anal. , No. 5 2797–2818 (May 2020)[15] Sperb, R.: Some complementary estimates in the dead core problem. Nonlinear Problems in AppliedMathematics. In honor of Ivar Stakgold on his 70th birthday, T. S. Angell, et al. (eds.), Philadelphia,(1996) 217–224.(P. Smyrnelis) Basque Center for Applied Mathematics, 48009 Bilbao, Spain
Email address , P. Smyrnelis: [email protected]@bcamath.org